Calculus highlightsCalculus highlightsfor AP/final reviewfor AP/final review

Calculus highlightsCalculus highlightsfor AP/final reviewfor AP/final review

*Common methods for evaluating finite limits analytically

(that don’t work with direct substitution):

1) Factor:

2) Rationalize:

*Common methods for evaluating finite limits analytically

(that don’t work with direct substitution):

1) Factor:

2) Rationalize:

Evaluating LimitsEvaluating Limits

limx→ a

x2 −a2

x−a=lim

x→ ax+a( ) =2a lim

x→ 3/2

2x−312x2 −16x−3

=

limx→ 0

x+a− ax

=limx→ 0

1x+a+ a

=1

2 a

limx→ 0

2x+5 − 5x

=

3) Clearing Fractions:

4) L’Hopital’s Rule:

3) Clearing Fractions:

4) L’Hopital’s Rule:

limx→ 0

1(x+a)−

1a

x=lim

x→ 0

−1a(x+a)

=−12a

limx→ 0

2(x+ 3)−

23

x=

limx→ c

f(x)g(x)

=limx→ c

f '(x)g'(x)

(for indeterminate forms of

continuous/differentiable functions)

(for indeterminate forms of

continuous/differentiable functions)

limx→ 0

sinxx

= limx→ 0

1−cosxx

=

limits at infinity and infinite limitslimits at infinity and infinite limits

*Evaluate limits of rational functions as x approaches

infinity by using the rules of horizontal asymptotes:

*Evaluate limits of rational functions as x approaches

infinity by using the rules of horizontal asymptotes:

limx→ ∞

axn +...bxm+...

=0,n<m

limx→ ∞

axn +...bxm+...

=ab,n=m

limx→ ∞

axn +...bxm+...

=±∞,n>m

Continuity, differentiability, and limits Continuity, differentiability, and limits

Continuity: f(x) is continuous if:

1) exists

2) f(c) is defined

3)

Continuity: f(x) is continuous if:

1) exists

2) f(c) is defined

3)

limx→ c

f(x)

limx→ c

f(x) = f(c)

Limits: exists if Limits: exists if limx→ c

f(x) limx→ c+

f(x) =limx→ c−

f(x)

Differentiability: f(x) is differentiable if

exists.

*If f(x) is differentiable, then it is continuous.

*f(x) is not differentiable at any sharp turns, i.e.,

is not differentiable at x=0.

Differentiability: f(x) is differentiable if

exists.

*If f(x) is differentiable, then it is continuous.

*f(x) is not differentiable at any sharp turns, i.e.,

is not differentiable at x=0.

f '(x)= limΔx→ 0

f(x+Δx)− f(x)Δx

f (x)= x

Which of the following are true for the following graph.

I. exists for all values of c in the given

domain

II. f(x) is continuous on the given domain

III. f(x) is differentiable on the given domain

IV. f(c) is defined for all values of c in the given domain

Which of the following are true for the following graph.

I. exists for all values of c in the given

domain

II. f(x) is continuous on the given domain

III. f(x) is differentiable on the given domain

IV. f(c) is defined for all values of c in the given domain

limx→ c

f(x)

DerivativesDerivatives

*Power Rule: *Power Rule: d

dxxn⎡⎣ ⎤⎦=nx

n−1

d

dxf (x)g(x)[ ] = f(x)g'(x)+g(x) f '(x)*Product Rule: *Product Rule:

*Quotient Rule: *Quotient Rule: d

dx

f (x)

g(x)

⎡

⎣⎢

⎤

⎦⎥=

g(x) f '(x)− f(x)g'(x)g(x)[ ]

2

*Chain Rule: *Chain Rule: d

dxf (g(x))[ ] = f '(g(x))g'(x)

*Trigonometric Functions:*Trigonometric Functions:d

dxsin x[ ] =cosx

ddx

cosx[ ] =−sinx

ddx

tanx[ ] =sec2 x

ddx

cscx[ ] =−cscxcotx

ddx

secx[ ] =secxcotx

ddx

cotx[ ] =−csc2 x

*Exponential and Logarithmic Functions:*Exponential and Logarithmic Functions:

d

dxex⎡⎣ ⎤⎦=e

x

d

dxln x[ ] =

1x

*Implicit Differentiation: When differentiating a term of a

function that contains y, multiply by y’ or

*Implicit Differentiation: When differentiating a term of a

function that contains y, multiply by y’ or dy

dx

y=2−x3x+1

− 3x+1

FindFinddy

dx

y=cos 3x4( )sin 4x2( )

y=tan3(5x2 )

3xy2 −y=4x2 −6

y=3x2e x

y= lnx( )3/2

tangent linestangent lines*The derivative or represents the slope of

the graph of f(x) at any given value of x.

