derivatives - tfd215moodle.tfd215.org/pluginfile.php/64948/mod_resource... · ap multiple choice...

Derivatives Big Ideas Rule of Four: Numerically, Graphically, Analytically, and Verbally

Average rate of Change: yx

∆∆

Difference Quotient: ( ) ( ) ( ) ( ) ( ) ( )2

f a h f a f a f a h f a h f a hh h h

+ − − − + − −

Average rates of change May be used to estimate the derivative at a point Caution: Symmetric Difference Quotient as used on graphing technology may lead to incorrect answers. Instantaneous Rate of Change: Change in the dependent variable with respect to the independent variable Limit of the average rate of change

0

limx

yx∆ →

∆∆

Derivative of a function at f x a= is the limit of the Difference Quotient (this is a number):

0

( ) ( ) ( ) ( )( ) lim ( ) limh x a

f a h f a f x f af a f ah x a→ →

+ − −′ ′= =−

Slope of the curve at a point

Derived function (Derivative) is the function defined by0

( ) ( )( ) limh

f x h f xf xh→

+ −′ =

Differentiability Implies Continuity Displacement, Velocity, Acceleration, Speed (Speed is the absolute value of velocity) When is speed increasing? Tangent line: May intersect the function elsewhere May cross the function at the point of tangency Write the equation of the tangent (point slope form) ( ) ( )( )y f a f a x a′− = − Write the equation of the normal Tangent line approximation: ( ) ( )( )y f a f a x a′= + − Units: Identify the units and explain using the correct units Implicit differentiation

Mike Koehler 2 - 1 Derivatives

Table of Derivatives Power Rule

( ) 1n nd duu nudx dx

−=

Trigonometric Functions

( ) ( )

( ) ( )

( ) ( )

2 2

sin cos cos sin

tan sec cot csc

sec sec tan csc csc cot

d du d duu u u udx dx dx dxd du d duu u u udx dx dx dxd du d duu u u u u udx dx dx dx

= = −

= = −

= = −

Inverse Trigonometric Functions

( ) ( )

( ) ( )

( ) ( )

1 1

2 2

1 12 2

1 1

2 2

1 1sin cos1 11 1tan cot

1 11 1sec csc

1 1

d du d duu udx dx dx dxu ud du d duu udx dx dx dxu ud du d duu udx dx dx dxu u u u

− −

− −

− −

−= =

− −−

= =+ +

−= =

− −

Exponential Functions

( ) ( ) lnu u u ud du d due e a a adx dx dx dx

= =

Logarithmic Functions

( ) ( )1 1 1ln loglna

d du d duu udx u dx dx u a dx

= =

Product Rule: ( )d dv duuv u vdx dx dx

= +

Quotient Rule: 2

du dvv ud u dx dxdx v v

− =

2

lo dhi - hi dlolo

⋅ ⋅

Chain Rule: [ ]( ( )) ( ( )) ( )f g x f g x g x′ ′ ′= ⋅

dy dy du dy dy dtdx du dx dx dx dt

= ⋅ =


AP Multiple Choice Questions 2008 AB Multiple Choice Problems 6 13 2003 AB Multiple Choice 1.

If ( )23 1 , then dyy xdx

= + =

A) ( )223x B) ( )32 1x + C) ( )22 3 1x + D) ( )2 33 1x x + E) ( )2 36 1x x +

4.

2 3If , then 3 2

x dyyx dx+

= =+

A) 2

12 13(3 2)

xx++

B) 2

12 13(3 2)

xx−+

C) 2

5(3 2)x +

D) 2

5(3 2)x

−+

E) 23

9. If ( )3( ) ln 4 , then (0) isxf x x e f− ′= + +

A) 25

− B) 15

C) 14

D) 25

E) nonexistent

14.

