REVISED 2019 - Ms. Virgil's Classroom · 2020. 3. 16. · fxcc( ) 0 or fxcc() undefined Make an fcc number line. f concave upward: fxcc( ) 0! f concave downward: fxcc( ) 0 Average

REVISED 2019

AP Calculus Gary Taylor

ABTM Calc Memorization Sheet

2b

aV r dx

Derivatives

Prod. d

dxuv uv vu

Quot. 2

d

dx

t bt tb

b b

Chain d

dxf u f u u

or d

f g x f g x g xdx

1n nd

dxx nx

1ln

d

dx xx , ln

d u

dx uu

1

lnloga

d

dx x ax

x xd

dxe e , lnx x

d

dxa a a

sin cosd

dxx x

cos sind

dxx x

2tan secd

dxx x

2cot cscd

dxx x

sec sec tand

dxx x x

csc csc cotd

dxx x x

2

1arcsin

1

d

dxx

x

2

1arctan

1

d

dxx

x

0

( ) ( )( ) lim

x

f x x f xf x

x

ax

afxfaf

ax

)()(lim)(

0limh

f a h f af a

h

Integrals

1,1

1

nCn

xdxx

nn

1ln

xdx x C

lnu

udx u C

Cedxexx

ln

xx aa dx C

a

Cxdxx sincos

Cxdxx cossin

Cxdxx tansec2

Cxdxx coslntan

Cxdxx sinlncot

Cxdxxx sectansec

Cxdxx cotcsc2

Cxdxxx csccotcsc

2 2

1arctan

u

a a

udu C

a u

2 2arcsin

u

a

udu C

a u

.

Trig Identities sin

costan

x

xx

1cossin 22 xx2 2tan sec 1x x

Average Rate of Change: AROC( ) ( )f b f a

b a

(slope between two

pts.)

Inst. Rate of Change: IROC f c (slope at a single point)

Mean Value Thm.: ab

afbfcf

)()()( (find c where sec tanm m )

Average Value of a Function:

b

a

avg

f x dxf

b a

areaavg. height

width

Intermediate Value Thm. A function f that is continuous on ,a b takes

on every y-value between and .f a f b

Extreme Value Thm. A function f that is continuous on ,a b has both an absolute minimum and an absolute maximum on the interval.

Second Fundamental Theorem

v

u

d

dxf t dt f v v f u u

Area of Trapezoid 1 21

2w h hA

Trapezoidal Rule

1 2 3 11

22 2 ... 2 n nA w h h h h h

With unequal widths add areas of all

trapezoids.

Riemann Sum (add areas of rectangles)

Volume

Disc

2b

aV r dx

Washer

2 2b

aV R r dx Cross Section

b

aV A dx

Start plus Accumulation

b

af b f a f x dx

Speed = v t

Displ. = 2

1

t

tv t dt

Total Dist. = 2

1

t

tv t dt

Speed is increasing when

velocity and acceleration

have the same sign.

Fundamental Theorem

b

af x dx f b f a

Memorize #1

Formulas

1. Area Between 2 Curves f and g on [a, b] where .f g

( ) ( )b

aA f x g x dx or ( ) ( )

b

aA f y g y dy (for functions of y only!)

(top curve – bottom curve) (right curve – left curve)

2. Volumes of Revolution

Discs 2b

aV r dx or dy Washers

2 2b

aV R r dx or dy

3. Volumes of Known Cross Sections

b

aV Adx or dy where A represents the area of a representative cross section.

4. Position – Velocity – Acceleration s(t) or x(t) – v(t) – a(t)

( ) ( )d

dts t v t Ctvdtta )()(

( ) ( )d

dtv t a t Ctsdttv )()(

v(t) = 0 implies particle at rest.

v(t) > 0 implies particle moving to right. v(t) < 0 implies particle moving to left.

Speed = v t

Displacement = b

av t dt Total Distance Traveled =

b

av t dt *

* If you cannot use a calculator you must split this integral up where a change of direction

occurs. (Make a velocity number line.)

5. Tangent Lines

Tangent: )( 11 xxmyy use the derivative to find m

6. Using Derivative Information

First Derivative Test:

Relative Max. or Min. Points: 0)( xf or )(xf undefined to find Critical Numbers.

Make an f number line. f increasing: 0)( xf f decreasing: 0)( xf

Write a sentence to justify. For example, “ f has a relative maximum at x = 2 because f changes from

positive to negative.”

Second Derivative Test:

Relative Max. or Min. Points: Find the sign of the second derivative at critical numbers and use concavity to

determine whether they are at relative max. or min. points.

Candidate Test:

Find absolute extrema by finding f-values at each critical number and endpoints.

Points of Inflection:

( ) 0f x or ( )f x undefined Make an f number line.

f concave upward: ( ) 0f x f concave downward: ( ) 0f x

Average Value of a Function:

b

a

avg

f x dxf

b a

(average height on [a, b])

Average Rate of Change: ab

afbfm

)()( (this is just slope between two points)

Instantaneous Rate of Change: use the derivative (slope at one point)

Definition of the Derivative: h

xfhxf

x

xfxxfxf

hx

)()(lim

)()(lim)(

00

Definition of the Derivative at a Single Point: 0

( ) ( )( ) lim

h

f a h f af a

h

or

ax

afxfaf

ax

)()(lim)(

Intermediate Value Theorem (IVT)

If f is continuous on ,a b and k is any y-value between and f a f b ,

then there is at least one x-value c between a and b such that .f c k

In other words, f takes on every y-value between and f a f b .

Extreme Value Theorem (EVT)

If f is continuous on ,a b then f has both an absolute minimum and an absolute maximum on the interval.

Mean Value Theorem (MVT) There is a number c in ,a b such that

f b f a

f cb a

. (for f continuous on [a, b] and differentiable on (a, b))

Informally: The Mean Value Theorem states that given the right

conditions of continuity and differentiability, there will be at least one tangent line parallel to the secant line.

In still other words: The instantaneous rate of change (slope of tangent) will equal the average rate of change

(slope of secant) at least once.

Definition of Continuity

A function f is continuous at an x-value c if lim limx c x c

f x f x f c

.

Note: A function can be continuous but not differentiable. (Example: f x x at x = 0)

If a function is differentiable, it must be continuous.

Approximation of the Area Under a Curve

a) Riemann Sum: Draw a figure and add up the rectangle areas (left, right, midpoint)

b) Trapezoids: (with unequal widths) add up areas of all trapezoids 1 21

2A w h h

(with n equal widths) 1 2 3 11

22 2 ... 2 n nA w h h h h h

Differential Equations: Separate variables and integrate.