*The equation of the tangent line to the graph of f(x) at a

given point (x, y) is found by finding at that given

value, then plugging it into the point-slope equation for a

line.

*The derivative or represents the slope of

the graph of f(x) at any given value of x.

*The equation of the tangent line to the graph of f(x) at a

given point (x, y) is found by finding at that given

value, then plugging it into the point-slope equation for a

line.

dy

dxf '(x)

f '(x)

Find the linear approximation of f(0.2) at

x=0 for

Find the linear approximation of f(0.2) at

x=0 for f (x)= 3x2 −1( )

3

average rate of change and instantaneous rate of changeaverage rate of change and instantaneous rate of change

*Average Rate of Change/Approximate Rate of Change:*Average Rate of Change/Approximate Rate of Change:

f (b)− f(a)b−a (slope formula)(slope formula)

*Instantaneous Rate of Change/Exact Rate of Change:*Instantaneous Rate of Change/Exact Rate of Change:

f '(x)

Intermediate value theorem, rolle’s theorem, and mean value theoremIntermediate value theorem, rolle’s theorem, and mean value theorem

*Intermediate Value Theorem: If f(x) is continuous on [a,

b] and f(a)<k<f(b), then there exists at least one value of

c in [a, b] such that f(c)=k.

*Intermediate Value Theorem: If f(x) is continuous on [a,

b] and f(a)<k<f(b), then there exists at least one value of

c in [a, b] such that f(c)=k.*Rolle’s Theorem: If f(x) is continuous and differentiable,

and f(a)=f(b), then there exists at least one value of c in

(a, b) such that

*Rolle’s Theorem: If f(x) is continuous and differentiable,

and f(a)=f(b), then there exists at least one value of c in

(a, b) such that f '(c)=0

*Mean Value Theorem: If f(x) is continuous and

differentiable then there exists a value c in (a, b) such

that

*Mean Value Theorem: If f(x) is continuous and

differentiable then there exists a value c in (a, b) such

that f '(c)=

f(b)− f(a)b−a

related ratesrelated rates

*Process for solving related rates problems:

1) Write an equation to represent the problem.

2) Find the derivative (implicitly, with respect to t) for

the equation.

3) Plug in all known variables, including given rates.

4) Solve for the unknown.

*Process for solving related rates problems:

1) Write an equation to represent the problem.

2) Find the derivative (implicitly, with respect to t) for

the equation.

3) Plug in all known variables, including given rates.

4) Solve for the unknown.

Graphical analysisGraphical analysis

*Increasing/Decreasing Behavior:

f(x) is increasing if

f(x) is decreasing if

*Increasing/Decreasing Behavior:

f(x) is increasing if

f(x) is decreasing iff '(x) < 0

f '(x) > 0

*Concavity/Points of Inflection:

f(x) is concave upward if

f(x) is concave downward if

f(x) has a point of inflection if changes sign

*Concavity/Points of Inflection:

f(x) is concave upward if

f(x) is concave downward if

f(x) has a point of inflection if changes sign

f "(x) > 0

f "(x) < 0

f "(x)

*Absolute/Global Extrema on [a, b]:

1) Find critical numbers (where or is

undefined)

2) Plug in critical numbers and endpoints into f(x)

3) Find smallest y-value (absolute/global minimum)

Find largest y-value (absolute/global maximum)

*Absolute/Global Extrema on [a, b]:

1) Find critical numbers (where or is

undefined)

2) Plug in critical numbers and endpoints into f(x)

3) Find smallest y-value (absolute/global minimum)

Find largest y-value (absolute/global maximum)

f '(x)=0 f '(x)

*Relative/Local Extrema:*Relative/Local Extrema:

First Derivative Test:First Derivative Test:1) Find critical numbers (where or

is undefined)

1) Find critical numbers (where or

is undefined)

f '(x)=0 f '(x)

f '(x)

2) When changes from positive to negative

at a critical number, there is a relative maximum at

(x, f(x)).

When changes from negative to positive at a

critical number, there is a relative minimum at (x,

f(x)).

2) When changes from positive to negative

at a critical number, there is a relative maximum at

(x, f(x)).

When changes from negative to positive at a

critical number, there is a relative minimum at (x,

f(x)).

f '(x)

Second Derivative Test:Second Derivative Test:

1) Find critical numbers

2) f(x) has a relative maximum if at a

critical number.

f(x) has a relative minimum if at a

critical number .