2If sin(2 ), then dyy x xdx

= =

A) 2 cos(2 )x x B) 4 cos(2 )x x C) ( )2 sin(2 ) cos(2 )x x x+

D) ( )2 sin(2 ) cos(2 )x x x x− E) ( )2 sin(2 ) cos(2 )x x x x+ 24. Let f be the function defined by 3( ) 4 5 3f x x x= − + . Which of the following is an equation of the line

tangent to the graph of f at the point where 1x = − ? A) 7 3y x= − B) 7 7y x= + C) 7 11y x= + D) 5 1y x= − − E) 5 5y x= − −

26. What is the slope of the line tangent to the curve 2 23 2 6 2y x xy− = − at the point (3,2) ?

A) 0 B) 49

C) 79

D) 67

E) 53


27. Let f be the function defined by 3( )f x x x= + . If 1( ) ( ) and (2) 1g x f x g−= = , what is the value of (2)g ′ ?

A) 1

13 B) 1

4 C) 7

4 D) 4 E) 13

1998 AB Multiple Choice 28.

If ( )( ) tan 2 , then 6

f x x f π ′= =

A) 3 B) 2 3 C) 4 D) 4 3 E) 8 1997 AB Multiple Choice 7. ( )2 3cosd x

dx=

A) ( ) ( )2 3 36 sin cosx x x B) ( )2 36 cosx x C) ( )2 3sin x

D) ( ) ( )2 3 36 sin cosx x x− E) ( ) ( )3 32sin cosx x− 17.

If 2 2 25x y+ = , what is the value of2

2 at the point (4,3)d ydx

?

A) 2527

− B) 727

− C) 727

D) 34

E) 2527

1997 AB Multiple Choice 10.

If 2 1y xy x= + + , then when 1, isdyxdx

= −

A) 12

B) 12

− C) -1 D) -2 E) nonexistent


AP Free Response Questions 2006 AB6 a b The twice-differentiable function f is defined for all real numbers and satisfies the following conditions:

(0) 2, (0) 4, (0) 3f f f′ ′′= = − = . a) The function g is given by ( ) ( )axg x e f x= + for all real numbers, where a is a constant. Find

(0) and (0)g g′ ′′ in terms of a . b) The function h is given by ( ) cos( ) ( )h x kx f x= for all real numbers, where k is a constant. Find ( )h x′ and

write an equation for the line tangent to the graph of at 0h x = . 1977 AB4 Let and f g and their inverses 1 1 and f g− − be differentiable function and let the values of ,f g , and the derivatives

and f g′ ′ at 1 and 2x x= = be given in the table below. x ( )f x ( )g x ( )f x′ ( )g x′ 1 3 2 5 4 2 2 π 6 7

Determine the value of each of the following: a) The derivative of at 2f g x+ = b) The derivative of at 2f g x⋅ = c)

The derivative of at 2f xg

=

d) (1) where ( ) ( ( ))h h x f g x′ = e) The derivative of 1 at 2g x− =


Assessing Understanding of Speed 1. If velocity is negative and acceleration is positive, then speed is ____________. 2. If velocity is positive and speed is decreasing, then acceleration is __________. 3. If velocity is positive and decreasing, then speed is ______________________. 4. If speed is increasing and acceleration is negative, then velocity is __________. 5. If velocity is negative and increasing, then speed is ______________________. 6. If the particle is moving to the left and speed is decreasing, then acceleration is ____________________. From Curriculum Module: Calculus: Motion available at AP Central


Textbook Calculus, Finney, Demanna, Waits, Kennedy; Prentice Hal, 2012

Section Questions 3.1 26 27 31 32 33 42 3.2 39 3.3 38 QQ p 126 1 2 3 4 3.4 5 9 10 11 12 18 26 32 3.5 30 40 41 QQ p148 1 2 3 4 4.1 58 64 65 4.2 49 50 56 57 QQ p 169 1 2 3 4 4.3 27 28 29 4.4 54 QQ p 185 1 2 3 4

Handouts The following pages contain handouts with problems accumulated from various sources.


AP Calculus

Chapter 3 Section 1 Sketch the graph of the derivative of the function whose graph is shown.