Approximations using a Tangent Line: Use a convenient point to find a tangent line equation.

Plug in the given x-value to find an approximate y-value.

Derivatives of Inverse Functions:

For f and g inverse functions )(

1)(

afbg

where baf )( and abg )( .

Fundamental Theorem of Calculus or Start Plus Accumulation Method

( ) ( )b

af x dx f b f a or ( ) ( )

b

af b f a f x dx

Second Fundamental Theorem of Calculus:

( )x

a

d

dxf t dt f x Chain rule version:

v

u

d

dxf t dt f v v f u u (Where u and v are x-functions)

a bc

,b f b

,a f a

a bc

f a

k

f b

Memorize #2

Derivatives (in order of importance)

Product rule: 1. d

dxuv uv vu

Quotient rule: 2. 2

d

dx

t bt tb

b b

Chain rule: 3. d

dxf u f u u or

df g x f g x g x

dx

x form u form (chain rule)

Power Rule 4. 1n nd

dxx nx 1n n

d

dxu nu u

Log Rule 5. 1

lnd

dx xx ln

d u

dx uu

Exponential Rule 6. x xd

dxe e u u

d

dxe e u

Trig Rules 7. sin cosd

dxx x sin cos

d

dxu u u

8. cos sind

dxx x cos sin

d

dxu u u

9. 2tan secd

dxx x 2tan sec

d

dxu u u

Inverse Trig 10. 2

1arcsin

1

d

dxx

x

2arcsin

1

d

dx

uu

u

11. 2

1arctan

1

d

dxx

x

2arctan

1

d

dx

uu

u

Exponential Rule 12. lnx xd

dxa a a lnu u

d

dxa a u a

Log Rule 13. 1

lnloga

d

dx x ax

lnloga

d u

dx u au

Trig Rules 14. 2cot cscd

dxx x 2cot csc

d

dxu u u

15. sec sec tand

dxx x x sec sec tan

d

dxu u u u

16. csc csc cotd

dxx x x csc csc cot

d

dxu u u u

IMPLICIT DIFFERENTIATION can be used when it is inconvenient to solve for y.

CONTINUED

INTEGRALS (in order of importance)

x form u form (reverse chain rule)

Power Rule 1.

1,1

1

nCn

xdxx

nn

1,1

1

nCn

udxuu

nn

Log Rule 2. 1

lnx

dx x C lnu

udx u C

Exponential Rule 3. Cedxexx Cedxue

uu

Trig Rules 4. Cxdxx sincos Cudxuu sincos

5. Cxdxx cossin Cudxuu cossin

6. Cxdxx tansec2 Cudxuu tansec

2

7. Cxdxx coslntan Cudxuu coslntan

Inverse Trig Rules 8.2 2

11arctan

x

a adx C

a x

2 2

1arctan

u

a a

udx C

a u

9.2 2

1arcsin

x

adx C

a x

2 2 arcsin

u

a

udx C

a u

Trig Rule 10. Cxdxx sinlncot Cudxuu sinlncot

Exponential Rule 11. ln

xx aa dx C

a ln

uu aa u dx C

a

Trig Rules 12. Cxdxxx sectansec Cudxuuu sectansec

13. Cxdxx cotcsc2

Cudxuu cotcsc2

14. Cxdxxx csccotcsc Cudxuuu csccotcsc

AP Workshop – Gary Taylor

APTM Calculus Exam Review Tips for Students

• Practice released Free Response questions.

It is vital that students identify standard types of questions (Area-Volume, Solving

Differential Equations, Using a Derivative Graph, etc.).

• Focus on presentation.

Finding a correct answer may not be enough to earn any points. The process must be

clearly shown.

• Avoid simplification on FR. Answers like sin(π), 0.5(4 + 5 + 8), or ln(e2) earn full credit.

• Label units when necessary. Circle directions like “label units in your answer” on the first

read-through to avoid forgetting.

• Watch for FR parts with multiple problems. Circle the questions as you answer them to

make sure each question is answered.

• Write single sentence explanations when necessary. In general, don’t say “it”. In general

don’t use the word slope in the same sentence as the words increasing or decreasing.

• Use a pencil.

• Don’t leave two solutions without crossing one out.

• If you were unable to answer part (a) and you need the answer for part (b), go back and

make up a reasonable answer and use it.

• Don’t leave any Multiple Choice question blank. Mark any problems to be revisited if

time permits.

• If asked to give “the value of a function”, you are being asked to give a y-value.

• Round to 3 or more decimal places. Intermediate rounding (before the final answer)

may lead to an inaccurate answer.

• Be expecting to approximate the value of an integral with areas of rectangles or

trapezoids of unequal widths (possibly using values from a table).

• Understand the concept of increasing/decreasing speed.

• Be able to distinguish between “average value of a function” and “average rate of

change”.

• If possible take a full length “Mock Exam” a week or two before the actual AP Exam.

2012 MULTIPLE CHOICE (odd problems) ASSIGNMENT #1

1


2


3


4

29.


5

31.

33.

35.

37.


6

39.

41.

43.

45.

POSITION—VELOCITY—ACCELERATION ASSIGNMENT #2

7

1997 AB 1 Calculator Allowed

1. A particle moves along the x-axis so that its velocity at any time 0t is given by

2( ) 3 2 1.v t t t The position x(t) is 5 for t = 2.

(a) Write a polynomial expression for the position of the particle at any time 0t .

(b) For what values of t, 0 3t , is the particle’s instantaneous velocity the same as its

average velocity on the closed interval [0, 3]?

(c) Find the total distance traveled by the particle from time t = 0 until time t = 3.

2002 AB 3 (Form B) Calculator Allowed

3. A particle moves along the x-axis so that its velocity v at any time t, for 0 16t , is given by

2sin( ) 1.tv t e At time t = 0, the particle is at the origin.

(a) Sketch the graph of v(t) for 0 16t .

(b) During what intervals of time is the particle moving to the left? Give a reason for your

answer.

(c) Find the total distance traveled by the particle from t = 0 to t = 4.

(d) Is there any time t, 0 16t , at which the particle returns to the origin? Justify your

answer.

1983 AB 2 No Calculator

2. A particle moves along the x-axis so that at time t its position is given by

3 2( ) 6 9 11.x t t t t

(a) What is the velocity of the particle at t = 0?

(b) During what time intervals is the particle moving to the left?

(c) What is the total distance traveled by the particle from t = 0 to t = 2?

4. The graph shown is the velocity function for a particle moving on a straight line.

(a) When is the particle at rest?

(b) Find a(2), a(3), and a(5).

(c) Find the displacement of the particle

from t = 0 to t = 7 seconds.

