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.

लेखक Flashcards Open Source App

इस डेक के बारे में

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.

इस डेक के कार्ड

कार्ड 1
प्रश्न
What does
$\lim_{x \to a} f(x)=L$
say?
उत्तर
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.
कार्ड 2
प्रश्न
How can a table estimate
$\lim_{x \to a} f(x)$
?
उत्तर
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.
कार्ड 3
प्रश्न
When does direct substitution evaluate a limit?
उत्तर
When the function is continuous at the target input. Then

$$\lim_{x \to a} f(x)=f(a).$$
कार्ड 4
प्रश्न
Three conditions for continuity at
$x=a$
?
उत्तर
$f(a)$
exists,
$\lim_{x \to a}f(x)$
exists, and

$$\lim_{x \to a}f(x)=f(a).$$
कार्ड 5
प्रश्न
Intermediate Value Theorem: hypotheses and conclusion?
उत्तर
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
$(a,b)$
.
कार्ड 6
प्रश्न
When does a two-sided limit equal
$L$
?
उत्तर
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.
कार्ड 7
प्रश्न
How do you read a finite limit from a graph?
उत्तर
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.
कार्ड 8
प्रश्न
Limit law for a sum or difference?
उत्तर
If both component limits exist,

$$\lim_{x\to a}[f(x)\pm g(x)]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x).$$
कार्ड 9
प्रश्न
What makes a discontinuity removable?
उत्तर
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.
कार्ड 10
प्रश्न
Squeeze Theorem: usable form?
उत्तर
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
$\lim_{x\to a}f(x)=L$
.
कार्ड 11
प्रश्न
What does
$\lim_{x\to a}f(x)=+\infty$
mean?
उत्तर
$f(x)$
grows without bound above as
$x$
approaches
$a$
. It describes unbounded behavior, not a finite limit value.
कार्ड 12
प्रश्न
What must a table show for a left-hand limit?
उत्तर
Inputs less than the target and moving toward it. For
$\lim_{x\to a^-}f(x)$
, use
$x<a$
with
$x$
getting closer to
$a$
.
कार्ड 13
प्रश्न
Limit law for a product?
उत्तर
If both limits exist,

$$\lim_{x\to a}[f(x)g(x)]=\left(\lim_{x\to a}f(x)\right)\left(\lim_{x\to a}g(x)\right).$$
कार्ड 14
प्रश्न
Graph signature of a jump discontinuity?
उत्तर
The left- and right-hand limits are finite but unequal. The function approaches different heights from the two sides.
कार्ड 15
प्रश्न
Which theorem can guarantee a root on
$[a,b]$
?
उत्तर
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.
कार्ड 16
प्रश्न
Horizontal asymptote from a limit at infinity?
उत्तर
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.
कार्ड 17
प्रश्न
What does an open circle say about a graph's limit?
उत्तर
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.
कार्ड 18
प्रश्न
Limit law for a quotient—and its condition?
उत्तर
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
$0$
.
कार्ड 19
प्रश्न
What does continuity on
$[a,b]$
require at the endpoints?
उत्तर
Continuity on
$(a,b)$
, right-continuity at
$a$
, and left-continuity at
$b$
:

$$\lim_{x\to a^+}f(x)=f(a),\qquad \lim_{x\to b^-}f(x)=f(b).$$
कार्ड 20
प्रश्न
When is the Squeeze Theorem a natural choice?
उत्तर
When a hard or oscillating expression can be trapped between two simpler expressions with the same limit.
कार्ड 21
प्रश्न
Vertical asymptote from one-sided behavior?
उत्तर
If at least one one-sided limit at
$x=a$
is
$+\infty$
or
$-\infty$
, then
$x=a$
is a vertical asymptote.
कार्ड 22
प्रश्न
Limit at infinity of equal-degree rational functions?
उत्तर
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
$b_n\ne0$
.
कार्ड 23
प्रश्न
When can a limit pass through a continuous outer function?
उत्तर
If
$\lim_{x\to a}g(x)=L$
and
$f$
is continuous at
$L$
, then