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-2 -1 1 2

-2

-1

1

2

3

4

-2 -1 1 2

-4

-3

-2

-1

1

2

3

4


-2 -1 1 2 3 4 5 6 7 8

-4

-3

-2

-1

1

2

3

4

-2 -1 1 2 3 4 5 6 7 8

-2

-1

1

2

3

4

5

6

-4 -3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

7

8

-4 -3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

7

8

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4


AP Calculus

Chapter 3 Section 1 ANSWERS

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-2 -1 1 2

-2

-1

1

2

3

4


-2 -1 1 2 3 4 5 6 7 8

-4

-3

-2

-1

1

2

3

4

-2 -1 1 2 3 4 5 6 7 8

-2

-1

1

2

3

4

5

6

-4 -3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

7

8

-4 -3 -2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

7

8

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

-5 -4 -3 -2 -1 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4


AP Calculus Chapter 3 Section 1 1. What is an equation of the tangent line at 3x = , assuming that (3) 5 and (3) 2f f ′= = ? 2. Suppose that 5 2y x= + is the equation of the tangent line to the graph of ( ) at 3.y f x a= =

What is (3)f ? What is (3)f ′ ? 3. Suppose that ( )f x is a function such that 2(2 ) (2) 3 5f h f h h+ − = + .

a) What is '(2)f ? b) What is the slope of the secant line through ( ) ( )2, (2) and 6, (6)f f ?

In exercises 4 to 9, each of the limits represents a derivative '( )f a . Find ( ) and .f x a

( )

( )

14

3 3

0 5

1

0

2

0 0

22

0 1

5 125 1254. lim 5. lim5

1sin46 26. lim 7. lim

14

5 25 5 18. lim 9. lim

1 13 910. lim 11. lim

1

h x

h x

h x

h x

h x

h xh x

hx

h x

h x

h xh x

π

→ →

−

→ →

+

→ →

→ →

+ − −−

+ − −

−

− −

−− + −−

12. The cost of extracting T tons of ore from a copper mine is ( )C f T= dollars. What does it mean to say

'(2000) 100f = ?


13. The graph of the function ( )f x is show at the right.

Graph the function '( )f x on the same set

of axes.

14 – 19. The graph ( )y f x= is shown on the right. 14 – 18. For each of the following pairs of numbers, use the graph to decide which is larger. Explain your answer. 14. (3) or (4)?f f 15. (3) (2) or (2) (1)f f f f− − ?

16. (2) (1) (3) (1) or 2 1 3 1

f f f f− −− −

?

17. '(1) or '(4)f f ? 18. Average rate of change between

1 and 3x x= = or between 3 and 5x x= = .

19. Arrange the following quantities in ascending order: 0 '(2) '(3) (3) (2)f f f f− 20. Sketch the derivative of the graph of the

function shown on the right. Sketch on the same axes.


Answers 1 5 2( 3)y x− = − 12 '(2000) 100f = says that when 2000 tons of

have already been extracted from the mine, the cost per ton is increasing at a rate of $100 per ton.

2 (3) 17 '(3) 5f f= = 3 2

0 0

2

(2 ) (2) 3 5'(2) lim lim 5

(6) (2) (2 4) (2)Slope of secant = 6 2 4

3 4 5 44, 174

h h

f h f h hfh hf f f f

h

→ →

+ − += = =

− + −=

−⋅ + ⋅

= ∴ =

13

4 3( ) 5f x x a= = 14 (4)f 5 3( ) 5f x x a= = 15 (2) (1)f f− 6 ( ) sin( ) 6f x x a π= = 16 (2) (1)

2 1f f−

−

7 1 14( )f x x a−= = 17 '(1)f

8 ( ) 5 2xf x a= = 18 Between 1 and 3. 9 ( ) 5 0xf x a= = 19 0 '(3) (3) (2) '(2)f f f f− 10 2( ) 3f x x a= = − 11

2

1( ) 1f x ax

= = 20


AP Calculus Chapter 3 Sections 2 – 3 Work problems on a separate sheet of paper. Show all work. Sketch a picture where appropriate. 1. Given that the tangent line to ( )y f x= at the point ( )1,3− passes through the point ( )0,4 , find ( 1)f ′ − . 2. Given that f (3) = -1 and f '(3) = 5, find an equation for the line tangent to the graph of ( )y f x= at the point

where x = 3.