(d) Find the total distance traveled by the

particle from t = 0 to t = 7 seconds.

(e) At what time is the velocity 10 ft/sec?

(f) Give a piecewise function for v(t).

(g) If the position of the particle at time zero is 4 feet,

find the position of the particle at time 7 seconds.


1. A particle moves along the x-axis in such a way that its acceleration at time t for 0t is

given by 23

a tt

. When t = 1, the position of the particle is 6 and its velocity is 2.

(a) Write an equation for the velocity, v(t), of the particle for all 0t .

(b) Write an equation for the position, x(t), of the particle for all 0t .

(c) Find the position of the particle when t = e .

in sec

v t

ft

time in seconds

AREA, VOLUME ASSIGNMENT #3

8

Calculators are allowed on all problems.

2002 AB 2 (Form B)

1. Let R be the region bounded by the y-axis and the graphs of 3

21

xy

x

and 4 2y x .

(a) Find the area of region R.

(b) Find the volume of the solid generated when R is revolved about the x-axis.

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to

the x-axis is a square. Find the volume of this solid.

1996 AB 2

2. Let R be the region in the first quadrant under the graph of 1

yx

for 4 9x .


(b) If the line x = k divides the region R into two regions of equal area, what is the value

of k?

(c) Find the volume of the solid whose base is the region R and whose cross sections cut

by planes perpendicular to the x-axis are squares.

2004 AB 1 (Form B)

1. Let R be the region enclosed by the graph of 1y x , the vertical line x = 10, and the x-

axis.


(b) Find the volume of a solid generated when R is revolved about the horizontal line

y = 3.

(c) Find the volume of a solid generated when R is revolved about the vertical line

x = 10.

AREA, VOLUME ASSIGNMENT #3

9

2003 AB 1 (Form B)

1. Let f be the function given by 2 3( ) 4 ,f x x x and let be the line 18 3 ,y x where is the

tangent line to the graph of f. Let R be the region bounded by the x-axis and the graph of f,

and let S be the region bounded by the graph of f, the line , and the x-axis, as shown below.

(a) Show that line is tangent to the graph of ( )y f x at the point x = 3.

(b) Find the area of region S.

(c) Find the volume of the solid generated when R is revolved about the x-axis.

1995 AB 4

4. The shaded regions 1R and 2R shown below are enclosed by the graphs of

2( )f x x and

( ) 2xg x .

(a) Find the x- and y-coordinates of the three points of intersection of the graphs of f and g.

(b) Without using absolute value, set up an expression involving one or more integrals

that gives the total area enclosed by the graphs of f and g. Do not evaluate.

(c) Without using absolute value, set up an expression involving one or more integrals

that gives the volume of the solid generated by revolving region 1R about the line

y = 5. Do not evaluate.

Note: Figure not drawn to scale.

1R

2R

x

y

(10,-15)(14,-15)

FUNCTIONS ASSIGNMENT #4

10

2008 BC 5 No Calculator (AB use a Calculator)

2007 BC 4 No Calculator (AB skip part C)

2001 AB/BC 4 No Calculator

1997 AB 4 Calculator allowed

FUNCTIONS ASSIGNMENT #4

11




MIXED REVIEW A ASSIGNMENT #5

12

1981 AB 1 Calculator allowed for PART C ONLY

1. Let f be the function defined by 4 2( ) 3 2.f x x x

(a) Find the zeros of f.

(b) Write an equation of the line tangent to the graph of f at the point where x = 1.

(c) Find the x –coordinate of each point at which the line tangent to the graph of f is

parallel to the line 2 4.y x

1981 AB 6—BC 4 Calculator allowed

6. A particle moves along the x-axis so that at time t its position is given by 2( ) sin( )x t t for

1 1t .

(a) Find the velocity at time t.

(b) Find the acceleration at time t.

(c) For what values of t does the particle change direction?

(d) Find all values of t for which the particle is moving left.


1. A particle initially at rest moves along the x-axis so that its acceleration at any time 0t is

given by 2( ) 12 4a t t . The position of the particle when t = 1 is x(1) = 3.

(a) Find the values of t for which the particle is at rest.

(b) Write an expression for the position x(t) of the particle at any time 0t .

(c) Find the total distance traveled by the particle from t = 0 to t = 2.


5. Let R be the region in the first quadrant under the graph of 2 2

xy

x

for 0 6x .

(a) Find the area of R.

(b) If the line x = k divides R into two regions of equal area, what is the value of k?

(c) What is the average value of 2 2

xy

x

on the interval 0 6x ?


3. Let f be the function given by ( ) 3.f x x (a) Sketch the graph of f and shade the region R enclosed by the graph of f, the x-axis, and

the vertical line x = 6.

(b) Find the area of the region R described in part (a).

(c) Rather than using the line x = 6 as in part (a), consider the line x = w, where w can be

any number greater than 3. Let A(w) be the area of the region enclosed by the graph of

f, the x-axis, and the vertical line x = w. Write an integral expression for A(w).

(d) Let A(w) be described in part (c). Find the rate of change of A with respect to w when

w = 6.

USING DERIVATIVE INFORMATION ASSIGNMENT #6

13

1991 AB 5

5. Let f be a function that is even and continuous on the closed interval [-3, 3]. The function f

and its derivatives have the properties indicated in the table below.

x 0 0 1x 1 1 2x 2 2 3x

( )f x 1 Positive 0 Negative -1 Negative

( )f x Undefined Negative 0 Negative Undefined Positive

( )f x Undefined Positive 0 Negative Undefined Negative

(a) Find the x-coordinate of each point at which f attains an absolute maximum value or

an absolute minimum value. For each x-coordinate you give, state whether f attains an

absolute maximum or an absolute minimum.

(b) Find the x-coordinate of each point of inflection on the graph of f. Justify your

answer.

(c) Sketch the graph of a function with all the given characteristics of f.

1989 AB 5

Note: This is the graph of the derivative of f, not the graph of f

5. The figure above shows the graph of f , the derivative of a function f. The domain of f is the

set of all real numbers x such that 10 10x .

(a) For what values of x does the graph of f have a horizontal tangent?

(b) For what values of x in the interval (-10, 10) does f have a relative maximum?

Justify your answer.

(c) For what values of x is the graph of f concave downward? Justify your answer.

USING DERIVATIVE INFORMATION ASSIGNMENT #6

14


5. The graph of the function f consists of a semicircle and two line segments as shown above.

Let g be the function given by 0

( ) ( )x

g x f t dt .

(a) Find g(3).

(b) Find all values of x on the open interval (-2, 5) at which g has a relative maximum.


(c) Write an equation for the line tangent to the graph of g at x = 3.

