What does \lim_{x \to a} f(x)=L say?

The values of f(x) approach L as x approaches a from both sides. The statement doesn't require f(a)=L or even require f(a) to exist.

How can a table estimate \lim_{x \to a} f(x) ?

Use inputs approaching a from below and above, then look for a common output value. Values exactly at x=a don't determine the limit.

When does direct substitution evaluate a limit?

When the function is continuous at the target input. Then $$\lim_{x \to a} f(x)=f(a).$$

Three conditions for continuity at x=a ?

f(a) exists, \lim_{x \to a}f(x) exists, and $$\lim_{x \to a}f(x)=f(a).$$

Intermediate Value Theorem: hypotheses and conclusion?

If f is continuous on [a,b] and N lies between f(a) and f(b) , then some c in [a,b] satisfies f(c)=N . If N is strictly between the endpoint values, c lies in (a,b) .

When does a two-sided limit equal L ?

Exactly when both one-sided limits equal L : $$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L.$$ If the one-sided limits differ, the two-sided limit doesn't exist.

How do you read a finite limit from a graph?

Follow the graph toward the target x -value from both sides. The common approached y -value is the limit, regardless of a hole or a differently placed filled point.

Limit law for a sum or difference?

If both component limits exist, $$\lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x).$$

What makes a discontinuity removable?

The finite two-sided limit exists, but the function value is missing or differs from that limit. Redefining the function at one point can make it continuous.

Squeeze Theorem: usable form?

If g(x)\le f(x)\le h(x) near a and $$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L,$$ then \lim_{x\to a}f(x)=L .

What does \lim_{x\to a}f(x)=+\infty mean?

f(x) grows without bound above as x approaches a . It describes unbounded behavior, not a finite limit value.

What must a table show for a left-hand limit?

Inputs less than the target and moving toward it. For \lim_{x\to a^-}f(x) , use x<a with x getting closer to a .

Limit law for a product?

If both limits exist, $$\lim_{x\to a}[f(x)g(x)]=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right).$$

Graph signature of a jump discontinuity?

The left- and right-hand limits are finite but unequal. The function approaches different heights from the two sides.

Which theorem can guarantee a root on [a,b] ?

The Intermediate Value Theorem. If f is continuous on [a,b] and 0 lies between f(a) and f(b) , then f(c)=0 for some c in the interval.

Horizontal asymptote from a limit at infinity?

If \lim_{x\to\infty}f(x)=L or \lim_{x\to-\infty}f(x)=L , then y=L is a horizontal asymptote in that direction.

What does an open circle say about a graph's limit?

Only that the displayed point isn't included there. The limit depends on nearby behavior; it may still exist and equal the hole's height.

Limit law for a quotient—and its condition?

If both limits exist and the denominator limit is nonzero, $$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}.$$ The law doesn't apply when the denominator limit is 0 .

What does continuity on [a,b] require at the endpoints?

Continuity on (a,b) , right-continuity at a , and left-continuity at b : $$\lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b).$$

When is the Squeeze Theorem a natural choice?

When a hard or oscillating expression can be trapped between two simpler expressions with the same limit.

Vertical asymptote from one-sided behavior?

If at least one one-sided limit at x=a is +\infty or -\infty , then x=a is a vertical asymptote.

Limit at infinity of equal-degree rational functions?

The ratio of the leading coefficients: $$\lim_{x\to\pm\infty}\frac{a_nx^n+\cdots}{b_nx^n+\cdots}=\frac{a_n}{b_n}.$$ This assumes b_n\ne0 .

When can a limit pass through a continuous outer function?

If \lim_{x\to a}g(x)=L and f is continuous at L , then $$\lim_{x\to a}f(g(x))=f(L).$$

What makes a discontinuity infinite?

The function becomes unbounded on at least one side of the input, usually producing a vertical asymptote.

Left limit =2 and right limit =5 : two-sided limit?

It doesn't exist. A finite two-sided limit requires the two one-sided limits to agree.

Standard trigonometric limit behind \frac{\sin x}{x} ?

With angles in radians, $$\lim_{x\to0}\frac{\sin x}{x}=1.$$ Equivalent scaled forms follow by substitution.

Continuity of a composition?

If g is continuous at a and f is continuous at g(a) , then f\circ g is continuous at a .

Limit at infinity when a rational numerator has lower degree?

0 . If the numerator's degree is less than the denominator's, the denominator dominates as x\to\pm\infty .

What does the indeterminate form 0/0 tell you?

Direct substitution hasn't determined the limit. Simplify, use an appropriate limit method, or inspect another representation; 0/0 isn't the limit value.

When do opposite infinite one-sided limits give a two-sided limit?

They don't. For example, +\infty from the left and -\infty from the right mean the two-sided limit doesn't exist, though a vertical asymptote may be present.

How do you choose a parameter to make a piecewise function continuous?

Set the relevant one-sided formulas and the function value equal at the join, then solve the resulting equation for the parameter.

Value of \lim_{x\to0}\frac{1-\cos x}{x} ?

0 . Rationalizing gives a product involving \sin x/x and a factor that approaches 0 .

Can \lim_{x\to a}f(x) exist when f(a) doesn't?

Yes. A limit uses nearby values, so a hole at x=a can coexist with a finite two-sided limit.

What graph behavior makes a finite limit fail even without a jump?

Unbounded or persistent oscillating behavior near the input can prevent a finite limit. Check both sides rather than relying on the plotted point.

Average rate of change of f on [a,b] ?

$$\frac{f(b)-f(a)}{b-a}$$ It is the slope of the secant line through (a,f(a)) and (b,f(b)) .

Derivative at x=a using an increment h ?

$$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$ The derivative exists only if this finite limit exists.

Tangent-line equation to y=f(x) at x=a ?

$$y-f(a)=f'(a)(x-a)$$ This requires f'(a) to exist.

What does differentiability imply about continuity?

If f is differentiable at a , then f is continuous at a . The converse is false: continuity alone doesn't guarantee differentiability.

Power rule for derivatives?

$$\frac{d}{dx}x^n=nx^{n-1}$$ Apply it where the original real-valued power function and its derivative are defined.

Output units of f per input unit of x . A derivative is a rate of change, so its units are a quotient.

Derivative at x=a using x\to a ?

$$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$ This is equivalent to the h -form after setting h=x-a .

How does a graph of f show the sign of f' ?

f'(x)>0 where f rises as x increases, and f'(x)<0 where f falls. A horizontal tangent gives f'(x)=0 when the derivative exists.

Derivative of a constant?

$$\frac{d}{dx}C=0$$ A constant function has zero rate of change.

Derivative of \sin x ?

$$\frac{d}{dx}(\sin x)=\cos x$$ The angle must be measured in radians for the standard formula.

$$\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$$ Differentiating each factor and multiplying the results is not the product rule.

How can nearby table values estimate f'(a) ?

Use a difference quotient with inputs close to a . A symmetric estimate is $$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}.$$ Smaller h often helps, subject to the data's precision.

What does f''(x) measure?

The rate of change of f'(x) with respect to x . Its units are the units of f per square input unit.

Instantaneous rate of change of f at a ?

f'(a) . It is the limit of average rates over intervals shrinking to a , and geometrically it is the tangent-line slope.

Derivative of a sum or difference?

$$\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$$

Derivative of \cos x ?

$$\frac{d}{dx}(\cos x)=-\sin x$$ The standard formula assumes radians.

For g(x)\ne0 , $$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}.$$ The order in the numerator matters.

AP Calculus AB Flashcards: 8-Unit Formulas, Theorems & Concepts

Review all eight AP Calculus AB units with formula conditions, theorem hypotheses, notation, graph interpretation, and concept cards.

Von Flashcards Open Source App

Über dieses Lernkartenset

Review the formulas, theorem conditions, notation, and representation links that support all eight AP Calculus AB units. The cards practice short recall in both useful directions: a name or situation can cue a formula, condition, theorem, or interpretation, and selected formulas or hypotheses can cue their meaning or use. Graph, table, symbolic, and word-based cards connect derivatives and integrals to rates, accumulation, extrema, concavity, motion, area, and volume.

The sequence starts with limits and the derivative definition, then builds through derivative rules and applications before moving to integration, differential equations, and applications of integration. Related variants are separated in the final order so one card does not simply reveal the next.

This is an AB-only review deck. It excludes parametric, polar, and vector-valued calculus; infinite sequences and series; integration by parts; partial fractions; improper integrals; logistic differential equations; arc length; and other BC-only material. It also excludes copied exam questions, mark schemes, curriculum prose, full theorem proofs, and long multi-step practice problems. Flashcards support recall; learners should still solve original practice problems and complete timed exam practice.

Every prompt, answer, explanation, metadata field, and ordering choice is independently written from common mathematical knowledge after checking the current official scope. No College Board question, mark scheme, curriculum text, logo, or trade dress is copied. AP® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this product. Official course requirements: AP Calculus AB Course.