3. Let 23 1

( )1

x xf x

ax b x ≤

= + >

Find the values of a and b so that f will be differentiable at x = 1. 4. The width of a rectangle is increasing at a rate of 2 cm/sec, and its length is increasing at a rate of 3 cm/sec. At

what rate is the area of the rectangle increasing when its width is 4 cm and its length is 5 cm? (Hint: A l w= ⋅ ) 5. Suppose (2) 1, '(2) 4, (2) 5, and '(2) 3f f g g= − = = = − . Find the derivative at 2 of each of the following

functions.

a) ( ) 2 ( ) ( )p x f x g x= + b) ( ) ( ) ( )q x f x g x=

c) ( )( )( )

f xr xg x

= d) ( )( )( ) ( )

f xs xf x g x

=+

6. Let ( )( ) ( ) ( ), and ( )( )

f xh x f x g x j xg x

= = . Fill in the missing entries in the table below using the information

about f and g given in the following table.

x ( )f x ( )f x′ ( )g x ( )g x′ ( )h x′ ( )j x′ -2 1 -1 -3 4 -1 0 1 1 -2 0 -1 2 -2 -3/4


7. Let f be a function defined by 2( )f x Ax Bx C= + + with the following properties:

( )i (0) 2; (ii) '(2) 10; (iii) ''(10) 4f f f= = = . Find the values of A, B, and C. 8. Find c so that the line 4 3y x= + is tangent to the curve 2y x c= + . 9. Find the x-coordinate of the point on the graph of 2y x= where the tangent line is parallel to the secant line that

cuts the curve at x = -1 and x = 2. 10. Find the coordinate of all points on the graph of 21y x= − at which the tangent line passes through the point

(2,0). 11. When an oil tank is drained for cleaning, there are 2( ) 100,000 4000 40V t t t= − + gallons of oil left in the tank t

minutes after the drain valve is opened. (a) At what average rate does oil drain from the tank during the first 20 minutes? (b) At what rate does oil drain out of the tank 20 minutes after the drain valve is opened? 12. Find a function 2y ax bx c= + + whose graph has an x-intercept of 1, a y-intercept of –2, and a tangent line

with a slope of –1 at the y-intercept. 13. Find k if the curve 2y x k= + is tangent to the line 6y x= . 14. In 1990, the population of the Unites States was about 250 million and was increasing at the rate of about 3

million people per year. Per-capita income was about $15,000 and was growing at about $1000 per year. Both the U.S. population and U.S. per-capita income vary with time; their functions denoted by ( ) and ( )p t i t respectively. The product function ( ) ( ) ( )w t p t i t= describes the total annual U.S. income, another function of t. How fast was total annual income growing at time t = 1990?

15. Suppose that ( )f x is a function such that 2(2 ) (2) 4 3f h f h h+ − = − . What is '(2)f ?


ANSWERS

1 1

2 1 5( 3)y x+ = −

3 6 3a b= = −

4 2cm22 sec

5 17 17) 5 ) 23 ) )25 16

a b c d

6

x ( )f x ( )f x′ ( )g x ( )g x′ ( )h x′ ( )j x′ -2 1 -1 -3 4 7 1

9−

-1 0 -2 1 1 -2 -2 0 -1 2 -2 1 -5 -3/4

7 A = 2 B = 2 C = 2

8 c = 7

9 12

x =

10 2 3x = ± (3.732, -12.928) (.267(8), .928)

11 (a) gal gal3200 or draining at a rate of 3200min min− (b) gal(20) 2400 minV ′ = −

12 23 2y x x= − −

13 9k =

14 (1990) (1990) (1990) (1990) (1990)

3 million 15,000 250 million 1000=45 billion + 250 billion = 295 billion

w p i i p′ ′ ′= += ⋅ + ⋅

15 '(2) 3f = −


AP Calculus Chapter 3 Section 4.1 1. Suppose (1) 2f = and the average rate of change of f between 1 and 5 is 6. Find (5)f . 2. Suppose that (1) 2f = − and that ( ) 3f x′ ≤ for all x in [ 10,10]− .