(d) Find the x-coordinate of each point of inflection of the graph of g on the open interval

(-2, 5). Justify your answer.

2004 AB 4 (Form B)

4. The figure below shows the graph of f , the derivative of the function f, on the closed

interval 1 5x . The graph of f has horizontal tangent lines at x = 1 and x = 3. The

function f is twice differentiable with f (2) = 6.

(a) Find the x –coordinate of each of the points of inflection of the graph of f. Give a

reason for your answer.

(b) At what value of x does f attain its absolute minimum value on the closed interval

1 5x ? At what value of x does f attain its absolute maximum value on the closed

interval 1 5x ? Justify your answers.

(c) Let g be the function defined by ( ) ( ).g x xf x Find an equation for the line tangent

to the graph of g at x = 2.

1999 AB 5

5. The graph of the function f, consisting of three line segments, is given above. Let

1

( ) ( )x

g x f t dt .

(a) Compute g(4) and g(-2).

(b) Find the instantaneous rate of change of g, with respect to x at x = 1.

(c) Find the absolute minimum value of g on the closed interval [-2, 4]. Justify your

answer.

(d) The second derivative of g is not defined at x = 1 and x = 2. How many of these

values are x-coordinates of points of inflection of the graph of g? Justify your answer.

MIXED REVIEW B ASSIGNMENT #7

15


4. Consider the curve defined by the equation cos 1y y x for 0 2y .

(a) Find dy

dx in terms of y.

(b) Write an equation for each vertical tangent to the curve.

(c) Find 2

2

d y

dx in terms of y.

2002 AB 2 (Form B) Calculator Allowed

2. The number of gallons, P(t) of a pollutant in a lake changes at a rate of 0.2( ) 1 3 tP t e

gallons per day, where t is measured in days. There are 50 gallons of pollutant in the lake a

time t = 0. The lake is considered to be safe when it contains 40 gallons or less of pollutant.

(a) Is the amount of pollutant increasing at time t = 9? Why or why not?

(b) For what value of t will the number of gallons of pollutant be at a minimum? Justify

your answer.

(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify

your answer.

(d) An investigator used the tangent line approximation to P(t) at t = 0 as a model for the

amount of pollutant in the lake. At what time t does this model predict that the lake

becomes safe?

2003 AB 4 (Form B) No Calculator

4. A particle moves along the x-axis with velocity at time 0t given by 1( ) 1 tv t e .

(a) Find the acceleration of the particle at time t = 3.

(b) Is the speed of the particle increasing at time t = 3? Give a reason for your answer.

(c) Find all values of t at which the particle changes direction. Justify your answer.

(d) Find the total distance traveled by the particle over the time interval 0 3t .

MIXED REVIEW B ASSIGNMENT #7

16

1991 BC 3

3. Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of

siny x and cosy x .

(a) Find the area of R. (without calculator)

(b) Find the volume of the solid generated when R is revolved about the x-axis. (with

calculator)

(c) Find the volume of the solid whose base is R and whose cross sections cut by planes

perpendicular to the x-axis are squares. (with calculator).

1996 AB 1

1. The figure above shows the graph of f , the derivative of a function f. The domain of f is the

set of all real numbers x such that 3 5x .

(a) For what values of x does f have a relative maximum? Why?

(b) For what values of x does f have a relative minimum? Why?

(c) On what intervals is the graph of f concave upward? Use f to justify your answer.

(d) Suppose that f (1) = 0. Draw a sketch that shows the general shape of the graph of the

function f on the open interval 0 2x .

2012 MULTIPLE CHOICE (even problems) ASSIGNMENT #8

17


18


19

30.

32.

34.


20

36.

38.

40.


21

44.

42.

MIXED REVEW C ASSIGNMENT #9

22

2003 AB 5 (Form B)

5. Let f be the function defined on the closed interval [0, 7]. The graph of f, consisting of four

line segments, is shown above. Let g be the function given by 2

( ) ( )x

g x f t dt .

(a) Find g(3), (3)g , and (3)g .

(b) Find the average rate of change of g on the interval 0 3x .

(c) For how many values c, where 0 3c , is ( )g c equal to the average rate found in

part (b)? Explain your reasoning.

(d) Find the x-coordinate of each point of inflection of the graph of g on the interval

0 7x . Justify your answer.


2. The shaded region R is bounded by the graph of 2y x , and the line 4y , as shown in the

figure above.


(b) Find the volume of the solid generated by revolving R about the x-axis.

(c) There exists a number k, k > 4, such that when R is revolved about the line y = k, the

resulting solid has the same volume as the solid in part (b). Write, but do not solve, an

equation involving an integral expression that can be used to find the value of k.

MIXED REVEW C ASSIGNMENT #9

23


2. A particle moves along the x-axis so that its velocity at time t, 0 5t , is given by

( ) 3( 1)( 3).v t t t At time t = 2, the position of the particle is x(2) = 0.

(a) Find the minimum acceleration of the particle.

(b) Find the total distance traveled by the particle.

(c) Find the average velocity of the particle over the interval 0 5t .

1984 AB 4—BC 3

4. A function f is continuous on the closed interval [-3, 3] such that f (-3) = 4 and f (3) = 1. The

functions f and f have the properties given in the table below.

(a) What are the x-coordinates of all the absolute maximum and absolute minimum points

of f on the interval [-3, 3]? Justify your answer.

(b) What are the x-coordinates of all points of inflection of f on the interval [-3, 3]?


(c) On the axis provided, sketch a graph that satisfies the given properties of f.


4. Let f be the function given by 3 2( ) 6 ,f x x x p where p is an arbitrary constant.

(a) Write an expression for ( )f x and use it to find the relative maximum and minimum

values of f in terms of p. Justify your answer.

(b) Find the value of p such that the average value of f over the closed interval [-1, 2] is 1.

x 3 1x x = -1 1 1x x = 1 1 3x

( )f x Positive Fails to exist Negative 0 Negative

( )f x Positive Fails to exist Positive 0 Negative

RELATED RATE STORY PROBLEMS ASSIGNMENT #10

24


6. A tightrope is stretched 30 feet above the ground between the Jay and the Tee buildings,

which are 50 feet apart. A tightrope walker, walking at a constant rate of 2 feet per second

from point A to point B, is illuminated by the spotlight 70 feet above point A, as shown in the

diagram.

(a) How fast is the shadow of the tightrope walker’s feet moving along the ground when

she is midway between the buildings? (Indicate units of measure.)

(b) How far from point A is the tightrope walker when the shadow of her feet reaches the

base of the Tee Building? (Indicate units of measure.)

