P. 76 (Required: 5, 8, 9, 11)P. 90 (Required: 20, 21, 25, 26) Caroline Margiotta

I. P. 76

Use the definition of derivative to calculate f ′ (c).

5.  f(x) = x2 + 5x + 1, c = –2

6.  f(x) = x2 + 6x – 2, c = –4

7.  f(x) = x3 – 4x2 + x + 8, c = 1

8.  f(x) = x3 – x2 – 4x + 6, c = –1

9.  f(x) = –0.7x + 2, c = 3

10.  f(x) = 1.3x – 3, c = 4

11.  f(x) = 5, c = –1

II. P. 90

Use the definition of derivative in the f(x + h) form to find the equation of the derivative

function. In each case, show that your answer is consistent with the answer you would

get if you used the formula for the derivative of the power function.

19.  f(x) = 7x4

20.  g(x) = 5x3

21.  v(t) = 10t2 – 5t + 7

22.  s(t) = t4 – 6t2 + 3.7

Sketch a reasonable graph of the derivative function.

25.

26.

I. P. 90

Find the equation of the derivative function.

1.  f(x) = 5x4

2.  y = 11x8

3.  v = 0.007t–83

4.  v(x) = 

5.  M(x) = 1215

6.  f(x) = 4.7723

7.  y = 0.3x2 – 8x + 4

8.  r = 0.2x2 + 6x – 1

13.  v = (3x – 4)2 (Expand first.)

14.  u = (5x – 7)2 (Expand first.)

15.  f(x) = (2x + 5)3 (Be clever!)

16.  f(x) = (4x – 1)3 (Be clever!)

II. P. 97-99

5.  An object moves as in Problem 3, with displacement, x, given by

x = –t3 + 13t2 – 35t + 27 where x is in feet and t is in seconds.

a.  Find equations for the velocity and the acceleration.

b.  Find the velocity and acceleration at t = 1,t = 6, and t = 8. At each time, state

•  Whether x is increasing or decreasing, and at what rate

•  Whether the object is speeding up or slowing down, and how you decided

c.  At what times in the interval [0, 9] is x arelative maximum? Is x ever negative if t is in

this interval?

6.  A particle (a small object) moves as in Problem 4, with displacement, x, given by

x = t4 – 11t3 + 38t2 – 48t + 50

where x is in meters and t is in minutes.

a. Write equations for the velocity, v, and acceleration, a.

b. Is the object speeding up or slowing down at t = 1? At t = 3? At t = 5? How can you

tell?

c. Find all values of t for which v = 0.

d. Plot the graphs of x and v on the same screen, then sketch the results. Explain what

is true about the displacement whenever v = 0.

e. Plot the graph of a on the same screen you used in part d. At each time the

acceleration is zero, what is true about v? About x?

9.  Velocity from Displacement Problem: If you place-kick a football, its displacement

above the ground, d, in meters, is given by d(t) = 18t – 4.9t2 where t is time in seconds

since it was punted

(The coefficient 18 is the initial

upward velocity in meters per second.)

(Figure 3-5g)

a. Find the algebraic derivative,  (t). Use your answer to find  (1) and  (3). What

name from physics is given to  ?

b.    At these times, is the football going up or down? How fast? How does the graph in

Figure 3-5g confirm these conclusions?

c.  Use the   equation to find the velocity at time t = 4. Explain why the answer has a

meaning in the mathematical world but not in the real world.

7.  Car Problem: Calvin’s car runs out of gas as it is going up a hill. The car rolls to a

stop, then starts rolling backward. As it rolls, its displacement, d, in feet from the bottom

of the hill, at t, in seconds since Calvin’s car ran out of gas, is given by

d(t) = 99 + 30t – t2

a.  Plot graphs of d and   on the same screen. Use a window large enough to include

the point where the graph of d crosses the positive t-axis. Sketch the result.

b.  For what range of times is the velocity positive? How do you interpret this answer in

terms of Calvin’s motion on the hill?

c.  At what time did Calvin’s car stop rolling up and start rolling back? How far was it

from the bottom of the hill at this time?

d.  If Calvin doesn’t put on the brakes, when will he be back at the bottom of the hill?

e.  How far was Calvin from the bottom of the hill when his car ran out of gas?

Find the second derivative,

15.  y = 5x3

16.  y = 7x4

17.  y = 9x2 + x5

18.  y = 10x2 – 15x + 42

19.    Compound Interest Second Derivative Problem:

If you invest $1000 in an IRA (individual retirement account) that pays 10% APR

(annual percentage rate), then the amount, m(t), in dollars, you have in the account

aftertime t, in years, is given by the exponential function m(t) = 1000(1.1t)

Figure 3-5k shows the graph of this function.