a) Could (4) 8f = ? b) Could ( 7) 25f − = − ? c) Could (8) 25f = − ? 3. Amy takes a trip from Chicago to Milwaukee. Due to road construction, she drives the first 10 miles at a

constant speed of 20 mph. For the next 30 miles she maintains a constant speed of 60 mph and then stops at restaurant for 10 minutes for lunch. She drives the next 45 miles at a constant speed of 45 mph.

a) Draw a graph that shows her distance along the road from Chicago as a function of time. b) Draw a graph that shows her velocity as a function of time. c) What is her average speed for the trip (including her stop at the restaurant)?

4. A car travels for 20 minutes with an average velocity of 30 mph, and then for 30 minutes with an average velocity of 50 mph. What is the average velocity of the car?

5. A car travels for 30 miles with an average velocity of 40 mph and then for 30 minutes at 60 mph. What is the

average velocity of the car for the 60-mile trip? 6. A car is to travel 2 miles. It goes the first mile at an average velocity of 30 mph. The driver wishes to average

60 mph for the entire two-mile trip. Is this possible? Explain. 7. Water is flowing into a spherical tank at a constant rate. Let ( )V t be the volume of the water in the tank and

( )H t be the height of the water level at time t.

a) Give a physical interpretation of dVdt

and dHdt

.

b) Is dVdt

positive, negative or zero when the tank is one quarter full? Justify your answer.

c) Is dHdt

positive, negative or zero when the tank is one quarter full? Justify your answer.

d) Which of dVdt

and dHdt

is constant? Explain your answer.

8. The position ( )p t (in meters) of an object at time t (in seconds) along a line is given by 2( ) 3 1p t t= + .

a) Find the change in position of the object between t = 1 and t = 3. b) Find the average velocity of the object between t = 1 and t = 3. c) Find the instantaneous velocity of the object at t = 1. d) Find the equation of the line tangent to the graph of ( )p t at t = 1.


9. A cannon ball is shot upward with initial height 5 feet and initial velocity 288 feet per second. This translates

to about 200 mph, which is possible for an old-fashioned cannon. The height of the cannon ball is given by the equation 2( ) 16 288 5h t t t= − + + . Assuming free-fall conditions, answer the following questions. a) What is the average velocity from t = 5 to t = 8 seconds? b) Find an expression for the velocity of the cannonball. c) What is the instantaneous velocity at t = 8 seconds? d) When does the cannonball reach its maximum height? How high does it rise? e) How long does the cannonball remain airborne? How fast is it going when it hits the ground?

Answers

1 26 2 No, 7 is the maximum Yes, -26 is the minimum Yes 3 39.231 mph 4 42 mph 5 48 mph 6 No. Last mile took two minutes. Must go entire trip in 2 min.

7

gal ftmin min

Positive Positive dVdt

8

24 m 12 m/sec 6 m/sec

4 6( 1)y x− = −

9

80 ft./sec ( ) 32 288v t t= − +

32 ft./sec 9 sec 1301 ft. 18.0173 sec and -288.5536 ft./sec


AP Calculus Chapter 3 Section 4.2 Work problems on a separate sheet of paper. Show all work. Sketch a picture where appropriate. 1. The height of a ball in feet t seconds after it is thrown is given by 2( ) 16 38 74h t t t= − + + .

a) From what height is the ball thrown? b) Find a function for the velocity of the ball. What was the ball's initial velocity? Was it thrown up or down?

How can you tell? c) Was the ball's height increasing or decreasing at time t = 2? How fast was it moving? d) At what time did the ball reach its maximum height? How high was the ball at that time? e) How long was the ball in the air? How fast was it going when it hit the ground?

2. A particle moves along the x-axis so that its position in feet at any time 0t ≥ is given by 4 2( ) 2 4x t t t= − + .

a) Find an expression for the velocity of the particle at any time 0t ≥ . b) Find the average velocity of the particle for the first two second. c) Find the instantaneous velocity of the particle at t = 2 seconds. d) Find the values of t for which the particle is at rest. e) Find the position of the particle when it is at rest. f) Find an expression for the acceleration of the particle. When is the acceleration 0? g) Find the displacement of the particle from t = 0 to t = 3 seconds. h) Find the total distance traveled by the particle from 0 to 3t t= = seconds.