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

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

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

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

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

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

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

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

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

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

Differentiating each factor and multiplying the results is not the product rule.
कार्ड 46
प्रश्न
How can nearby table values estimate
$f'(a)$
?
उत्तर
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.
कार्ड 47
प्रश्न
What does
$f''(x)$
measure?
उत्तर
The rate of change of
$f'(x)$
with respect to
$x$
. Its units are the units of
$f$
per square input unit.
कार्ड 48
प्रश्न
Instantaneous rate of change of
$f$
at
$a$
?
उत्तर
$f'(a)$
. It is the limit of average rates over intervals shrinking to
$a$
, and geometrically it is the tangent-line slope.
कार्ड 49
प्रश्न
Derivative of a sum or difference?
उत्तर
$$\frac{d}{dx}[f(x)\pm g(x)]=f'(x)\pm g'(x)$$
कार्ड 50
प्रश्न
Derivative of
$\cos x$
?
उत्तर
$$\frac{d}{dx}(\cos x)=-\sin x$$

The standard formula assumes radians.
कार्ड 51
प्रश्न
Quotient rule?
उत्तर
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.
कार्ड 52
प्रश्न
Common notations for the first derivative?
उत्तर
$f'(x)$
,
$y'$
,
$\dfrac{dy}{dx}$
, and
$\dfrac{d}{dx}f(x)$
. They describe the same derivative in different contexts.
कार्ड 53
प्रश्न
Derivative of
$e^x$
?
उत्तर
$$\frac{d}{dx}e^x=e^x$$
कार्ड 54
प्रश्न
What graph features can make
$f$
nondifferentiable?
उत्तर
A discontinuity, corner, cusp, vertical tangent, or other failure of the difference-quotient limit. Continuity is necessary but not sufficient.
कार्ड 55
प्रश्न
Derivative of
$\tan x$
?
उत्तर
Where
$\tan x$
is defined,

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

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

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

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

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

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

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

When
$a=1$
, the derivative is
$0$
.
कार्ड 62
प्रश्न
How can a graph estimate
$f'(a)$
?
उत्तर
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.
कार्ड 63
प्रश्न
Derivative of
$\sec x$
?
उत्तर
Where
$\sec x$
is defined,

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

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

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

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

Use the four table entries at the same input.
कार्ड 67
प्रश्न
Derivative of
$\cot x$
?
उत्तर
Where
$\cot x$
is defined,

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

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

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

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

Differentiate the outer function at the unchanged inner input, then multiply by the inner derivative.
कार्ड 72
प्रश्न
How do you identify inner and outer functions in a composite?
उत्तर
Find the expression evaluated first—that is the inner function. The operation applied to its result is the outer function.
कार्ड 73
प्रश्न
Core rule when differentiating an implicit equation in
$x$
and
$y$
?
उत्तर
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.
कार्ड 74
प्रश्न
Derivative of an inverse function at
$x$
?
उत्तर
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))}.\$
कार्ड 75
प्रश्न
Derivative of
$\arcsin x$
?
उत्तर
For
$|x|<1$
,

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

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

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

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

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

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

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

The extra factor is the chain rule.
कार्ड 82
प्रश्न
Slope of a tangent to an implicit curve
$F(x,y)=0$
?
उत्तर
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.
कार्ड 83
प्रश्न
Why must
$f'(a)\ne0$
to use
$[f^{-1}]'(f(a))=1/f'(a)$
?
उत्तर
Because the reciprocal slope would be undefined when
$f'(a)=0$
. The inverse may have a vertical tangent or fail to be differentiable there.
कार्ड 84
प्रश्न
Derivative of
$\arccos x$
?
उत्तर
For
$|x|<1$
,

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

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

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

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

This combines the power rule with the chain rule.
कार्ड 95
प्रश्न
Vertical tangent on an implicit curve: derivative clue?
उत्तर
$dy/dx$
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.
कार्ड 96
प्रश्न
Table formula for an inverse derivative at
$x=b$
?
उत्तर
Find
$a$
in the table with
$f(a)=b$
. If
$f'(a)\ne0$
, then