2002 AB 6 (Form B) No calculator

6. Ship A is traveling due west toward Lighthouse Rock at a speed of 15 kilometers per hour

(km/hr). Ship B is traveling due north away from Lighthouse Rock at a speed of 10km/hr.

Let x be the distance between Ship A and Lighthouse Rock at time t, and let y be the distance

between Ship B and Lighthouse Rock at time t, as shown in the figure above.

(a) Find the distance, in kilometers, between Ship A and Ship B when x = 4 km and

y = 3 km.

(b) Find the rate of change, in km/hr, of the distance between the two ships when x = 4 km

and y = 3 km.

(c) Let be the angle shown in the figure. Find the rate of change in , in radians per hour, when x = 4 km and y = 3 km.


5. The volume V of a cone ( 21

3V r h ) is increasing at the rate of 28 cubic cm per second.

At the instant when the radius r of the cone is 3 cm, its volume is 12 cubic cm and the radius is

increasing at 1

2cm per second.

(a) At the instant when the radius of the cone is 3 cm, what is the rate of change of the

area of its base?

(b) At the instant when the radius of the cone is 3 cm, what is rate of change of its

height h?

(c) At the instant when the radius of the cone is 3 cm, what is the instantaneous rate of

change of the area of its base with respect to its height h?

RELATED RATE STORY PROBLEMS ASSIGNMENT #10

25


5. As shown in the figure below, water is draining from a conical tank with height 12 feet and

diameter 8 feet into a cylindrical tank that has a base with area 400 square feet. The depth h, in feet, of the water in the conical tank is changing at the rate of (h – 12) feet per minute.

(The volume V of a cone with radius r and height h is 21

.3

V r h )

(a) Write an expression for the volume of water in the conical tank as a function of h.

(b) At what rate is the volume of water in the conical tank changing when h = 3? Indicate

units of measure.

(c) Let y be the depth, in feet, of the water in the cylindrical tank. At what rate is y

changing when h = 3? Indicate units of measure.


5. An oil storage tank has the shape shown above, obtained by revolving the curve 49

625y x

from x = 0 to x = 5 about the y-axis, where x and y are measured in feet. Oil flows into the

tank at the constant rate of 8 cubic feet per minute.

(a) Find the volume of the tank. Indicate units of measure.

(b) To the nearest minute, how long would it take to fill the tank if the tank was empty

initially?

(c) Let h be the depth, in feet, of oil in the tank. How fast is the depth of the oil in the

tank increasing when h = 4? Indicate units of measure.

OTHER RATE PROBLEMS ASSIGNMENT #11

26

You may use a calculator on the entire assignment.

2015 AB/BC 1

2017 AB 2

OTHER RATE PROBLEMS ASSIGNMENT #11

27

2008 AB/BC 2

1999 AB 3

3. The rate at which water flows out a pipe, in gallons per hour, is given by a differentiable

function R of time t. The table below shows the rate as measured every 3 hours for a 24-hour

period.

(a) Use a midpoint Riemann sum with 4 subdivisions of equal

length to approximate 24

0( )R t dt . Using correct units,

explain the meaning of your answer in terms of water flow.

(b) Is there some time t, 0 24t , such that ( ) 0R t ? Justify

your answer.

(c) The rate of water flow, R(t) can be approximated by

21

( ) 768 2379

Q t t t . Use Q(t) to approximate the

average rate of water flow during the 24-hour time period.

Indicate the units of measure.

1997 AB 6

6. Let v(t) be the velocity, in feet per second, of a skydiver at time t seconds, 0t . After her

parachute opens, her velocity satisfies the differential equation 2 32dv

vdt

, with initial

condition v(0) = -50.

(a) Use separation of variables to find an expression for v in terms of t, where t is

measured in seconds.

(b) Terminal velocity is defined as lim ( )t

v t

. Find the terminal velocity of the skydiver to

the nearest foot per second.

(c) It is safe to land when her speed is 20 feet per second. At what time t does she reach

this speed?

t

(hours)

R(t)

(gallons per hour)

0 9.6

3 10.4

6 10.8

9 11.2

12 11.4

15 11.3

18 10.7

21 10.2

24 9.6

2000 FREE RESPONSE ASSIGNMENT #12

28

AB 1/BC 1 Calculator Allowed

Let R be the shaded region in the first quadrant enclosed by the graphs of 2xy e , 1 cosy x , and

the y-axis, as shown in the figure above.

(a) Find the area of the region R.

(b) Find the volume of the solid generated when the

region R is revolved about the x-axis.

(c) The region R is the base of a solid. For this solid, each

cross section perpendicular to the x-axis is a square.

Find the volume of this solid.

AB 2/BC 2 Calculator allowed

Two runners A and B, run on a straight racetrack for

0 10t seconds. The graph at the right which consists of two

line segments, shows the velocity, in meters per second, of

Runner A. the velocity, in meters per second, of Runner B is

given by the function v defined by 24

( )2 3

tv t

t

(a) Find the velocity of Runner A and the velocity of Runner B

at time t = 2 seconds. Indicate the units of measure.

(b) Find the acceleration of Runner A and the acceleration of

Runner B at time t = 2 seconds. Indicate the units of measure.

(c) Find the total distance run by Runner A and the total distance run by Runner B over the time

interval 0 10t seconds. Indicate the units of measure.

AB 3

The figure at the right shows the graph of f , the

derivative of the function f , for 7 7x . The

graph of f has horizontal tangent lines at x = -3,

x = 2, and x = 5, and a vertical tangent line at x = 3.

(a) Find all the values of x, 7 7x , at which f attains

a relative minimum. Justify your answer.

(b) Find all the values of x, 7 7x , at which f attains

a relative maximum. Justify your answer.

(c) Find all the values of x, 7 7x , at which ( ) 0.f x

(d) At what values of x, 7 7x , does f attain an absolute maximum? Justify your answer.


29

AB-4 Calculator allowed

Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out

of the tank at the rate of 1t gallons per minute, for 0 120t minutes. At time t = 0, the tank

contains 30 gallons of water.

(a) How many gallons of water leak out of the tank from time t = 0 and t = 3 minutes?

(b) How many gallons of water are in the tank at time t = 3 minutes?

(c) Write an expression for A(t), the total number of gallons of water in the tank at time t.

(d) At what time t, 0 120t , is the amount of water in the tank a maximum? Justify your

answer.

AB 5/BC 5

Consider the curve given by 2 3 6xy x y .

(a) Show that 2 2

3

3

2

dy x y y

dx xy x

.

(b) Find all the points on the curve whose x-coordinate is 1, and write an equation for the tangent

line at each of these points.

(c) Find the x-coordinate of each point on the curve where the tangent line is vertical.

AB 6

Consider the differential equation 2

2

3y

dy x

dx e .