1) Find critical numbers

2) f(x) has a relative maximum if at a

critical number.

f(x) has a relative minimum if at a

critical number .

f "(x) > 0

f "(x) < 0

For ,

find

(a) the intervals on which f(x) is

increasing or decreasing

(b) the intervals on which f(x) is

concave upward or concave

downward

(c) the points of inflection of f(x)

(d) any relative extrema of f(x)

(e) the absolute extrema of f(x) on [-2,

2]

For ,

find

(a) the intervals on which f(x) is

increasing or decreasing

(b) the intervals on which f(x) is

concave upward or concave

downward

(c) the points of inflection of f(x)

(d) any relative extrema of f(x)

(e) the absolute extrema of f(x) on [-2,

2]

f (x)=2x3 −2x2 −12x+5

OptimizationOptimization*Process for solving optimization problems:

1) Draw and label a sketch, if applicable.

2) Write a primary equation to be optimized.

3) Use any secondary equations to rewrite the

primary equation in terms of one variable.

4) Apply First Derivative Test, Second Derivative

Test, or absolute extrema test to finding the

maximum or minimum value.

*Process for solving optimization problems:

1) Draw and label a sketch, if applicable.

2) Write a primary equation to be optimized.

3) Use any secondary equations to rewrite the

primary equation in terms of one variable.

4) Apply First Derivative Test, Second Derivative

Test, or absolute extrema test to finding the

maximum or minimum value.

Riemann sums and trapezoidal sumsRiemann sums and trapezoidal sums

*Riemann Sum: Approximates the area under a curve

with a finite number of rectangles that intercept the

graph at their right endpoint, left endpoint, or midpoint.

*Trapezoidal Sum: Approximates the area under a curve

with a finite number of trapezoids.

*Riemann Sum: Approximates the area under a curve

with a finite number of rectangles that intercept the

graph at their right endpoint, left endpoint, or midpoint.

*Trapezoidal Sum: Approximates the area under a curve

with a finite number of trapezoids.

Given the following table of values, find

R(3), L(3), M(3), and T(3).

Given the following table of values, find

R(3), L(3), M(3), and T(3).

x -1 2 7 10

f(x) 6 -4 2 0

Given , find R(4), L(4), M(4),

and T(4) on the interval

Given , find R(4), L(4), M(4),

and T(4) on the interval

f (x)=cos x2( )[0,2π ]

IntegralsIntegrals

*Definition of a Definite Integral: When the limit as the

number of rectangles approaches infinity of a Riemann

Sum is found, this represents the area under the curve

bound by the x-axis, or the definite integral of the

function on a given interval. A definite integral is also

used to find the “total amount accumulated” of something

given its rate of change.

*Definition of a Definite Integral: When the limit as the

number of rectangles approaches infinity of a Riemann

Sum is found, this represents the area under the curve

bound by the x-axis, or the definite integral of the

function on a given interval. A definite integral is also

used to find the “total amount accumulated” of something

given its rate of change.

*Power Rule: *Power Rule: xn dx∫ =

xn+1

n+1+C

*U-Substitution: *U-Substitution: f (u)du∫ =F(u)+C

*Exponential and Logarithmic Functions:*Exponential and Logarithmic Functions:

ex dx∫ =ex +C

1

xdx∫ =ln x +C

*Trigonometric Functions:*Trigonometric Functions:sin xdx =−cosx+C∫cosxdx∫ =sinx+C

tanxdx∫ =−lncosx +C

cscxdx∫ =−lncscx+cotx +C

secxdx∫ =lnsecx+ tanx +C

cotxdx∫ =lnsinx +C

sec2 xdx∫ =tanx+C

cscxcotxdx=∫ −cscx+C

secxtanxdx∫ =secx+C

csc2 xdx∫ =−cotx+C

−x2 (5 − 3x3)dx =∫

tan 2x( )sec2(2x)dx=∫

ln x( )5

3xdx=∫

e x

xdx∫ =

*Special Properties of Integrals:*Special Properties of Integrals:

f (x)dx =0a

a

∫

f(x)dxb

a

∫ =− f(x)dxa

b

∫

Fundamental theorems of calculusFundamental theorems of calculus

*First Fundamental Theorem of Calculus:*First Fundamental Theorem of Calculus:

f (x)dxa

b

∫ =F(b)−F(a)

f '(x)dx = f (b)− f (a)a

b

∫-or--or-

-or- (with U-

Substitution)

-or- (with U-

Substitution)f (g(x))g '(x)dx = f(u)du

g(a)

g(b)

∫a

b

∫

*Second Fundamental Theorem of Calculus:*Second Fundamental Theorem of Calculus:

d

dxf (t)dt

a

x

∫⎡

⎣⎢

⎤

⎦⎥= f(x)

ddx

f(t)dta

g(x)

∫⎡

⎣⎢

⎤

⎦⎥= f(g(x))g'(x)-or--or-

Given and F(3)=-10, find F(6).Given and F(3)=-10, find F(6).f (x)dx =223

6

∫

ln t dt−5

ex2

∫ =

average value of a functionaverage value of a function

*Average Value of a Function:*Average Value of a Function:

1

b−af(x)dx

a

b

∫

The following graph shows the number of

cellphone sales an AT&T representative makes

at each hour of a given workday. Find the

average number of sales the representative

makes during [1, 8].