Karten in diesem Lernkartenset

Karte 1
Frage
What does
$\lim_{x \to a} f(x)=L$
say?
Antwort
The values of
$f(x)$
approach
$L$
as
$x$
approaches
$a$
from both sides. The statement doesn't require
$f(a)=L$
or even require
$f(a)$
to exist.
Karte 2
Frage
How can a table estimate
$\lim_{x \to a} f(x)$
?
Antwort
Use inputs approaching
$a$
from below and above, then look for a common output value. Values exactly at
$x=a$
don't determine the limit.
Karte 3
Frage
When does direct substitution evaluate a limit?
Antwort
When the function is continuous at the target input. Then

$$\lim_{x \to a} f(x)=f(a).$$
Karte 4
Frage
Three conditions for continuity at
$x=a$
?
Antwort
$f(a)$
exists,
$\lim_{x \to a}f(x)$
exists, and

$$\lim_{x \to a}f(x)=f(a).$$
Karte 5
Frage
Intermediate Value Theorem: hypotheses and conclusion?
Antwort
If
$f$
is continuous on
$[a,b]$
and
$N$
lies between
$f(a)$
and
$f(b)$
, then some
$c$
in
$[a,b]$
satisfies
$f(c)=N$
. If
$N$
is strictly between the endpoint values,
$c$
lies in
$(a,b)$
.
Karte 6
Frage
When does a two-sided limit equal
$L$
?
Antwort
Exactly when both one-sided limits equal
$L$
:

$$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L.$$

If the one-sided limits differ, the two-sided limit doesn't exist.
Karte 7
Frage
How do you read a finite limit from a graph?
Antwort
Follow the graph toward the target
$x$
-value from both sides. The common approached
$y$
-value is the limit, regardless of a hole or a differently placed filled point.
Karte 8
Frage
Limit law for a sum or difference?
Antwort
If both component limits exist,

$$\lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x).$$
Karte 9
Frage
What makes a discontinuity removable?
Antwort
The finite two-sided limit exists, but the function value is missing or differs from that limit. Redefining the function at one point can make it continuous.
Karte 10
Frage
Squeeze Theorem: usable form?
Antwort
If
$g(x)\le f(x)\le h(x)$
near
$a$
and

$$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L,$$

then
$\lim_{x\to a}f(x)=L$
.
Karte 11
Frage
What does
$\lim_{x\to a}f(x)=+\infty$
mean?
Antwort
$f(x)$
grows without bound above as
$x$
approaches
$a$
. It describes unbounded behavior, not a finite limit value.
Karte 12
Frage
What must a table show for a left-hand limit?
Antwort
Inputs less than the target and moving toward it. For
$\lim_{x\to a^-}f(x)$
, use
$x<a$
with
$x$
getting closer to
$a$
.
Karte 13
Frage
Limit law for a product?
Antwort
If both limits exist,

$$\lim_{x\to a}[f(x)g(x)]=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right).$$
Karte 14
Frage
Graph signature of a jump discontinuity?
Antwort
The left- and right-hand limits are finite but unequal. The function approaches different heights from the two sides.
Karte 15
Frage
Which theorem can guarantee a root on
$[a,b]$
?
Antwort
The Intermediate Value Theorem. If
$f$
is continuous on
$[a,b]$
and
$0$
lies between
$f(a)$
and
$f(b)$
, then
$f(c)=0$
for some
$c$
in the interval.
Karte 16
Frage
Horizontal asymptote from a limit at infinity?
Antwort
If
$\lim_{x\to\infty}f(x)=L$
or
$\lim_{x\to-\infty}f(x)=L$
, then
$y=L$
is a horizontal asymptote in that direction.
Karte 17
Frage
What does an open circle say about a graph's limit?
Antwort
Only that the displayed point isn't included there. The limit depends on nearby behavior; it may still exist and equal the hole's height.
Karte 18
Frage
Limit law for a quotient—and its condition?
Antwort
If both limits exist and the denominator limit is nonzero,

$$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}.$$

The law doesn't apply when the denominator limit is
$0$
.
Karte 19
Frage
What does continuity on
$[a,b]$
require at the endpoints?
Antwort
Continuity on
$(a,b)$
, right-continuity at
$a$
, and left-continuity at
$b$
:

$$\lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b).$$
Karte 20
Frage
When is the Squeeze Theorem a natural choice?
Antwort
When a hard or oscillating expression can be trapped between two simpler expressions with the same limit.
Karte 21
Frage
Vertical asymptote from one-sided behavior?
Antwort
If at least one one-sided limit at
$x=a$
is
$+\infty$
or
$-\infty$
, then
$x=a$
is a vertical asymptote.
Karte 22
Frage
Limit at infinity of equal-degree rational functions?
Antwort
The ratio of the leading coefficients:

$$\lim_{x\to\pm\infty}\frac{a_nx^n+\cdots}{b_nx^n+\cdots}=\frac{a_n}{b_n}.$$

This assumes
$b_n\ne0$
.
Karte 23
Frage
When can a limit pass through a continuous outer function?
Antwort
If
$\lim_{x\to a}g(x)=L$
and
$f$
is continuous at
$L$
, then

$$\lim_{x\to a}f(g(x))=f(L).$$
Karte 24
Frage
What makes a discontinuity infinite?
Antwort
The function becomes unbounded on at least one side of the input, usually producing a vertical asymptote.
Karte 25
Frage
Left limit
$=2$
and right limit
$=5$
: two-sided limit?
Antwort
It doesn't exist. A finite two-sided limit requires the two one-sided limits to agree.
Karte 26
Frage
Standard trigonometric limit behind
$\frac{\sin x}{x}$
?
Antwort
With angles in radians,

$$\lim_{x\to0}\frac{\sin x}{x}=1.$$

Equivalent scaled forms follow by substitution.
Karte 27
Frage
Continuity of a composition?
Antwort
If
$g$
is continuous at
$a$
and
$f$
is continuous at
$g(a)$
, then
$f\circ g$
is continuous at
$a$
.
Karte 28
Frage
Limit at infinity when a rational numerator has lower degree?
Antwort
$0$
. If the numerator's degree is less than the denominator's, the denominator dominates as
$x\to\pm\infty$
.
Karte 29
Frage
What does the indeterminate form
$0/0$
tell you?
Antwort
Direct substitution hasn't determined the limit. Simplify, use an appropriate limit method, or inspect another representation;
$0/0$
isn't the limit value.
Karte 30
Frage
When do opposite infinite one-sided limits give a two-sided limit?
Antwort
They don't. For example,
$+\infty$
from the left and
$-\infty$
from the right mean the two-sided limit doesn't exist, though a vertical asymptote may be present.
Karte 31
Frage
How do you choose a parameter to make a piecewise function continuous?
Antwort
Set the relevant one-sided formulas and the function value equal at the join, then solve the resulting equation for the parameter.
Karte 32
Frage
Value of
$\lim_{x\to0}\frac{1-\cos x}{x}$
?
Antwort
$0$
. Rationalizing gives a product involving
$\sin x/x$
and a factor that approaches
$0$
.
Karte 33
Frage
Can
$\lim_{x\to a}f(x)$
exist when
$f(a)$
doesn't?
Antwort
Yes. A limit uses nearby values, so a hole at
$x=a$
can coexist with a finite two-sided limit.
Karte 34
Frage
What graph behavior makes a finite limit fail even without a jump?
Antwort
Unbounded or persistent oscillating behavior near the input can prevent a finite limit. Check both sides rather than relying on the plotted point.
Karte 35
Frage
Average rate of change of
$f$
on
$[a,b]$
?
Antwort
$$\frac{f(b)-f(a)}{b-a}$$

It is the slope of the secant line through
$(a,f(a))$
and
$(b,f(b))$
.
Karte 36
Frage
Derivative at
$x=a$
using an increment
$h$
?
Antwort
$$f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$

The derivative exists only if this finite limit exists.
Karte 37
Frage
Tangent-line equation to
$y=f(x)$
at
$x=a$
?
Antwort
$$y-f(a)=f'(a)(x-a)$$

This requires
$f'(a)$
to exist.
Karte 38
Frage
What does differentiability imply about continuity?
Antwort
If
$f$
is differentiable at
$a$
, then
$f$
is continuous at
$a$
. The converse is false: continuity alone doesn't guarantee differentiability.
Karte 39
Frage
Power rule for derivatives?
Antwort
$$\frac{d}{dx}x^n=nx^{n-1}$$

Apply it where the original real-valued power function and its derivative are defined.
Karte 40
Frage
Units of
$f'(x)$
?
Antwort
Output units of
$f$
per input unit of
$x$
. A derivative is a rate of change, so its units are a quotient.
Karte 41
Frage
Derivative at
$x=a$
using
$x\to a$
?
Antwort
$$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$