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

Use the product rule outside and the chain rule on the composite factor.
कार्ड 99
प्रश्न
Why can
$d^2y/dx^2$
depend on both
$x$
and
$y$
?
उत्तर
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.
कार्ड 100
प्रश्न
A quantity
$y$
changes through
$u$
, which changes with
$x$
. How are the rates connected?
उत्तर
When the functions are differentiable, the chain rule gives

$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}.$$
कार्ड 101
प्रश्न
What local property lets a function have an inverse derivative?
उत्तर
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.
कार्ड 102
प्रश्न
Derivative of
$a^{g(x)}$
?
उत्तर
For
$a>0$
,

$$\frac{d}{dx}a^{g(x)}=a^{g(x)}\ln(a)\,g'(x).$$
कार्ड 103
प्रश्न
How should
$Q'(t)$
be interpreted in context?
उत्तर
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.
कार्ड 104
प्रश्न
Position, velocity, and acceleration relationships?
उत्तर
For position
$s(t)$
,

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

For
$x$
close to
$a$
,
$f(x)\approx L(x)$
.
कार्ड 107
प्रश्न
L’Hospital’s Rule: basic conditions?
उत्तर
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.
कार्ड 108
प्रश्न
If distance is in meters and time in seconds, units of acceleration?
उत्तर
Meters per second squared,
$\text{m}/\text{s}^2$
. Acceleration is the rate of change of velocity with respect to time.
कार्ड 109
प्रश्न
Speed in terms of velocity?
उत्तर
$$\text{speed}=|v(t)|$$

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

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

It is most reliable for small
$|\Delta x|$
where the function is well approximated by its tangent.
कार्ड 122
प्रश्न
When may L’Hospital’s Rule be applied more than once?
उत्तर
When the derivative quotient still has
$0/0$
or
$\infty/\infty$
form and the rule's conditions continue to hold.
कार्ड 123
प्रश्न
What must a contextual derivative sentence include?
उत्तर
The changing quantity, the instant or input, the rate's sign or direction when relevant, and correct output-per-input units.
कार्ड 124
प्रश्न
Velocity negative and acceleration positive: what happens?
उत्तर
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.
कार्ड 125
प्रश्न
How should a negative related rate be interpreted?
उत्तर
The measured quantity is decreasing at that instant. Keep the sign and state the units; don't silently report only its magnitude.
कार्ड 126
प्रश्न
When is local linearity a sound approximation tool?
उत्तर
When
$f$
is differentiable near the base point and the target input is close enough that curvature has limited effect.
कार्ड 127
प्रश्न
Can L’Hospital’s Rule handle a one-sided limit?
उत्तर
Yes, if the required indeterminate form and differentiability conditions hold on that side and the derivative-quotient limit exists or is infinite.
कार्ड 128
प्रश्न
When is speed increasing?
उत्तर
When velocity and acceleration have the same sign, so
$v(t)a(t)>0$
.
कार्ड 129
प्रश्न
Volume changes with time: notation for its rate?
उत्तर
$dV/dt$
. Its units are cubic length units per time unit.
कार्ड 130
प्रश्न
Why are similar triangles useful in related rates?
उत्तर
They can reduce several changing lengths to one constraint before differentiation, keeping the rate equation tied to the geometry.
कार्ड 131
प्रश्न
Meaning of
$dy=f'(x)dx$
in approximation?
उत्तर
$dy$
is the tangent-line estimate of the actual output change
$\Delta y$
caused by an input change
$dx$
.
कार्ड 132
प्रश्न
What conclusion does L’Hospital’s Rule permit?
उत्तर
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.
कार्ड 133
प्रश्न
When is speed decreasing?
उत्तर
When velocity and acceleration have opposite signs, so
$v(t)a(t)<0$
.
कार्ड 134
प्रश्न
What does a tangent slope read from a contextual graph represent?
उत्तर
The instantaneous rate of the vertical quantity with respect to the horizontal quantity, with vertical units per horizontal unit.
कार्ड 135
प्रश्न
Does
$v(t)=0$
guarantee a particle changes direction?
उत्तर
No. It is momentarily at rest, but direction changes only if velocity changes sign across that time.
कार्ड 136
प्रश्न
How do you translate “
$Q$
increases by 3 units per minute” into derivative notation?
उत्तर
$dQ/dt=3$
in the stated time interval or at the stated instant. “Decreases by 3” would give
$dQ/dt=-3$
.
कार्ड 137
प्रश्न
Extreme Value Theorem: hypothesis and conclusion?
उत्तर
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
$[a,b]$
.
कार्ड 138
प्रश्न
What is a critical number of
$f$
?
उत्तर
A number
$c$
in the domain of
$f$
where
$f'(c)=0$
or
$f'(c)$
doesn't exist.
कार्ड 139
प्रश्न
First derivative test for a local maximum?
उत्तर
$f'$
changes from positive to negative at the critical point, so
$f$
changes from increasing to decreasing.
कार्ड 140
प्रश्न
Second-derivative sign for concave up?
उत्तर
If
$f''(x)>0$
on an interval, then
$f$
is concave up there and
$f'$
is increasing.
कार्ड 141
प्रश्न
If the graph of
$f'$
is above the
$x$
-axis, what does
$f$
do?
उत्तर
$f$
is increasing because
$f'(x)>0$
.
कार्ड 142
प्रश्न
First step in an optimization model?
उत्तर
Define the objective quantity and the constraint, including the feasible domain. Rewrite the objective as a function of one variable before differentiating.
कार्ड 143
प्रश्न
Mean Value Theorem: hypotheses and conclusion?
उत्तर
If
$f$
is continuous on
$[a,b]$
and differentiable on
$(a,b)$
, then some
$c$
in
$(a,b)$
satisfies