3. A particle moves along the x-axis so that its position in meters at any time 0t ≥ is given by

3 2( ) 4 18 15 1x t t t t= − + − . a) Where is the particle at time t = 0? Where is the particle at time t = 3? b) Find an expression for the velocity at any time 0t ≥ . c) Find all values of t for which the particle is at rest. d) Find the total distance traveled by the particle from 0 to 3t t= = seconds.

4. A particle moves along the x-axis so that its position at any time 0t ≥ is given by 3 2( ) 3x t t t t= − − + . 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]?

5. The position of a ball rolling down an inclined plane 8 meters long is given by the formula 2( ) .2 .6s t t t= + ,

where s is the number of meters traveled after t seconds.

a) How far has the ball traveled after 2 seconds? b) How fast is the ball traveling after 2 seconds? Indicate the units of measure. c) What is the average velocity of the ball on the interval from 1 to 4t t= = seconds? d) Write an equation for v , the velocity of the ball at any time t , and use it to compute the velocity of the ball

at the instant that it reaches the end of the inclined plane.


6. A particle moves along the y-axis so that its position, measured in feet, at any time 0t ≥ second is given by

( ) sin( ) cos( ) 2y t t t t= + + .

a) Find the position of the particle at time 0t = . b) Write an expression for the velocity of the particle in term of t . c) For what values of , 0 5t t≤ ≤ , is the particle moving upward? d) Write an expression for the acceleration of the particle in term of t . e) For 0t > , find the position of the particle the first time the velocity is zero. f) Find the total distance the particle travels over the interval 0 5t≤ ≤ .

Answers 1a 74 feet 4 1.786 seconds 1b ( ) 32 38 38 ft sec up velocity postitivev t t= − + 1c Decreasing 26ft sec 5a 2 meters 1d 38 32=19 16 seconds 96.5625 feet 5b 1.4meters sec 1e 3.6416 seconds -78.613 ft sec 5c 1.6meters sec 5d ( ) .4 .6 (5) 2.6m secv t t v= + =

2a 3( ) 4 4v t t t= − 2b 4 ft sec 6a (0) 3 fty = 2c 24 ft sec 6b ( ) cos( )v t t t=

2d At rest at 0 sec and 1 sec 6c 30 or 5 seconds2 2

t tπ π< < < ≤

2e Position is 3 ft and 4 ft 6d sin( ) cos( )a v t t t′= = − +

2f 2( ) 12 4 1/ 3 .577 seca t t t= − = = 6e 2 3.571 feet2 2

y π π = + =

2g 63 feet 6f 1.56404 5.83279 .73124 8.12807 feet+ + = 2h 65 feet 3a 1 meter -10 meter− 3b 2( ) 12 36 15v t t t= − + 3c .5 sec 2.5 sec 3d 23 meters


AP Calculus Chapter 4 Section 1 1. Let ( ) ( ( ))h x f g x= . Use the information about f and g given in the table below to fill in the missing

information about h and h'. x ( )f x ( )f x′ ( )g x ( )g x′ ( )h x ( )h x′ 1 1 2 4 3 2 2 1 3 4 3 4 3 1 2 4 3 4 2 1

2. Assume that g is a function such that ( )g x′ exists for all x . Find the derivative of the function f . Answers will involve and g g ′ .

a. ( )( ) ( ) nf x g x= b. ( )( ) nf x g x= c. ( )( ) sin ( )f x g x= d. ( )( ) sin( )f x g x= e. ( )( ) tan ( )f x g x= e. ( )( ) tan( )f x g x=

3. Use the figures below to evaluate the derivatives.

a.

( )3

( )x

d f g xdx =

b.

( )7

( )x

d f g xdx =

c. ( )

3

( )x

d g f xdx =

d.