This is equivalent to the
$h$
-form after setting
$h=x-a$
.
Karte 42
Frage
How does a graph of
$f$
show the sign of
$f'$
?
Antwort
$f'(x)>0$
where
$f$
rises as
$x$
increases, and
$f'(x)<0$
where
$f$
falls. A horizontal tangent gives
$f'(x)=0$
when the derivative exists.
Karte 43
Frage
Derivative of a constant?
Antwort
$$\frac{d}{dx}C=0$$

A constant function has zero rate of change.
Karte 44
Frage
Derivative of
$\sin x$
?
Antwort
$$\frac{d}{dx}(\sin x)=\cos x$$

The angle must be measured in radians for the standard formula.
Karte 45
Frage
Product rule?
Antwort
$$\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$$

Differentiating each factor and multiplying the results is not the product rule.
Karte 46
Frage
How can nearby table values estimate
$f'(a)$
?
Antwort
Use a difference quotient with inputs close to
$a$
. A symmetric estimate is

$$f'(a)\approx\frac{f(a+h)-f(a-h)}{2h}.$$

Smaller
$h$
often helps, subject to the data's precision.
Karte 47
Frage
What does
$f''(x)$
measure?
Antwort
The rate of change of
$f'(x)$
with respect to
$x$
. Its units are the units of
$f$
per square input unit.
Karte 48
Frage
Instantaneous rate of change of
$f$
at
$a$
?
Antwort
$f'(a)$
. It is the limit of average rates over intervals shrinking to
$a$
, and geometrically it is the tangent-line slope.
Karte 49
Frage
Derivative of a sum or difference?
Antwort
$$\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$$
Karte 50
Frage
Derivative of
$\cos x$
?
Antwort
$$\frac{d}{dx}(\cos x)=-\sin x$$

The standard formula assumes radians.
Karte 51
Frage
Quotient rule?
Antwort
For
$g(x)\ne0$
,

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}.$$

The order in the numerator matters.
Karte 52
Frage
Common notations for the first derivative?
Antwort
$f'(x)$
,
$y'$
,
$\dfrac{dy}{dx}$
, and
$\dfrac{d}{dx}f(x)$
. They describe the same derivative in different contexts.
Karte 53
Frage
Derivative of
$e^x$
?
Antwort
$$\frac{d}{dx}e^x=e^x$$
Karte 54
Frage
What graph features can make
$f$
nondifferentiable?
Antwort
A discontinuity, corner, cusp, vertical tangent, or other failure of the difference-quotient limit. Continuity is necessary but not sufficient.
Karte 55
Frage
Derivative of
$\tan x$
?
Antwort
Where
$\tan x$
is defined,

$$\frac{d}{dx}(\tan x)=\sec^2x.$$

Angles are in radians.
Karte 56
Frage
What does the derivative function
$f'$
assign to each input?
Antwort
The instantaneous rate of change—or tangent slope—of
$f$
at that input, wherever the derivative exists.
Karte 57
Frage
Derivative of
$\ln x$
?
Antwort
For
$x>0$
,

$$\frac{d}{dx}\ln x=\frac1x.$$

More generally,
$d(\ln|x|)/dx=1/x$
for
$x\ne0$
.
Karte 58
Frage
How does the power rule handle roots or negative powers?
Antwort
Rewrite them as
$x^n$
and apply
$nx^{n-1}$
on intervals where the real-valued expression is defined. Domain restrictions still matter.
Karte 59
Frage
Derivative of
$\csc x$
?
Antwort
Where
$\csc x$
is defined,

$$\frac{d}{dx}(\csc x)=-\csc x\cot x.$$

Angles are in radians.
Karte 60
Frage
If
$f'(x)>0$
throughout an interval, what does
$f$
do there?
Antwort
$f$
is increasing on that interval.
Karte 61
Frage
Derivative of
$a^x$
for a constant base?
Antwort
For
$a>0$
,

$$\frac{d}{dx}a^x=a^x\ln a.$$

When
$a=1$
, the derivative is
$0$
.
Karte 62
Frage
How can a graph estimate
$f'(a)$
?
Antwort
Estimate the slope of the tangent line at
$x=a$
, using two convenient points on a drawn tangent when available. A steeper tangent has a derivative with larger magnitude.
Karte 63
Frage
Derivative of
$\sec x$
?
Antwort
Where
$\sec x$
is defined,

$$\frac{d}{dx}(\sec x)=\sec x\tan x.$$

Angles are in radians.
Karte 64
Frage
If
$f''(x)>0$
, how is
$f'$
changing?
Antwort
$f'$
is increasing. This is also the derivative condition associated with
$f$
being concave up.
Karte 65
Frage
Derivative of
$\log_a x$
?
Antwort
For
$x>0$
,
$a>0$
, and
$a\ne1$
,

$$\frac{d}{dx}\log_a x=\frac{1}{x\ln a}.$$
Karte 66
Frage
Product rule from a table at
$x=a$
?
Antwort
For
$h=fg$
,

$$h'(a)=f'(a)g(a)+f(a)g'(a).$$

Use the four table entries at the same input.
Karte 67
Frage
Derivative of
$\cot x$
?
Antwort
Where
$\cot x$
is defined,

$$\frac{d}{dx}(\cot x)=-\csc^2x.$$

Angles are in radians.
Karte 68
Frage
Why isn't
$|x|$
differentiable at
$x=0$
?
Antwort
Its left-hand slope is
$-1$
and right-hand slope is
$1$
. The one-sided derivative limits disagree, creating a corner.
Karte 69
Frage
Constant-multiple rule?
Antwort
For a constant
$c$
,

$$\frac{d}{dx}[c f(x)]=c f'(x).$$
Karte 70
Frage
Quotient rule from a table at
$x=a$
?
Antwort
For
$h=f/g$
with
$g(a)\ne0$
,

$$h'(a)=\frac{f'(a)g(a)-f(a)g'(a)}{[g(a)]^2}.$$
Karte 71
Frage
Chain rule for
$f(g(x))$
?
Antwort
$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$

Differentiate the outer function at the unchanged inner input, then multiply by the inner derivative.
Karte 72
Frage
How do you identify inner and outer functions in a composite?
Antwort
Find the expression evaluated first—that is the inner function. The operation applied to its result is the outer function.
Karte 73
Frage
Core rule when differentiating an implicit equation in
$x$
and
$y$
?
Antwort
Treat
$y$
as a differentiable function of
$x$
. Every derivative of an expression involving
$y$
gains a factor of
$dy/dx$
by the chain rule.
Karte 74
Frage
Derivative of an inverse function at
$x$
?
Antwort
If
$f$
is differentiable and one-to-one near
$f^{-1}(x)$
, with
$f'(f^{-1}(x))\ne0$
,

$\[f^{-1}]'(x)=\frac{1}{f'(f^{-1}(x))}.\$
Karte 75
Frage
Derivative of
$\arcsin x$
?
Antwort
For
$|x|<1$
,

$$\frac{d}{dx}(\arcsin x)=\frac{1}{\sqrt{1-x^2}}.$$
Karte 76
Frage
Notation for the third derivative of
$f$
?
Antwort
$f'''(x)$
or
$\dfrac{d^3f}{dx^3}$
. The exponent on
$d$
indicates derivative order; it is not an ordinary power.
Karte 77
Frage
If
$h(x)=f(g(x))$
, what table entries give
$h'(a)$
?
Antwort
$$h'(a)=f'(g(a))g'(a)$$

Use
$g(a)$
to find the input needed for the table entry of
$f'$
.
Karte 78
Frage
For
$x^2+y^2=r^2$
, what is
$dy/dx$
?
Antwort
Where
$y\ne0$
,

$$\frac{dy}{dx}=-\frac{x}{y}.$$

Differentiate to get
$2x+2y(dy/dx)=0$
.
Karte 79
Frage
If
$f(a)=b$
, how do you find
$[f^{-1}]'(b)$
?
Antwort
Provided
$f'(a)\ne0$
,

$\[f^{-1}]'(b)=\frac{1}{f'(a)}.\$

The inverse swaps the input-output pair
$(a,b)$
.
Karte 80
Frage
Derivative of
$\arctan x$
?
Antwort
For every real
$x$
,

$$\frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}.$$
Karte 81
Frage
Derivative of
$e^{g(x)}$
?
Antwort
$$\frac{d}{dx}e^{g(x)}=e^{g(x)}g'(x)$$