The following graph shows the number of

cellphone sales an AT&T representative makes

at each hour of a given workday. Find the

average number of sales the representative

makes during [1, 8].

particle motionparticle motion

x '(t)=v(t)v'(t) =a(t)x"(t) =a(t)

*Position, Velocity, and Acceleration:*Position, Velocity, and Acceleration:

v(t)dt∫ =x(t)+C

a(t)dt∫ =v(t)+C

a(t)dt∫ =x(t)+C∫

*Speed:*Speed: speed =v(t)*Total Distance

Traveled:

*Total Distance

Traveled:v(t) dt∫

*A particle moves left when v(t)<0 and right when

v(t)>0.

*A particle changes direction when v(t) changes sign.

*A particle stops when v(t)=0.

*A particle speeds up when v(t) and a(t) are the same

sign and slows down when v(t) and a(t) are opposite

signs.

*A particle is farthest to the left when x(t) is minimized

and farthest to the right when x(t) is maximized.

*A particle moves left when v(t)<0 and right when

v(t)>0.

*A particle changes direction when v(t) changes sign.

*A particle stops when v(t)=0.

*A particle speeds up when v(t) and a(t) are the same

sign and slows down when v(t) and a(t) are opposite

signs.

*A particle is farthest to the left when x(t) is minimized

and farthest to the right when x(t) is maximized.

solving differential equations through separation of variablessolving differential equations through separation of variables

*Differential equations are equations containing *Differential equations are equations containing dy

dx

*To solve a differential equation means to integrate to

find the original equation in terms of only x and y. We

can do this by first separating the variables, then

integrating both sides.

*To solve a differential equation means to integrate to

find the original equation in terms of only x and y. We

can do this by first separating the variables, then

integrating both sides.

SolveSolve xydy

dx=−2

Area between two curvesArea between two curves

*To find the area between two curves, integrate the

difference between the larger function and the smaller

function (top function - bottom function).

*To find the area between two curves, integrate the

difference between the larger function and the smaller

function (top function - bottom function).

Volumes of revolutionVolumes of revolution

*Disk Method: To find the volume of the solid formed by

revolving a region about a horizontal line that is

adjacent to the region, use .

To find the volume of the solid formed by revolving a

region about a vertical line that is adjacent to the region,

use .

*Disk Method: To find the volume of the solid formed by

revolving a region about a horizontal line that is

adjacent to the region, use .

To find the volume of the solid formed by revolving a

region about a vertical line that is adjacent to the region,

use .

π R(x)[ ]2dx

a

b

∫

π R(y)[ ]2dy

a

b

∫

*Washer Method: To find the volume of the solid formed

by revolving a region about a horizontal line that is not

adjacent to the region, use .

*Washer Method: To find the volume of the solid formed

by revolving a region about a horizontal line that is not

adjacent to the region, use .

To find the volume of the solid formed by revolving a

region

about a vertical line that is not adjacent to the region,

use

To find the volume of the solid formed by revolving a

region

about a vertical line that is not adjacent to the region,

use

π (R(x))2 − (r(x))2⎡⎣ ⎤⎦a

b

∫ dx

π (R(y))2 − (r(y))2⎡⎣ ⎤⎦a

b

∫ dy

Cross-sectional volumesCross-sectional volumes

*To find the volume of the solid formed by lying cross-

sections in the form of a geometric shape perpendicular

to the x-axis in a bounded region, use

, where A is the area formula for the given geometric

shape and f(x)-g(x) is the height of the representative

rectangle in the region. Be sure you consider what

quantity f(x)-g(x) represents in the area formula.

*To find the volume of the solid formed by lying cross-

sections in the form of a geometric shape perpendicular

to the x-axis in a bounded region, use

, where A is the area formula for the given geometric

shape and f(x)-g(x) is the height of the representative

rectangle in the region. Be sure you consider what

quantity f(x)-g(x) represents in the area formula.

A f (x)−g(x)[ ]a

b

∫ dx