(a) Find a solution ( )y f x to the differential equation satisfying 1

(0)2

f .

(b) Find the domain and range of the function f found in part (a).

2012 Free Response ASSIGNMENT #13

30

A graphing calculator is required for problems 1 and 2.


31

Do not use a calculator for problems 3-6.


32


33

Calculators may be used on the first three problems only.

1. Let R and S be the regions in the first quadrant shown in the figure

The region R is bounded by the x-axis and the graphs of 32y x

and tany x . The region S is bounded by the y-axis and the graphs

of 32y x and tany x .


(b) Find the area of S.

(c) Find the volume of the solid generated when S is revolved about the x-axis.

2. The temperature, in degrees Celsius (°C), of water in the pond is a differentiable function W of time t. The table shows the water temperature as recorded every 3 days over

a 15-day period.

(a) Use data from the table to find an approximation for (12)W . Show the

computations that lead to your answer. Indicate units of measure.

(b) Approximate the average temperature, in degrees Celsius, of the water over the time interval 0 ≤ 𝑡 ≤ 15

days by using the trapezoidal approximation with subintervals of length 3t days.

(c) A student proposes the function P, given by 3( ) 20 10

t

P t te

, as a model for the temperature of the water

in the pond at time t, where t is measured in days and P(t) is measured in degrees Celsius. Find 12P . Using appropriate units, explain the meaning of your answer in terms of water temperature.

(d) Use the function P defined in part (c) to find the average value, in degrees Celsius, of P(t) over time the

interval 0 15t days.

3. A car is traveling on a straight road with velocity

55 ft / sec at time 0t . For 0 18t seconds, the

car’s acceleration a t , in 2/ft sec , is the piecewise

linear function defined by the graph shown.

(a) Is the velocity of the car increasing at t = 2 seconds? Why or why not?

(b) At what time in the interval 0 18t , other than t = 0, is the velocity of the car 55 ft/sec? Why?

(c) On the time interval 0 18t , what is the car’s absolute maximum velocity, in ft/sec, and at what time does it occur? Justify your answer.

(d) At what time in the interval 0 18t , if any, is the car’s velocity equal to zero? Justify your answer.

t

(days)

W(t)

(°C)

0 20

3 31

6 28

9 24

12 22

15 21

(2,15)

(10,-15) (14,-15)

(18,15)

2( / )

a t

ft sec

t seconds


34

No calculator is allowed for these problems.

4. Let h be a function defined for all 𝑥 ≠ 0 such that ℎ(4) = −3 and the derivative of h is given by

2 2x

h xx

for all 𝑥 ≠ 0.

(a) Find all the values of x for which the graph of h has a horizontal tangent, and determine whether h has a local maximum, a local minimum, or neither at each of these values. Justify your answer.

(b) On what intervals, if any, is the graph of h concave up? Justify your answer.

(c) Write an equation for the line tangent to the graph of h at x = 4.

(d) Does the line tangent to the graph of h at x = 4 lie above or below the graph of h for x > 4? Why?

5. A cubic polynomial function f defined by 3 2( ) 4f x x ax bx k where a, b and k are constants. The

function f has a local minimum at x = -1, and the graph of f has a point of inflection x = -2.

(a) Find the values of a and b.

(b) If 1

0( ) 32f x dx , what is the value of k?

6. The function f is differentiable for all real numbers. The point 1

3,4

is on the graph of ( )y f x , and the slope

of each point ,x y on the graph is given by 2 6 2dy

y xdx

.

(a) Find

2

2

d y

dx and evaluate it at the point

13,

4

.

(b) Find ( )y f x by solving the differential equation 2 6 2dy

y xdx

with the initial condition 1

34

f .


35

Calculators may be used on the first three problems only.


36

No calculator is allowed for these problems.

SLOPE FIELDS ASSIGNMENT #16 NAME_________________

37

1.

The calculator drawn slope field for the

differential equation dy

xydx

is shown in the

figure below. The solution curve passing

through the point (0, 1) is also shown.

2. The calculator drawn slope field for the differential

equation dy

x ydx

is shown in the figure below.

(a) Sketch the solution curve through the point (0, 2). (a) Sketch the solution curve through the point (0, 2).

(b) Sketch the solution curve through the point (0, -1) (b) Sketch the solution curve through the point (0, -2)

Draw a slope field for each of the following differential equations. Show a segment at each indicated point.

3. 1

dyx

dx

4. 2

dyy

dx **Note the scales!!!

5. dy y

dx x

(a) Sketch a solution curve which

passes through the point (1, 0)


passes through the point (0, -1).


passes through the point (2, -1).

For problems 6-8, find the equations of the solution curves you sketched in problems 3-5. Each equation should be

expressed in the form of ( )y f x . Use your graphing calculator to graph each of your equations for problems 6-8

to see if those graphs match your solution curves drawn in problems 3-5.

6. 1

dyx

dx

7. 2

dyy

dx

8. dy y

dx x

x

y

x

y

x

y

•


38

9. At right is a slope field for the differential equation x

dye

dx

.

(a) Sketch the solution curve passing through the point (0, 0)

(b) Find a particular solution in the form of ( )y f x to the

differential equation xdy

edx

.

(c) Without the use of a calculator (instead use transformations to

the graph of xy e ), determine whether your equation from

part b represents the function which you graphed in part a.

10. The slope field for a differential equation is shown at the

right. Which statement is true for solutions of the differential

equation?

I. For 0x all solutions are decreasing

II. All solutions level off near the x-axis.

III. For 0y all solutions are increasing

(a) I only (b) II only (c) III only (d) II and III only (e) I, II and III

11.

The slope field for the differential equation 2 2

4 2

dy x y y

dx x y

will have horizontal segments when

(a) 2 ,y x only (b) 2 ,y x only (c) 2 ,y x only (d) 0y , only (e) 0y or 2y x


39

12.

Which one of the following could be the graph of the solution of

the differential equation whose slope field is shown at right.

(E) (A) (B)

(C) (D)

13.

Which statement is true about the solutions y(x), of a differential

equation whose slope field is shown at the right.

I. If y (0) > 0 then lim ( ) 0.x

y x

II. If 2 (0) 0y then lim ( ) 2.x

y x

III. If y (0) < -2 then lim ( ) 2.x

y x

(A) I ONLY (B) II ONLY (C) III ONLY (D) II and III ONLY (E) I, II, and III

14. Shown at the right is the slope field for which of the

following differential equations?

(A) 1dy

xdx

(B) 2dy

xdx

(C) dy

x ydx


40

15. Consider the differential equation given by 2

dy xy

dx .