The extra factor is the chain rule.
Karte 82
Frage
Slope of a tangent to an implicit curve
$F(x,y)=0$
?
Antwort
Differentiate the relation with respect to
$x$
, solve for
$dy/dx$
, then substitute the point. Confirm the point lies on the curve and the resulting slope is defined.
Karte 83
Frage
Why must
$f'(a)\ne0$
to use
$[f^{-1}]'(f(a))=1/f'(a)$
?
Antwort
Because the reciprocal slope would be undefined when
$f'(a)=0$
. The inverse may have a vertical tangent or fail to be differentiable there.
Karte 84
Frage
Derivative of
$\arccos x$
?
Antwort
For
$|x|<1$
,

$$\frac{d}{dx}(\arccos x)=-\frac{1}{\sqrt{1-x^2}}.$$
Karte 85
Frage
How do you find
$d^2y/dx^2$
for an implicit relation?
Antwort
Differentiate the first-derivative equation again with respect to
$x$
, include
$dy/dx$
factors, then substitute the known expression for
$dy/dx$
if needed.
Karte 86
Frage
Derivative of
$\ln(g(x))$
?
Antwort
Where
$g(x)>0$
,

$$\frac{d}{dx}\ln(g(x))=\frac{g'(x)}{g(x)}.$$

For
$\ln|g(x)|$
, the same derivative holds where
$g(x)\ne0$
.
Karte 87
Frage
Derivative of
$y^n$
when
$y=y(x)$
?
Antwort
$$\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}$$

The
$dy/dx$
factor comes from the chain rule.
Karte 88
Frage
How are tangent slopes of inverse graphs related?
Antwort
At reflected points
$(a,b)$
and
$(b,a)$
, the slopes are reciprocals when both are defined and nonzero.
Karte 89
Frage
Derivative of
$\arcsin(g(x))$
?
Antwort
$$\frac{d}{dx}\arcsin(g(x))=\frac{g'(x)}{\sqrt{1-[g(x)]^2}},\qquad |g(x)|<1.$$
Karte 90
Frage
Derivative of
$\sin(g(x))$
?
Antwort
$$\frac{d}{dx}\sin(g(x))=\cos(g(x))g'(x)$$
Karte 91
Frage
Horizontal tangent on an implicit curve: derivative condition?
Antwort
$dy/dx=0$
at a valid point, with the derivative defined there. In a fraction for
$dy/dx$
, the numerator is typically zero while the denominator is nonzero.
Karte 92
Frage
How do you differentiate
$f^{-1}(x)$
without solving for the inverse?
Antwort
Use the reciprocal derivative formula and the matching original input: find
$a$
with
$f(a)=x$
, then compute
$1/f'(a)$
.
Karte 93
Frage
Difference between
$f''(x)$
and
$[f'(x)]^2$
?
Antwort
$f''(x)$
is the derivative of
$f'(x)$
. The expression
$[f'(x)]^2$
is the square of the first derivative; they are generally unrelated.
Karte 94
Frage
Derivative of
$[g(x)]^n$
?
Antwort
$$\frac{d}{dx}[g(x)]^n=n[g(x)]^{n-1}g'(x)$$

This combines the power rule with the chain rule.
Karte 95
Frage
Vertical tangent on an implicit curve: derivative clue?
Antwort
$dy/dx$
becomes unbounded or undefined while the curve still has a vertical tangent. In a simplified derivative fraction, the denominator is often zero and the numerator nonzero.
Karte 96
Frage
Table formula for an inverse derivative at
$x=b$
?
Antwort
Find
$a$
in the table with
$f(a)=b$
. If
$f'(a)\ne0$
, then

$\[f^{-1}]'(b)=\frac{1}{f'(a)}.\$
Karte 97
Frage
Derivative of
$\arctan(g(x))$
?
Antwort
$$\frac{d}{dx}\arctan(g(x))=\frac{g'(x)}{1+[g(x)]^2}$$
Karte 98
Frage
How do product and chain rules combine in
$f(x)g(h(x))$
?
Antwort
$$\frac{d}{dx}[f(x)g(h(x))]=f'(x)g(h(x))+f(x)g'(h(x))h'(x).$$

Use the product rule outside and the chain rule on the composite factor.
Karte 99
Frage
Why can
$d^2y/dx^2$
depend on both
$x$
and
$y$
?
Antwort
An implicit relation may never solve explicitly for
$y$
. After differentiating twice and replacing
$dy/dx$
, the result can naturally remain a function of both coordinates.
Karte 100
Frage
A quantity
$y$
changes through
$u$
, which changes with
$x$
. How are the rates connected?
Antwort
When the functions are differentiable, the chain rule gives

$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$
Karte 101
Frage
What local property lets a function have an inverse derivative?
Antwort
The function must be one-to-one near the point, differentiable there, and have a nonzero derivative. Then the inverse is locally differentiable with reciprocal slope.
Karte 102
Frage
Derivative of
$a^{g(x)}$
?
Antwort
For
$a>0$
,

$$\frac{d}{dx}a^{g(x)}=a^{g(x)}\ln(a)\,g'(x).$$
Karte 103
Frage
How should
$Q'(t)$
be interpreted in context?
Antwort
At time
$t$
, the quantity
$Q$
changes at an instantaneous rate of
$Q'(t)$
output units per unit of time. Include the quantity, time, direction or sign, and units.
Karte 104
Frage
Position, velocity, and acceleration relationships?
Antwort
For position
$s(t)$
,

$$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$$
Karte 105
Frage
Central idea of a related-rates problem?
Antwort
Write an equation connecting the changing quantities, differentiate it with respect to time, then use the data for the specified instant.
Karte 106
Frage
Linearization of
$f$
near
$x=a$
?
Antwort
$$L(x)=f(a)+f'(a)(x-a)$$

For
$x$
close to
$a$
,
$f(x)\approx L(x)$
.
Karte 107
Frage
L’Hospital’s Rule: basic conditions?
Antwort
For a quotient with differentiable numerator and denominator near the target, denominator derivative nonzero nearby, and indeterminate form
$0/0$
or
$\infty/\infty$
, the original limit equals the limit of the derivative quotient when that latter limit exists or is infinite.
Karte 108
Frage
If distance is in meters and time in seconds, units of acceleration?
Antwort
Meters per second squared,
$\text{m}/\text{s}^2$
. Acceleration is the rate of change of velocity with respect to time.
Karte 109
Frage
Speed in terms of velocity?
Antwort
$$\text{speed}=|v(t)|$$

Velocity includes direction; speed is nonnegative magnitude.
Karte 110
Frage
Why do
$x$
and
$y$
gain
$dx/dt$
and
$dy/dt$
in related rates?
Antwort
They are functions of time. Differentiating an expression such as
$x^2$
with respect to
$t$
gives
$2x(dx/dt)$
by the chain rule.
Karte 111
Frage
Differential approximation connecting
$dx$
and
$dy$
?
Antwort
$$dy=f'(x)\,dx$$

For a small change
$\Delta x$
, the actual change satisfies
$\Delta y\approx f'(x)\Delta x$
.
Karte 112
Frage
Which indeterminate forms directly allow L’Hospital’s Rule?
Antwort
$0/0$
and
$\infty/\infty$
. Other indeterminate forms must first be rewritten as an appropriate quotient.
Karte 113
Frage
How do you estimate an instantaneous contextual rate from a table?
Antwort
Use a nearby difference quotient, preferably with times on both sides of the target. Report the result with output units per time unit.
Karte 114
Frage
What does positive acceleration say about velocity?
Antwort
Velocity is increasing. It does not by itself say the object moves forward or speeds up; the sign of velocity also matters.
Karte 115
Frage
Related rates: when should numerical values be substituted?
Antwort
After differentiating the general relationship with respect to time. Substituting fixed values too early can erase the rates being sought.
Karte 116
Frage
How does concavity predict linearization error?
Antwort
Near the tangency point, a concave-up graph generally lies above its tangent line, so the linearization underestimates. A concave-down graph generally lies below, so it overestimates.
Karte 117
Frage
Why can't L’Hospital’s Rule be applied directly to a product?
Antwort
The rule applies to quotients with
$0/0$
or
$\infty/\infty$
form. Rewrite an indeterminate product such as
$0\cdot\infty$
as a quotient first.
Karte 118
Frage
When is a particle moving in the positive direction?
Antwort
When
$v(t)>0$
. Position then increases as time increases.
Karte 119
Frage
How can velocity show a change of direction?
Antwort
Velocity changes sign. A time with
$v(t)=0$
is only a candidate; confirm the sign differs on the two sides.
Karte 120
Frage
First equation to seek in a geometric related-rates problem?
Antwort
A geometric constraint involving the changing quantities, such as a Pythagorean, area, volume, or similar-triangle relation.
Karte 121
Frage
Tangent-line approximation of
$f(a+\Delta x)$
?
Antwort
$$f(a+\Delta x)\approx f(a)+f'(a)\Delta x$$