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

$$L_n=\sum_{i=1}^{n} f(x_{i-1})\Delta x.$$
कार्ड 177
प्रश्न
What does
$\int_a^b f(x)\,dx$
represent geometrically?
उत्तर
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.
कार्ड 178
प्रश्न
Fundamental Theorem of Calculus: evaluate a definite integral?
उत्तर
If
$f$
is continuous on
$[a,b]$
and
$F$
is an antiderivative of
$f$
, then

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$\int\frac{g'(x)}{g(x)}\,dx=\ln|g(x)|+C.$$
कार्ड 217
प्रश्न
What algebraic rewrites often reveal a basic antiderivative?
उत्तर
Split sums, factor out constants, simplify rational powers, and rewrite radicals as exponents. Then apply known antiderivative rules term by term.
कार्ड 218
प्रश्न
What units does
$\int_a^b r(t)\,dt$
have?
उत्तर
Rate units multiplied by input units. For example, liters per minute integrated over minutes gives liters.
कार्ड 219
प्रश्न
What is a differential equation?
उत्तर
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.
कार्ड 220
प्रश्न
How does a verbal rate statement become a differential equation?
उत्तर
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
$dy/dt=ky$
.
कार्ड 221
प्रश्न
How do you verify that
$y=f(x)$
solves a differential equation?
उत्तर
Differentiate
$f$
as needed, substitute
$y$
and its derivatives into the equation, and confirm both sides agree on the claimed interval.
कार्ड 222
प्रश्न
General solution versus particular solution?
उत्तर
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.
कार्ड 223
प्रश्न
What does one segment in a slope field show?
उत्तर
At
$(x,y)$
, its slope equals the value of
$dy/dx$
given by the differential equation at that point.
कार्ड 224
प्रश्न
What units does the constant
$k$
have in
$dy/dt=ky$
?
उत्तर
Inverse time units, such as per hour. That makes the exponent
$kt$
dimensionless.
कार्ड 225
प्रश्न
Euler's method update formula?
उत्तर
With step size
$\Delta x$
,

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

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

$$g(y)\,dy=f(x)\,dx.$$
कार्ड 227
प्रश्न
What is an initial value problem?
उत्तर
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.
कार्ड 228
प्रश्न
What is an isocline in a slope field?
उत्तर
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
$k$
.
कार्ड 229
प्रश्न
What does step size mean in Euler's method?
उत्तर
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.
कार्ड 230
प्रश्न
General solution of
$dy/dt=ky$
?
उत्तर
$$y=Ce^{kt}$$