(a) On the axes provided below, sketch a slope field for the given differential equation at the nine points

indicated.

x

y

(b) Find the particular solution ( )y f x to the given differential equation with the initial condition

(0) 3f .

16.

Consider the differential equation 2 4

dyy x

dx .

x

y

(a) The slope field for the given differential equation is

provided. Sketch the solution curve that passes

through the point (0, 1) and sketch the solution curve

that passes through the point (0, -1).

(b) Find the value of b for which 2y x b is a solution to the given differential equation. Justify your

answer.

(c) Let g be the function that satisfies the given differential equation with the initial condition g (0) =0.

Does the graph of g have a local extremum at the point (0, 0) ? If so, is the point a local maximum

or a local minimum? Justify your answer.


41

17. Consider the differential equation given by 2

1dy

x ydx

.

(a) On the axes provided, sketch a slope field for the given differential equation at the eleven points

indicated.

x

y

(b) Use the slope field for the given differential equation to explain why a solution could not have the

graph shown below.

x

y


42

Match each slope field with the equation that the slope field could represent.

(A)

(B)

(C)

(D)

(E) (F)

(G)

(H) 18. 1y

20. 2

1y

x

22. 2y x

24. cosy x

19. y x

21. 31

6y x

23. siny x

25. lny x

Match the slope fields with their differential equations.

(A)

(B) (C)

(D)

(E)

26. cosdy

xdx

28. dy x

dx y

30. dy

ydx

27. dy

x ydx

29. 1

12

dyx

dx

MULTIPLE CHOICE PRACTICE (2017 Course Description) ASSIGNMENT #17

43


44


45


46


47

21. Let f be defined as follows, where 0a .

2 2

, ,

0 .

x afor x a

f x x a

for x a

Which of the following are true about f ?

I. lim ( )x a

f x

exists.

II. f a exists.

III. f x is continuous at .x a

(A) None (B) I only (C) II only (D) I and II only (E) I, II and III

22. A particle moves along the x-axis so that at any time 0t its velocity is given by ln 1 2 1v t t t . The

total distance traveled by the particle from 0t to 2t is

(A) 0.67 (B) 1.10 (C) 1.30 (D) 1.54 (E) 2.67


48

Do not use a calculator.


49


50


51

You may use a calculator on the remaining problems.

29.

31.

33.


52

35.

37.

39.

41.


53

43.

45.

.


54

Calculators are allowed on problems 1-3.


55

No calculators are allowed on problems 4-6.

(-5,-2)

(-3,2)

(0,1)(2,1)

(4,-1)

x

y


56

A calculator is allowed on

the first three problems only.


57


58

A calculator is allowed on the first three problems only.


59

Do not use a calculator on the remaining problems.


60

FREE RESPONSE TYPE PRACTICE PROBLEMS ASSIGNMENT #22

61

−

1. It’s a Bug’s Life (No Calculator)

A bug moves along a vertical piece of string in

such a way that its velocity with respect to

time can be represented by the graph shown at

right for 0 20t minutes.

(14,6)

Time in Minutes

a. During what interval(s) of time is the bug moving down (negative direction) the string? Justify

your answer.

b. During what interval(s) of time is the bug resting (that is, not moving)? Justify your answer.

c. During what interval(s) is the bug’s acceleration positive? Justify your answer.

d. Find 20

0( )v t dt .

e. Assuming that the bug started (t = 0) its journey at a height 40 inches from the top of the string,

where did the bug end its journey? (t = 20)

f. Find 20

0( )v t dt .

g. Using the correct units, explain the meaning of the integral from part f.

h. Find the average velocity of the bug over the interval 0 20t .

i. Find the average acceleration of the bug over the interval 0 20t .


62

2. Calculator allowed. A particle moves along the x-axis so that its velocity at any time 1t is

given by 2

3

4( )

tv t

t

The position ( )x t is 0 for t = 1.

a. Write an expression for the acceleration of the particle at any time 1t .

b. Write an expression for the position of the particle at any time 1t .

c. For what value(s) of t does the particle change direction? Justify your answer.

d. During what interval(s) of time is the acceleration of the particle positive? Justify your answer.

e. Find the average velocity of the particle on the interval 21,e .

f. Find the total distance traveled by the particle from time t = 1 to time 2t e .

g. During what interval(s) of time is the particle moving to the left? (Hint: Refer back to part c.)


h. Find lim ( )t

v t

and lim ( )t

x t

3.

Calculator allowed

Let R be the region in the first quadrant bounded by the graphs of ( ) 2cosf x x , 1

( )2

g x x ,

and the y – axis.

a. Sketch the graphs of ( )f x , ( )g x , and shade region R. (1st quadrant only)

b. Set-up an integral for the area of region R.

c. Set-up an integral for the volume of a solid whose base is R and whose cross sections cut by

planes perpendicular to the x – axis are squares.

d. Set up an integral for the volume of the solid formed when R is revolved about the x – axis.

e. Set-up an integral for the volume of the solid formed when R is revolved about the line y = -2.

f. Set-up an integral for the volume of the solid formed when R is revolved about the line y = k.

(where k is a constant such that 2k .

g. Find the area of region R.

h. Suppose c is a constant such that the vertical line x = c divides the region R into two regions of

equal area. Find the value of c.


63

4. No Calculator allowed.

The function ( )f x is an even function which is

continuous on the interval [-4, 4]. ( )f x and

( )f x both exist on the open intervals (-4, 0)

and (0, 4). The graph of ( )f x for the interval

(0, 4) is shown at the right. (This is only a

portion of the graph of ( )f x ).

x

y

a. Find all the values of x on the open interval (-4, 4) at which ( )f x has a relative maximum.


b. Find all the values of x on the open interval (-4, 4) at which ( )f x has a relative minimum. Justify

your answer.

c. Find the x – coordinate for each point of inflection for the graph of ( )f x . Justify your answer.

d. Now, suppose ( 4) (0) (4) 1f f f . Graph ( )f x .

e. At what x-value(s) does the absolute maximum value of ( )f x occur? Why?

Suppose the table below indicates values of ( )f x for selected values of x.

x 0 .5 1 1.5 2 2.5 3 3.5 4

( )f x 1 1.6 2 1.6 1 0.3 0 0.3 1

f.

Use the trapezoidal rule with 4 subintervals on [0, 4] to approximate 4

0( )f x dx .

g. Approximate 4

0( )f x dx with a Riemann Sum, using midpoints of 4 subintervals.

h. Approximate 4

4( )f x dx

using your answers above.

FUNCTION REVIEW WORKSHEET ASSIGNMENT #23

64

Let f x be the function represented by the graph shown.