It is most reliable for small
$|\Delta x|$
where the function is well approximated by its tangent.
Karte 122
Frage
When may L’Hospital’s Rule be applied more than once?
Antwort
When the derivative quotient still has
$0/0$
or
$\infty/\infty$
form and the rule's conditions continue to hold.
Karte 123
Frage
What must a contextual derivative sentence include?
Antwort
The changing quantity, the instant or input, the rate's sign or direction when relevant, and correct output-per-input units.
Karte 124
Frage
Velocity negative and acceleration positive: what happens?
Antwort
The particle moves in the negative direction while its velocity increases toward zero. Its speed decreases as long as velocity and acceleration have opposite signs.
Karte 125
Frage
How should a negative related rate be interpreted?
Antwort
The measured quantity is decreasing at that instant. Keep the sign and state the units; don't silently report only its magnitude.
Karte 126
Frage
When is local linearity a sound approximation tool?
Antwort
When
$f$
is differentiable near the base point and the target input is close enough that curvature has limited effect.
Karte 127
Frage
Can L’Hospital’s Rule handle a one-sided limit?
Antwort
Yes, if the required indeterminate form and differentiability conditions hold on that side and the derivative-quotient limit exists or is infinite.
Karte 128
Frage
When is speed increasing?
Antwort
When velocity and acceleration have the same sign, so
$v(t)a(t)>0$
.
Karte 129
Frage
Volume changes with time: notation for its rate?
Antwort
$dV/dt$
. Its units are cubic length units per time unit.
Karte 130
Frage
Why are similar triangles useful in related rates?
Antwort
They can reduce several changing lengths to one constraint before differentiation, keeping the rate equation tied to the geometry.
Karte 131
Frage
Meaning of
$dy=f'(x)dx$
in approximation?
Antwort
$dy$
is the tangent-line estimate of the actual output change
$\Delta y$
caused by an input change
$dx$
.
Karte 132
Frage
What conclusion does L’Hospital’s Rule permit?
Antwort
Under its conditions,

$$\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}.$$

It does not say the two quotients are equal as functions.
Karte 133
Frage
When is speed decreasing?
Antwort
When velocity and acceleration have opposite signs, so
$v(t)a(t)<0$
.
Karte 134
Frage
What does a tangent slope read from a contextual graph represent?
Antwort
The instantaneous rate of the vertical quantity with respect to the horizontal quantity, with vertical units per horizontal unit.
Karte 135
Frage
Does
$v(t)=0$
guarantee a particle changes direction?
Antwort
No. It is momentarily at rest, but direction changes only if velocity changes sign across that time.
Karte 136
Frage
How do you translate “
$Q$
increases by 3 units per minute” into derivative notation?
Antwort
$dQ/dt=3$
in the stated time interval or at the stated instant. “Decreases by 3” would give
$dQ/dt=-3$
.
Karte 137
Frage
Extreme Value Theorem: hypothesis and conclusion?
Antwort
If
$f$
is continuous on the closed interval
$[a,b]$
, then
$f$
has at least one absolute minimum value and at least one absolute maximum value on
$[a,b]$
.
Karte 138
Frage
What is a critical number of
$f$
?
Antwort
A number
$c$
in the domain of
$f$
where
$f'(c)=0$
or
$f'(c)$
doesn't exist.
Karte 139
Frage
First derivative test for a local maximum?
Antwort
$f'$
changes from positive to negative at the critical point, so
$f$
changes from increasing to decreasing.
Karte 140
Frage
Second-derivative sign for concave up?
Antwort
If
$f''(x)>0$
on an interval, then
$f$
is concave up there and
$f'$
is increasing.
Karte 141
Frage
If the graph of
$f'$
is above the
$x$
-axis, what does
$f$
do?
Antwort
$f$
is increasing because
$f'(x)>0$
.
Karte 142
Frage
First step in an optimization model?
Antwort
Define the objective quantity and the constraint, including the feasible domain. Rewrite the objective as a function of one variable before differentiating.
Karte 143
Frage
Mean Value Theorem: hypotheses and conclusion?
Antwort
If
$f$
is continuous on
$[a,b]$
and differentiable on
$(a,b)$
, then some
$c$
in
$(a,b)$
satisfies

$$f'(c)=\frac{f(b)-f(a)}{b-a}.$$
Karte 144
Frage
Candidates test for absolute extrema on
$[a,b]$
?
Antwort
Assuming
$f$
is continuous on
$[a,b]$
, evaluate
$f$
at every critical number in
$(a,b)$
and at both endpoints. The largest value is the absolute maximum; the smallest is the absolute minimum.
288 Karten
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Karte 145
Frage
First derivative test for a local minimum?
Antwort
$f'$
changes from negative to positive at the critical point, so
$f$
changes from decreasing to increasing.
Karte 146
Frage
What must happen at an inflection point?
Antwort
The graph's concavity changes. A zero or undefined value of
$f''$
is only a candidate; verify a concavity change.
Karte 147
Frage
If
$f'$
has a local maximum, what can that say about
$f$
?
Antwort
$f''$
may change from positive to negative there, so
$f$
may change from concave up to concave down. Confirm the sign change rather than relying only on the point.
Karte 148
Frage
How do you confirm an optimization answer is absolute?
Antwort
Compare objective values at all feasible critical points and relevant endpoints, or use a justified monotonicity argument over the feasible domain.
Karte 149
Frage
Rolle’s Theorem: hypotheses and conclusion?
Antwort
If
$f$
is continuous on
$[a,b]$
, differentiable on
$(a,b)$
, and
$f(a)=f(b)$
, then some
$c$
in
$(a,b)$
satisfies
$f'(c)=0$
.
Karte 150
Frage
Difference between absolute and relative extrema?
Antwort
An absolute maximum is at least every other value on the stated domain, and an absolute minimum is at most every other value. A relative extremum makes the corresponding comparison only with nearby values.
Karte 151
Frage
If
$f$
is continuous at a critical number
$c$
and
$f'$
is positive on both sides, is there a local extremum?
Antwort
No. The function is increasing through
$c$
, so it has no local extremum there.
Karte 152
Frage
Second derivative test for a local minimum?
Antwort
If
$f'(c)=0$
and
$f''(c)>0$
, then
$f$
has a local minimum at
$c$
.
Karte 153
Frage
Zeros of
$f'$
correspond to what features of
$f$
?
Antwort
Horizontal tangents where
$f'$
exists. They are critical numbers and possible extrema, but each needs further sign or value analysis.
Karte 154
Frage
Implicit relation: how can
$dy/dx$
reveal local behavior?
Antwort
Its sign shows whether the relation's local branch rises or falls as
$x$
increases; zeros and undefined values mark possible horizontal or vertical tangents.
Karte 155
Frage
Which theorem links an average slope to an instantaneous slope?
Antwort
The Mean Value Theorem, provided the function is continuous on the closed interval and differentiable on its interior.
Karte 156
Frage
How can an implicit derivative locate a horizontal tangent?
Antwort
At a valid point on the relation, find where the simplified numerator of
$dy/dx$
is zero while its denominator is nonzero, then confirm the local branch has the expected behavior.
Karte 157
Frage
Derivative-sign chart: where is
$f$
decreasing?
Antwort
On intervals where
$f'(x)<0$
.
Karte 158
Frage
Second derivative test for a local maximum?
Antwort
If
$f'(c)=0$
and
$f''(c)<0$
, then
$f$
has a local maximum at
$c$
.
Karte 159
Frage
If
$f'$
is increasing, what is the concavity of
$f$
?
Antwort
$f$
is concave up on that interval, assuming the relevant derivatives exist.
Karte 160
Frage
Why must an optimization domain be stated?
Antwort
The context can exclude algebraic candidates and add boundary cases. Absolute extrema depend on the feasible inputs, not just on where the derivative is zero.
Karte 161
Frage
Which theorem guarantees absolute extrema, not where they occur?
Antwort
The Extreme Value Theorem. Continuity on a closed interval guarantees at least one maximum value and one minimum value, but finding them requires analysis.
Karte 162
Frage
Can
$f'(c)$
fail to exist at a local extremum?
Antwort
Yes. Corners, cusps, or other nondifferentiable domain points can be local extrema, which is why critical numbers include points where
$f'$
is undefined.
Karte 163
Frage
For a function continuous at
$c$
, what same-sign pattern in
$f'$
rules out a local extremum there?
Antwort
If
$f'$
is positive on both sides of
$c$
, or negative on both sides, then
$f$
keeps the same monotonic direction through
$c$
and has no local extremum there.
Karte 164
Frage
If
$f'(c)=0$
and
$f''(c)=0$
, what does the second derivative test conclude?
Antwort
Nothing. The test is inconclusive; use a first-derivative sign change, higher analysis, or direct comparison.
Karte 165
Frage
If the graph of
$f'$
crosses from negative to positive, what feature does
$f$
have?
Antwort
A local minimum at the crossing input, provided the input is in the domain of
$f$
.
Karte 166
Frage
How can an implicit derivative locate a vertical tangent?
Antwort
Find valid points where the simplified
$dy/dx$
denominator is zero and numerator is nonzero, then confirm the curve has the expected local behavior.
Karte 167
Frage
Which extra condition turns the Mean Value Theorem into Rolle’s Theorem?
Antwort
$f(a)=f(b)$
. The average slope becomes zero, so the guaranteed instantaneous slope is also zero.
Karte 168
Frage
Why are endpoints included in the candidates test?
Antwort
An absolute maximum or minimum on a closed interval can occur at an endpoint even though no two-sided derivative test applies there.
Karte 169
Frage
If
$f'(x)=0$
throughout an interval, what is
$f$
there?
Antwort
$f$
is constant on that interval, assuming the usual continuity and differentiability conditions that let the Mean Value Theorem apply.
Karte 170
Frage
Second-derivative sign for concave down?
Antwort
If
$f''(x)<0$
on an interval, then
$f$
is concave down there and
$f'$
is decreasing.
Karte 171
Frage
Graph of
$f'$
has a local minimum: possible effect on
$f$
?
Antwort
$f''$
may change from negative to positive, so
$f$
may change from concave down to concave up. Verify the sign change.
Karte 172
Frage
What should the final line of an optimization solution state?
Antwort
The requested maximum or minimum quantity in context, with units and the feasible input that produces it when relevant.
Karte 173
Frage
Can Rolle’s Theorem be used if
$f$
has a corner inside
$(a,b)$
?
Antwort
No. A corner breaks differentiability on the open interval, so the theorem's hypotheses are not satisfied.
Karte 174
Frage
How do
$f''$
zeros help analyze a graph?
Antwort
They are candidates for changes in concavity. Test the sign of
$f''$
on both sides; a zero alone doesn't guarantee an inflection point.
Karte 175
Frage
What does the accumulation function
$F(x)=\int_a^x f(t)\,dt$
measure?
Antwort
The signed net accumulation of
$f$
from
$a$
to
$x$
. Contributions above the axis are positive; contributions below it are negative.
Karte 176
Frage
Left Riemann sum on equal subintervals?
Antwort
If
$\Delta x=(b-a)/n$
and
$x_i=a+i\Delta x$
, then