( )7

( )x

d g f xdx =

4. Let ( ) ( ( )) and ( ) ( ) ( )h x f g x j x f x g x= = ⋅ . Fill in the missing entries in the table below.

x ( )f x ( )f x′ ( )g x ( )g x′ ( )h x ( )h x′ ( )j x ( )j x′ -1 3 2 1 0 -1/2 3 0 0 1/2 -1 1 0 -1/2 1 -5 0 2 0


5. Find the derivatives of ( ) ( )2 2sin and siny x y x= = . 6. Assume that (0) 2 and (0) 3f f ′= = . Find the derivatives of ( ) ( )3( ) and 7 at 0f x f x x = . 7. Find the derivative of tan(5 )y x=

8. Find the derivative of ( ) 43 cos( )y x x−

= + . 9. Find the derivative of ( )3 3tan ( ) tany x x= + 10. Some values of the derivative a function f are shown in the table below. No explicit formula for f is given.

x 0 1 2 3 4 5 6 ( )f x′ 4 -3 2 0 -1 -6 1

a. Let ( ) ( 3)g x f x= + . For which values of x can you evaluate ( )g x′ ? Evaluate ( )g x′ for these values. b. Let ( ) 3 ( )h x f x= . For which values of x can you evaluate ( )h x′ ? Evaluate ( )h x′ for these values. c. Let ( ) (3 )j x f x= . For which values of x can you evaluate ( )j x′ ? Evaluate ( )j x′ for these values. d. Let ( ) ( ) 3k x f x= − . For which values of x can you evaluate ( )k x′ ? Evaluate ( )k x′ for these values.

11. Suppose that f is a function such that ( ) 0f x′ < for all x. If ( )( ) ( )g x f f x= , will ( )g x′ be less than, equal to,

or greater than 0. Explain your answer.

12. If 2 2( ) 2 ( ) 3 ( ), ( ) ( ), and ( ) - ( )h x f x g x f x g x g x f x′ ′= − = = , then ( )h x′ = A) 10 ( ) ( )f x g x− B) 2 ( ) ( )f x g x− C) 2 ( ) ( )f x g x D) 10 ( ) ( )f x g x E) 2 ( ) ( )f x g x′ ′

13. ( )2If ( ) , ( ) , and ( ) 5 1, then dyg x x y f g x f x xdx

′= = = − =

A) 22 5 1x x − B) 5

5 1x

x − C) 2

5

5 1

xx −

D) 2 25 1x x − E) 2

2 5 1xx −


Answers

1

x ( )f x ( )f x′ ( )g x ( )g x′ ( )h x ( )h x′ 1 1 2 4 3 3 12 2 2 1 3 4 4 12 3 4 3 1 2 1 4 4 3 4 2 1 2 1

2

a 1( ( )) '( )nn g x g x− ⋅ b 1'( )( )n ng x nx − c cos( ( )) '( )g x g x d '(sin( )) cos( )g x x e 2sec ( ( )) '( )g x g x f 2'(tan( ))sec ( )g x x

3

a. 12

b. 0

c. -1 d. 12

4

x ( )f x ( )f x′ ( )g x ( )g x′ ( )h x ( )h x′ ( )j x ( )j x′ -1 3 2 1 1/10 0 -1/2 3 2.3 0 0 1/2 -1 1 3 2 0 -1/2 1 0 -5 0 4 0 2 0 0

5 ( )22 cos 2sin( ) cos( )x x x x

6 36 21

7 ( )25sec 5dy xdx

=

8 ( )( ) 52 34 sin( ) 3 cos( )y x x x x−

′ = − +

9 ( )( )2 2 3 2 23 sec sec ( ) tan ( )dy x x x xdx

= +

10

a) For 0,1,2,3 Values: 0,-1,-6, and 1 b) For 0,1,2,3,4,5,6 Values: 12,-9, 6, 0,-3,-18, 3 c) For 0, 1, 2 Values: 12, 0, 3 d) For 0,1,2,3,4,5,6 Values: 4,-3, 2, 0,-1,-6, 1

11 Greater than zero.

( ) ( ( )) ( ) ( )( )g x f f x f x′ ′ ′= → − − = + 12 D 13 A


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