for a constant
$C$
. The zero solution is included by
$C=0$
.
कार्ड 231
प्रश्न
Core method for solving a separable differential equation?
उत्तर
Separate the variables, integrate both sides, include a constant of integration, and solve for
$y$
when practical. Then apply any initial condition.
कार्ड 232
प्रश्न
How should a solution curve follow a slope field?
उत्तर
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.
कार्ड 233
प्रश्न
How is Euler's method repeated?
उत्तर
At each new point, recompute the slope from the differential equation, multiply by
$\Delta x$
, and add that change to the current
$y$
-value.
कार्ड 234
प्रश्न
Why is one integration constant enough after integrating both sides?
उत्तर
Two constants can be combined:
$C_2-C_1$
is still an arbitrary constant. Write a single
$C$
.
कार्ड 235
प्रश्न
Solution of
$dy/dt=ky$
with
$y(0)=y_0$
?
उत्तर
$$y(t)=y_0e^{kt}.$$
कार्ड 236
प्रश्न
Can one differential equation have infinitely many solutions?
उत्तर
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.
कार्ड 237
प्रश्न
What is an equilibrium solution of
$dy/dx=F(y)$
?
उत्तर
A constant solution
$y=c$
where
$F(c)=0$
. In the slope field, the segments along that horizontal line have zero slope.
कार्ड 238
प्रश्न
When does a forward Euler estimate tend to lie below the true solution?
उत्तर
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.
कार्ड 239
प्रश्न
What can be lost when dividing to separate variables?
उत्तर
Equilibrium solutions that make the divided factor zero. Check those constant solutions directly in the original differential equation.
कार्ड 240
प्रश्न
In
$dy/dt=ky$
, what do the signs of
$k$
mean?
उत्तर
For a positive quantity,
$k>0$
produces exponential growth,
$k<0$
produces exponential decay, and
$k=0$
keeps the quantity constant.
कार्ड 241
प्रश्न
How can a table of slopes identify the matching differential equation?
उत्तर
Test representative
$(x,y)$
entries in each candidate equation. Match zeros, signs, and repeated slope patterns before checking exact numerical values.
कार्ड 242
प्रश्न
How does the sign of
$dy/dx$
describe a solution?
उत्तर
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.
कार्ड 243
प्रश्न
How can a differential equation determine a solution's concavity?
उत्तर
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.
कार्ड 244
प्रश्न
Why must a differential-equation solution include an interval or domain?
उत्तर
Algebraic steps may introduce restrictions, and the formula must remain differentiable and satisfy the original equation throughout the interval containing the initial point.
कार्ड 245
प्रश्न
Doubling time for exponential growth
$y=y_0e^{kt}$
?
उत्तर
For
$k>0$
,

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

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

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

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

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

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

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

$$V=\int_a^b A(x)\,dx.$$
कार्ड 255
प्रश्न
Mean Value Theorem for Integrals: hypotheses and conclusion?
उत्तर
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
$(a,b)$
.
कार्ड 256
प्रश्न
Velocity and acceleration from position
$s(t)$
?
उत्तर
$$v(t)=s'(t),\qquad a(t)=v'(t)=s''(t).$$
कार्ड 257
प्रश्न
Cross-sectional area when each slice is a square?
उत्तर
If the base segment has length
$s(x)$
, then

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

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

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

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

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

Split the interval wherever
$v(t)=0$
and its sign changes.
कार्ड 263
प्रश्न
Cross-sectional area when each slice is a rectangle?
उत्तर
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.
कार्ड 264
प्रश्न
How do you determine bounds for area between curves?
उत्तर
Use the stated region boundaries or solve the curve-intersection equations. Check whether additional intersections inside the interval require splitting.
कार्ड 265
प्रश्न
How is a rotation radius measured from a horizontal axis
$y=k$
?
उत्तर
As vertical distance:
$|y-k|$
. For washers, identify which boundary stays farther from the axis over the interval.
कार्ड 266
प्रश्न
When is a particle moving to the right or left?
उत्तर
It moves right where
$v(t)>0$
and left where
$v(t)<0$
. Position alone does not determine direction.
कार्ड 267
प्रश्न
Cross-sectional area when the diameter of a semicircle is
$d(x)$
?
उत्तर
The radius is
$d(x)/2$
, so

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

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

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

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

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