$$L_n=\sum_{i=1}^{n} f(x_{i-1})\Delta x.$$
Karte 177
Frage
What does
$\int_a^b f(x)\,dx$
represent geometrically?
Antwort
Signed area between the graph and the
$x$
-axis from
$a$
to
$b$
, when
$f$
is integrable. Regions below the axis subtract from regions above it.
Karte 178
Frage
Fundamental Theorem of Calculus: evaluate a definite integral?
Antwort
If
$f$
is continuous on
$[a,b]$
and
$F$
is an antiderivative of
$f$
, then

$$\int_a^b f(x)\,dx=F(b)-F(a).$$
Karte 179
Frage
Derivative of
$F(x)=\int_a^x f(t)\,dt$
?
Antwort
If
$f$
is continuous, then

$$F'(x)=f(x).$$

This connects accumulation with instantaneous rate.
Karte 180
Frage
Why do all antiderivatives of the same function differ by a constant?
Antwort
If
$F'=f$
and
$G'=f$
on an interval, then
$(F-G)'=0$
, so
$F-G=C$
on that interval.
Karte 181
Frage
Right Riemann sum on equal subintervals?
Antwort
If
$\Delta x=(b-a)/n$
and
$x_i=a+i\Delta x$
, then

$$R_n=\sum_{i=1}^{n} f(x_i)\Delta x.$$
Karte 182
Frage
How does reversing integral bounds change the value?
Antwort
It changes the sign:

$$\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx.$$
Karte 183
Frage
Net Change Theorem?
Antwort
If
$Q'(t)$
is the rate of change of a quantity, then

$$Q(b)-Q(a)=\int_a^b Q'(t)\,dt.$$
Karte 184
Frage
Derivative of
$\int_a^{g(x)} f(t)\,dt$
?
Antwort
If
$f$
is continuous on an interval containing
$a$
and the range of
$g$
, and
$g$
is differentiable, then

$$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt=f(g(x))g'(x).$$
Karte 185
Frage
Power rule for antiderivatives?
Antwort
For
$n\ne-1$
,

$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C.$$
Karte 186
Frage
Midpoint Riemann sum on equal subintervals?
Antwort
With midpoint
$m_i=(x_{i-1}+x_i)/2$
,

$$M_n=\sum_{i=1}^{n} f(m_i)\Delta x.$$
Karte 187
Frage
How can an integral be split at an interior point
$c$
?
Antwort
For
$a\le c\le b$
,

$$\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx.$$
Karte 188
Frage
Derivative of
$G(x)=\int_x^b f(t)\,dt$
?
Antwort
If
$f$
is continuous, then

$$G'(x)=-f(x).$$

The variable lower bound produces the negative sign.
Karte 189
Frage
Antiderivative of
$1/x$
?
Antwort
On any interval not crossing zero,

$$\int \frac1x\,dx=\ln|x|+C.$$
Karte 190
Frage
Trapezoidal approximation on equal subintervals?
Antwort
$$T_n=\frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right].$$
Karte 191
Frage
How do geometric regions help evaluate a definite integral?
Antwort
Replace each familiar region with its geometric area, then attach a positive sign above the axis and a negative sign below it.
Karte 192
Frage
Basic antiderivatives of sine and cosine?
Antwort
$$\int \sin x\,dx=-\cos x+C,\qquad \int \cos x\,dx=\sin x+C.$$
Karte 193
Frage
Definite integral as a limit of Riemann sums?
Antwort
For an integrable function and sample points
$x_i^*$
,

$$\int_a^b f(x)\,dx=\lim_{\max\Delta x_i\to0}\sum_i f(x_i^*)\Delta x_i.$$
Karte 194
Frage
Constant-multiple rule for integrals?
Antwort
For a constant
$k$
,

$$\int_a^b kf(x)\,dx=k\int_a^b f(x)\,dx.$$

The analogous rule holds for indefinite integrals.
Karte 195
Frage
What pattern suggests
$u$
-substitution?
Antwort
A composite expression paired with its derivative, such as
$f(g(x))g'(x)$
. Set
$u=g(x)$
so
$du=g'(x)\,dx$
.
Karte 196
Frage
How should bounds change in a definite
$u$
-substitution?
Antwort
If
$u=g(x)$
, replace the
$x$
-bounds
$a,b$
with
$u$
-bounds
$g(a),g(b)$
. Then finish entirely in
$u$
, or return to
$x$
before applying the original bounds.
Karte 197
Frage
What condition makes
$F(x)=\int_a^x f(t)\,dt$
differentiable with
$F'(x)=f(x)$
?
Antwort
Continuity of
$f$
on an interval containing
$a$
and
$x$
is the standard AP Calculus condition.
Karte 198
Frage
Sum-and-difference rule for definite integrals?
Antwort
For integrable
$f$
and
$g$
,

$$\int_a^b [f(x)\pm g(x)]\,dx=\int_a^b f(x)\,dx\pm\int_a^b g(x)\,dx.$$
Karte 199
Frage
Basic antiderivative of
$e^x$
?
Antwort
$$\int e^x\,dx=e^x+C.$$
Karte 200
Frage
Basic antiderivatives of
$\sec^2x$
and
$\csc^2x$
?
Antwort
$$\int\sec^2x\,dx=\tan x+C,\qquad \int\csc^2x\,dx=-\cot x+C.$$
Karte 201
Frage
For an increasing integrable function, how do left and right sums compare with the integral?
Antwort
On the same partition, the left sum is an underestimate and the right sum is an overestimate. Reverse those conclusions when the function is decreasing.
Karte 202
Frage
How does concavity predict trapezoidal and midpoint error?
Antwort
For a concave-up function, trapezoidal sums overestimate and midpoint sums underestimate. For a concave-down function, the directions reverse.
Karte 203
Frage
Why might polynomial long division help before integrating a rational function?
Antwort
When the numerator's degree is at least the denominator's, division rewrites the expression as a polynomial plus a simpler proper rational function.
Karte 204
Frage
What denominator pattern suggests an arctangent antiderivative?
Antwort
After completing the square and scaling, a form like