1. 6

limx

f x

2. 4

limx

f x

3. 4

limx

f x

4. 8

limx

f x

5. limx

f x

6. 2

limx

f x

7. Is f continuous at 2x ? 8. Is f differentiable at 2x ?

9. Is f continuous at 6x ? 10. Does 6f exist?

11. Does 6f exist? 12. 4f 13. 6f

Let 0

t

g t f x dx on the interval 0 4t . (Using the function graphed above.)

14. 4g 15. g t 16. 2g

17. g t 18. 1g

19. Find the relative extrema of g in 0,4 . Specify max. or min. and give a reason.

20. Find the points of inflection of g in 0,4 . Give a reason.

21. 2g t 22. 2d

dtg t

Let j x be a function defined on the interval 0 4x such that j x f x which is graphed above.

23. If 2 5, find 3 .j j 24. If 2 5, find 0 .j j

Let 3

31

1xh x dt

t .

25. h x 26. 8h

27. 2

3

3x

x

dt dt

dx

x

y

APTM Calculus Workshop Gary Taylor

Tricky Limit Problems

The limit problems on this worksheet are inspired by these AP test problems:

AP Calculus BC Practice Exam, 2016: #86

AP Calculus BC International Exam, 2016, #88

AP Calculus AB International Exam, 2017, #15

( )y f x= ( )y g x= ( )y h x= ( )y j x=

Find the following limits or state that the limit does not exist.

1. ( )2

limx

j x→−

= 2. ( )1

limx

j x→

=

3. ( )( )3

limx

f g x→

= 4. ( )0

lim 2x

f x→

+ =

5. ( ) ( )( )0

lim 2x

g x f x→

• + = 6. ( )( )( )22

lim 1 6x

f x→

− − =

7. ( ) ( )( )2

limx

h x f x→

+ = 8. ( )( )2

limx

j j x→−

=

9. ( )

1.5

1lim

2 3x

g x

x→

−=

− 10. ( )( )

0lim 1x

g f x→

+

−

x

y

−

x

y

−

x

y

− −

−

x

y

APTM Calculus Workshop – Gary Taylor—taylormathconsulting.com

Existence Theorems

Intermediate Value Theorem (IVT)

If f is continuous on ,a b and k is any y-value between ( ) ( ) and f a f b ,

then there is at least one x-value c between a and b such that ( ) .f c k=

In other words, f takes on every y-value between ( ) ( ) and f a f b .

Extreme Value Theorem (EVT)

If f is continuous on ,a b then f has both an absolute (global) minimum and an absolute (global) maximum

on the interval.

In practice, the standard method of finding these max/min values is by the candidate test. The candidates are

the critical values (where f is zero or undefined) as well as the endpoints of the interval. The candidate test

involves finding function values for each candidate to identify the maximum and/or minimum.

Mean Value Theorem (MVT)

If f is continuous on ,a b and differentiable on ( ),a b , then there is a

number c in ( ),a b such that ( )( ) ( )f b f a

f cb a

− =

−.

Informally: The Mean Value Theorem states that given the right conditions of continuity and differentiability,

there will be at least one tangent line parallel to the secant line.

In still other words: The instantaneous rate of change (slope of tangent) will equal the average rate of change

(slope of secant) at least once.

Definition of Continuity

A function f is continuous at an x-value c if and only if ( ) ( ) ( )lim limx c x c

f x f x f c− +→ →

= = .

Note: A function can be continuous but not differentiable. (Example: ( )f x x= at x = 0)

If a function is differentiable, it must be continuous.

a bc

( )f a

k

( )f b

tangent slope

(inst. rt. of ch.) secant slope

(avg. rt. of ch.) a bc

( )( ),b f b

( )( ),a f a

APTM Calculus Workshop – Gary Taylor—taylormathconsulting.com

Exercises:

For Problems 1-10: Given f is differentiable on

the closed interval 1,10 with values shown.

1. Find the average rate of change of f on the interval 1,10 .

Fill in the blanks for problems 2-5:

2. There exists a c in ( )1,10 such that ( )1

3f c = because f is ____________ on the interval ______(since it

is given that f is _____________) by the ________Value Theorem.

3. There exists a c in ( )1,10 such that ( ) 2f c = because f is _____________ on the interval ______(since it

is given that f is _____________) and ______ < ____ < ______ by the __________Value Theorem.

4. f attains both an global maximum and a global minimum on the interval 1,10 because f is

_____________ on the interval ______(since it is given that f is _____________) by the

__________Value Theorem.

5. Since ( ) ( )3 10f f= there exists a c in ( )3,10 such that ( )f c = ___ because f is _______________ on

the interval ______(since it is given that f is _____________) by the __________Value Theorem.

6. If ( ) 0f x on ( )3,10 what must be true at the c-value from problem 5? Give a reason for your answer.

7. Find the smallest interval on which it is certain that ( ) 2f c = for some c by the IVT.

8. Find the smallest interval on which it is certain that ( ) 1f c = − for some c by the MVT.

9. True or False? ( )0 7f x on 1,10 by the EVT.

10. Could there be a c on ( )6,9 where ( ) 8f c = ? Explain why the IVT does not guarantee the existence of this

c-value

11. Sketch an example of a function which is defined at every x-value on a closed interval but which does not

have an absolute maximum on the interval. Why is this not a violation of the EVT.

12. Find the average rate of change of ( )g x x= on the interval 1,3− . Explain why the MVT does not

guarantee a c-value where the instantaneous rate of change of g equals this average rate of change.

13. If ( )2 3h = − and ( )5 3h = and h is defined at every x-value in the interval ( )2,5 sketch a graph on which

h has no x-intercept in the interval ( )2,5 . Why is this not a violation of the IVT?

14. The function f (x) shown is continuous at x = 0 because

( ) ( )lim limx x

f x f→ →

= = .

15. Explain why the function is not continuous at x =1.

16. If ( )g x is differentiable at x = 4 and ( )4 7g = , then ( )4

limx

g x→

= ____ because ( )g x is ______________ at

x = 4.

17. If ( )h x is continuous on 1,5− and ( ) 0h x = at x = 2 only and ( )h x is defined except at x = 4, identify

the candidates for the x-values at which h has absolute extrema.

18. If ( )

2

1 3

2 2

4 3, 3

, 3

x x xh x

x x

− + =

−

for the function in problem 17, find the global maximum and minimum values

of h on 1,5− .

( )

1 3 6 9 10

0 3 7 5 3

x

f x

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REVISED 2019 - Ms. Virgil's Classroom · 2020. 3. 16. · fxcc( ) 0 or fxcc() undefined Make an fcc number line. f concave upward: fxcc( ) 0! f concave downward: fxcc( ) 0 Average