Use the numerical derivative feature of your graphing calculator to find the values of the

derivatives (5) and  (10). Explain what these numbers represent, and why   (10)

>   (5).

Use the numerical derivative feature again to find the second derivatives,  (5) and 

(10). You can do this by storing the numerical derivative function as y2, then finding the

numerical derivative of y2. What are the units of the second derivative? What real-world

quantity does the second derivative represent?

Page 102 (Required: 7 & additional problems)11/10/2012 21:30

7. For each function below, state which is the inside function and which is the outside

one. Addition: Then find the derivative.

a.  f(x) = sin 3x

b.  h(x) = sin3 x

c.  g(x) = sin x3

d.  r(x) = 2cos x

e.  q(x) = 1/(tanx)

f.  L(x) = log (sec x)

Additional Problems:

If f(x)=sinx, find the 17th derivative of f(x).

If f(x)=cosx, find the 27th derivative of f(x).

Review 3.2-3.6:Page 124 (Required: R4, R5)Page 126 (Required: T1-T4, T12-T15, T20) Caroline Margiotta

I. Page 124

R4:

a.  Write the definition of derivative (the h or x form).

b.  What single word means “find the derivative”?

c.  Write the property for the derivative of a power function.

d.  Prove the property of the derivative of a constant times a function.

e.  Prove the property of the derivative of a sum of two functions.

f. How do you pronounce dy/dx ? How do you pronounce d/dx(y)? What do these

symbols mean?

g.  Find an equation for the derivative function.

i.  f(x) = 7x9/5

ii.  g(x) = 7x–4 – (x^2 /6) – x + 7

iii . h(x) = 73

h.    Compare the exact value of (32) in part g with the numerical derivative at that point.

i.    On a copy of Figure 3-10a, sketch the graph of the derivative function for the function

shown.

Figure 3-10a

R5e.  Spaceship Problem: A spaceship is approaching Mars. It fires its retrorockets,

which causes it to slow down, stop, rise up again, then come back down. Its

displacement, y, in kilometers, from the surface is found to be

 y = –0.01t3 + 0.9t2 – 25t + 250

where t is time, in seconds, since the retrorocket was fired.

i.  Write equations for the velocity and acceleration of the spaceship.

ii.  Find the acceleration at time t = 15. At that time, is the spaceship speeding

up or slowing down? How do you tell?

iii.  Find by direct calculation the values of t at which the spaceship is

stopped.

iv.  When does the spaceship touch the surface of Mars? What is its velocity

at that time? Describe what you think will happen to the spaceship at that

time.

II. Page 126

Part I: NO CALCULATORS ALLOWED!

T1.  Write the definition of derivative at a point. Write the definition of derivative as a

function.

T2.  Prove directly from the definition of derivative that if f(x) = 3x4, then (x) = 12x3.

T3.  Sketch the graph of a function for which (5) = 2. Sketch a line tangent to the graph

at that point. What would happen to the graph and the tangent line if you were to zoom

in on the point where x = 5? What is the name of the property that expresses this

relationship between the graph and the tangent line?

T4.  Amos must evaluate (5) where f(x) = 7x. He substitutes 5 for x, getting f(5) = 35.

Then he differentiates the 35 and gets (5) = 0 (which equals the score his instructor

gives him for the problem!). What mistake did Amos make? What is the correct answer?

For problems T5-T11, Find an equation for the derivative function.

T5.  f(x) = (7x + 3)15

T6.  g(x) = cos (x5)

T7.  

T8.  u = 36x

T9.  u = cos (sin5 7x)

T10.  y = 60x2/3 – x + 25

T11.

if y = e9x

T12. Estimate the value of the derivative,  , at x = 1for the graph shown in Figure 3-

10c.

T13.  If y = 3 + 5x–1.6, where y is the displacement of a moving object, find equations for

the velocity function and the acceleration function. What special name is given to the

acceleration function that expresses its relationship to the original displacement

function?

T14.  Find an equation for the anti-derivative of (x) = 72x5/4.

Part II: GRAPHING CALCULATORS ALLOWED!

T15.  Find an equation for f(x) if (x) = 5 sin x and f(0) = 13.

T20.  How can you tell from the velocity and acceleration of a moving object that the object is slowing

down at a particular value of t?