$$\int\frac{1}{1+u^2}\,du=\arctan u+C.$$
Karte 205
Frage
Basic antiderivatives of
$\sec x\tan x$
and
$\csc x\cot x$
?
Antwort
$$\int\sec x\tan x\,dx=\sec x+C,\qquad \int\csc x\cot x\,dx=-\csc x+C.$$
Karte 206
Frage
How does an initial condition determine an antiderivative?
Antwort
First find the family
$F(x)+C$
. Substitute the given point, such as
$y(a)=b$
, and solve for
$C$
.
Karte 207
Frage
Should a definite-integral answer include
$+C$
?
Antwort
No. A definite integral is one number or quantity. The constant of integration belongs to an indefinite integral or general antiderivative.
Karte 208
Frage
Why does an indefinite integral include
$+C$
?
Antwort
Differentiation loses additive constants. The
$+C$
represents every function with the stated derivative.
Karte 209
Frage
When is
$F(x)=\int_a^x f(t)\,dt$
increasing?
Antwort
Where
$F'(x)=f(x)>0$
. It is decreasing where
$f(x)<0$
.
Karte 210
Frage
How is the concavity of
$F(x)=\int_a^x f(t)\,dt$
determined?
Antwort
Since
$F'(x)=f(x)$
,
$F$
is concave up where
$f$
is increasing and concave down where
$f$
is decreasing, assuming the needed derivatives exist.
Karte 211
Frage
How is
$\int_a^b |f(x)|\,dx$
interpreted?
Antwort
As total geometric area between
$f$
and the
$x$
-axis. Split at zeros of
$f$
and make every regional contribution nonnegative.
Karte 212
Frage
What constant-factor check completes many
$u$
-substitutions?
Antwort
Compare
$du=g'(x)\,dx$
with the remaining factor. Multiply or divide by a constant so the integrand contains exactly the needed differential.
Karte 213
Frage
How do you recover
$\Delta x$
from a sigma-form Riemann sum on
$[a,b]$
?
Antwort
Identify the factor multiplying each function value. For
$n$
equal subintervals, it should be

$$\Delta x=\frac{b-a}{n}.$$
Karte 214
Frage
Riemann sum for unequal subinterval widths?
Antwort
If
$x_{i-1}$
to
$x_i$
has width
$\Delta x_i$
and sample point
$x_i^*$
, use

$$\sum_i f(x_i^*)\Delta x_i.$$
Karte 215
Frage
Does continuity guarantee integrability on a closed interval?
Antwort
Yes. A function continuous on
$[a,b]$
is integrable there. Some discontinuous functions are also integrable, so continuity is sufficient, not necessary.
Karte 216
Frage
Antiderivative pattern for
$g'(x)/g(x)$
?
Antwort
Where
$g(x)\ne0$
,

$$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C.$$
Karte 217
Frage
What algebraic rewrites often reveal a basic antiderivative?
Antwort
Split sums, factor out constants, simplify rational powers, and rewrite radicals as exponents. Then apply known antiderivative rules term by term.
Karte 218
Frage
What units does
$\int_a^b r(t)\,dt$
have?
Antwort
Rate units multiplied by input units. For example, liters per minute integrated over minutes gives liters.
Karte 219
Frage
What is a differential equation?
Antwort
An equation that relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true on an interval.
Karte 220
Frage
How does a verbal rate statement become a differential equation?
Antwort
Name the changing quantity and its independent variable, translate its rate as a derivative, and express that derivative using the stated relationship. For example, “rate proportional to
$y$
” becomes
$dy/dt=ky$
.
Karte 221
Frage
How do you verify that
$y=f(x)$
solves a differential equation?
Antwort
Differentiate
$f$
as needed, substitute
$y$
and its derivatives into the equation, and confirm both sides agree on the claimed interval.
Karte 222
Frage
General solution versus particular solution?
Antwort
A general solution contains an arbitrary constant and represents a family of solution curves. A particular solution uses an initial condition to determine that constant.
Karte 223
Frage
What does one segment in a slope field show?
Antwort
At
$(x,y)$
, its slope equals the value of
$dy/dx$
given by the differential equation at that point.
Karte 224
Frage
What units does the constant
$k$
have in
$dy/dt=ky$
?
Antwort
Inverse time units, such as per hour. That makes the exponent
$kt$
dimensionless.
Karte 225
Frage
Euler's method update formula?
Antwort
With step size
$\Delta x$
,

$$y_{n+1}=y_n+F(x_n,y_n)\Delta x$$

for
$dy/dx=F(x,y)$
.
Karte 226
Frage
What makes a first-order differential equation separable?
Antwort
It can be rearranged so all
$y$
factors accompany
$dy$
and all
$x$
factors accompany
$dx$
, such as

$$g(y)\,dy=f(x)\,dx.$$
Karte 227
Frage
What is an initial value problem?
Antwort
A differential equation paired with a value such as
$y(x_0)=y_0$
. It asks for a solution through
$(x_0,y_0)$
; depending on the equation, there may be zero, one, or multiple such solutions.
Karte 228
Frage
What is an isocline in a slope field?
Antwort
A curve along which the differential equation gives the same slope. For
$dy/dx=F(x,y)$
, an isocline satisfies
$F(x,y)=k$
for a constant
$k$
.
Karte 229
Frage
What does step size mean in Euler's method?
Antwort
It is the horizontal change
$\Delta x$
used in each tangent-line step. The sign of
$\Delta x$
determines whether the approximation moves right or left.
Karte 230
Frage
General solution of
$dy/dt=ky$
?
Antwort
$$y=Ce^{kt}$$

for a constant
$C$
. The zero solution is included by
$C=0$
.
Karte 231
Frage
Core method for solving a separable differential equation?
Antwort
Separate the variables, integrate both sides, include a constant of integration, and solve for
$y$
when practical. Then apply any initial condition.
Karte 232
Frage
How should a solution curve follow a slope field?
Antwort
It must pass through its initial point and remain tangent to the short field segments it crosses. It should not be drawn by connecting segment endpoints.
Karte 233
Frage
How is Euler's method repeated?
Antwort
At each new point, recompute the slope from the differential equation, multiply by
$\Delta x$
, and add that change to the current
$y$
-value.
Karte 234
Frage
Why is one integration constant enough after integrating both sides?
Antwort
Two constants can be combined:
$C_2-C_1$
is still an arbitrary constant. Write a single
$C$
.
Karte 235
Frage
Solution of
$dy/dt=ky$
with
$y(0)=y_0$
?
Antwort
$$y(t)=y_0e^{kt}.$$
Karte 236
Frage
Can one differential equation have infinitely many solutions?
Antwort
Yes. A differential equation commonly describes a family of solution functions. An initial condition may select one particular solution when the relevant conditions support uniqueness.
Karte 237
Frage
What is an equilibrium solution of
$dy/dx=F(y)$
?
Antwort
A constant solution
$y=c$
where
$F(c)=0$
. In the slope field, the segments along that horizontal line have zero slope.
Karte 238
Frage
When does a forward Euler estimate tend to lie below the true solution?
Antwort
When the solution is concave up over the step, its tangent line lies below the curve. For concave down, the tangent-line estimate tends to lie above.
Karte 239
Frage
What can be lost when dividing to separate variables?
Antwort
Equilibrium solutions that make the divided factor zero. Check those constant solutions directly in the original differential equation.
Karte 240
Frage
In
$dy/dt=ky$
, what do the signs of
$k$
mean?
Antwort
For a positive quantity,
$k>0$
produces exponential growth,
$k<0$
produces exponential decay, and
$k=0$
keeps the quantity constant.
Karte 241
Frage
How can a table of slopes identify the matching differential equation?
Antwort
Test representative
$(x,y)$
entries in each candidate equation. Match zeros, signs, and repeated slope patterns before checking exact numerical values.
Karte 242
Frage
How does the sign of
$dy/dx$
describe a solution?
Antwort
The solution is increasing where
$dy/dx>0$
and decreasing where
$dy/dx<0$
. At one point,
$dy/dx=0$
gives a horizontal tangent; a constant
$y=c$
is an equilibrium only when the derivative equation gives zero all along that level.
Karte 243
Frage
How can a differential equation determine a solution's concavity?
Antwort
Differentiate the equation with respect to the independent variable to obtain
$y''$
, using the chain rule for any
$y$
-dependence. Then use the sign of
$y''$
along the solution.
Karte 244
Frage
Why must a differential-equation solution include an interval or domain?
Antwort
Algebraic steps may introduce restrictions, and the formula must remain differentiable and satisfy the original equation throughout the interval containing the initial point.
Karte 245
Frage
Doubling time for exponential growth
$y=y_0e^{kt}$
?
Antwort
For
$k>0$
,

$$T_d=\frac{\ln2}{k}.$$

It is independent of the initial amount.
Karte 246
Frage
How can a slope field reveal whether
$dy/dx$
depends only on
$y$
?
Antwort
Slopes repeat horizontally: every point at the same height has the same segment slope.
Karte 247
Frage
How is an initial condition used after separation?
Antwort
Substitute the given
$x$
and
$y$
values into the integrated relationship to solve for the arbitrary constant, then state the resulting particular solution.
Karte 248
Frage
How do units check a model
$dy/dt=F(t,y)$
?
Antwort
The right side must have the same units as
$dy/dt$
: units of
$y$
per unit of
$t$
. A mismatch signals an incorrect translation or parameter unit.
Karte 249
Frage
Why should a separated solution be checked in the original equation?
Antwort
Division, logarithms, square roots, or algebraic rearrangement can lose or introduce branches. Direct substitution confirms the formula and its stated domain.
Karte 250
Frage
Half-life for exponential decay
$y=y_0e^{kt}$
?
Antwort
For
$k<0$
,

$$T_{1/2}=\frac{\ln(1/2)}{k}=\frac{\ln2}{|k|}.$$
Karte 251
Frage
Average value of
$f$
on
$[a,b]$
?
Antwort
For integrable
$f$
and
$a<b$
,

$$f_{\text{avg}}=\frac{1}{b-a}\int_a^b f(x)\,dx.$$
Karte 252
Frage
Displacement from velocity
$v(t)$
on
$[a,b]$
?
Antwort
$$s(b)-s(a)=\int_a^b v(t)\,dt.$$

Velocity below zero contributes negative displacement.
Karte 253
Frage
Area between vertical curves
$y=f(x)$
and
$y=g(x)$
?
Antwort
On intervals where
$f(x)\ge g(x)$
,

$$A=\int_a^b [f(x)-g(x)]\,dx.$$

Think top minus bottom.
Karte 254
Frage
Volume from known cross-sectional area
$A(x)$
?
Antwort
If slices are perpendicular to the
$x$
-axis,

$$V=\int_a^b A(x)\,dx.$$
Karte 255
Frage
Mean Value Theorem for Integrals: hypotheses and conclusion?
Antwort
If
$f$
is continuous on
$[a,b]$
, then some
$c\in[a,b]$
satisfies

$$f(c)=\frac{1}{b-a}\int_a^b f(x)\,dx.$$

If
$a<b$
, a point can also be chosen in
$(a,b)$
.
Karte 256
Frage
Velocity and acceleration from position
$s(t)$
?
Antwort
$$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$$
Karte 257
Frage
Cross-sectional area when each slice is a square?
Antwort
If the base segment has length
$s(x)$
, then

$$A(x)=[s(x)]^2.$$
Karte 258
Frage
How do you find accumulation from an inflow rate and an outflow rate?
Antwort
Integrate the net rate:

$$Q(b)-Q(a)=\int_a^b [r_{\text{in}}(t)-r_{\text{out}}(t)]\,dt.$$
Karte 259
Frage
Area between horizontal curves written as
$x=R(y)$
and
$x=L(y)$
?
Antwort
On intervals where
$R(y)\ge L(y)$
,

$$A=\int_c^d [R(y)-L(y)]\,dy.$$

Think right minus left.
Karte 260
Frage
Disc-method volume formula?
Antwort
For radius
$R(x)$
and slices perpendicular to the
$x$
-axis,

$$V=\pi\int_a^b [R(x)]^2\,dx.$$
Karte 261
Frage
What units does average value have?
Antwort
The same units as the original function. Integration adds an input unit, and division by interval length removes it.
Karte 262
Frage
Total distance traveled from velocity
$v(t)$
?
Antwort
$$\text{distance}=\int_a^b |v(t)|\,dt.$$

Split the interval wherever
$v(t)=0$
and its sign changes.
Karte 263
Frage
Cross-sectional area when each slice is a rectangle?
Antwort
If the slice has base
$b(x)$
and height
$h(x)$
, then

$$A(x)=b(x)h(x).$$

Use the stated relationship to express both in the integration variable.
Karte 264
Frage
How do you determine bounds for area between curves?
Antwort
Use the stated region boundaries or solve the curve-intersection equations. Check whether additional intersections inside the interval require splitting.
Karte 265
Frage
How is a rotation radius measured from a horizontal axis
$y=k$
?
Antwort
As vertical distance:
$|y-k|$
. For washers, identify which boundary stays farther from the axis over the interval.
Karte 266
Frage
When is a particle moving to the right or left?
Antwort
It moves right where
$v(t)>0$
and left where
$v(t)<0$
. Position alone does not determine direction.
Karte 267
Frage
Cross-sectional area when the diameter of a semicircle is
$d(x)$
?
Antwort
The radius is
$d(x)/2$
, so

$$A(x)=\frac12\pi\left(\frac{d(x)}2\right)^2=\frac{\pi}{8}[d(x)]^2.$$
Karte 268
Frage
Why must an area integral be split where curves intersect?
Antwort
The top/bottom or right/left relationship may switch there. Splitting keeps each integrand nonnegative and prevents signed cancellation.
Karte 269
Frage
How can a velocity table approximate displacement?
Antwort
Use a left, right, midpoint, or trapezoidal sum for
$\int v(t)\,dt$
. Each term is velocity times a time width.
Karte 270
Frage
Washer-method volume formula?
Antwort
For outer radius
$R(x)$
and inner radius
$r(x)$
,

$$V=\pi\int_a^b ([R(x)]^2-[r(x)]^2)\,dx.$$
Karte 271
Frage
How do you recover position from velocity and an initial position?
Antwort
If
$s(a)$
is known,

$$s(t)=s(a)+\int_a^t v(u)\,du.$$
Karte 272
Frage
How do you choose between vertical and horizontal area slices?
Antwort
Choose the direction that describes the region with fewer pieces. Vertical slices use top minus bottom; horizontal slices use right minus left.
Karte 273
Frage
Cross-sectional area of an equilateral triangle with side
$s(x)$
?
Antwort
$$A(x)=\frac{\sqrt3}{4}[s(x)]^2.$$
Karte 274
Frage
How can a table approximate the average value of
$f$
on
$[a,b]$
?
Antwort
First approximate
$\int_a^b f(x)\,dx$
with an appropriate Riemann or trapezoidal sum, then divide by
$b-a$
.
Karte 275
Frage
Single expression for area between two curves?
Antwort
When the functions are integrable,

$$A=\int_a^b |f(x)-g(x)|\,dx.$$

For hand evaluation, split where their order changes.
Karte 276
Frage
How is a rotation radius measured from a vertical axis
$x=k$
?
Antwort
As horizontal distance:
$|x-k|$
. With
$dy$
slices, write the relevant boundaries as
$x$
-functions of
$y$
.
Karte 277
Frage
How can a rate table approximate total change with unequal time gaps?
Antwort
Multiply a representative rate on each interval by that interval's actual width, then sum. Do not assume a common
$\Delta t$
when the table is uneven.
Karte 278
Frage
When should a volume integral use
$dy$
?
Antwort
When slices perpendicular to the
$y$
-axis make the cross-sectional area easiest to express as
$A(y)$
. Then use
$V=\int_c^d A(y)\,dy$
.
Karte 279
Frage
What signals that a washer, not a disc, is needed?
Antwort
The rotated region leaves a hole around the axis. Subtract the inner circular area from the outer circular area.
Karte 280
Frage
What base length is used for cross sections over a planar region?
Antwort
Use the distance across the base region within each slice: top minus bottom for vertical slices, or right minus left for horizontal slices.
Karte 281
Frage
Why must total distance split at velocity sign changes?
Antwort
Distance accumulates speed
$|v|$
, not signed velocity. A single integral of
$v$
would cancel motion in opposite directions.
Karte 282
Frage
When does an accumulated quantity reach a local maximum?
Antwort
When its net rate changes from positive to negative. A zero rate alone is only a candidate; check the sign change.
Karte 283
Frage
What distinguishes area from a definite integral?
Antwort
Area is nonnegative. A definite integral is signed and may be zero or negative because regions below the axis subtract.
Karte 284
Frage
How do position, velocity, and acceleration graphs correspond?
Antwort
Velocity is the slope of position, and acceleration is the slope of velocity. Conversely, signed area under velocity gives position change, and signed area under acceleration gives velocity change.
Karte 285
Frage
How do you interpret
$\int_a^b [r(t)-c(t)]\,dt$
in context?
Antwort
As net change: total amount added by
$r$
minus total amount removed by
$c$
over the interval. State the resulting quantity and units.
Karte 286
Frage
What units does a volume integral
$\int A(x)\,dx$
have?
Antwort
Cubic units. Cross-sectional area contributes square units and slice thickness contributes one more length unit.
Karte 287
Frage
How can a graph of a rate reveal the largest accumulated value?
Antwort
Compare accumulation at endpoints and at times where the rate is zero or undefined. Compute signed areas to track the quantity's changes from its initial value.
Karte 288
Frage
Why should a contextual integral answer include a sentence?
Antwort
The number alone does not say whether it represents displacement, distance, area, volume, or net change. Name the quantity, interval, and units.

Abstract calculus graph with a glowing orange curve, tangent line, shaded area under the curve, and layered contour lines on a dark grid.

288 Karten