homework homework assignment #11 read section 3.3 page 139, exercises: 1 – 73 (eoo), 71 rogawski...

Homework

Homework Assignment #11 Read Section 3.3 39, Exercises: 1 – 73 (EOO), 71

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 139Compute f ′ (x) using the limit definition.


1. 4 3f x x

0 0

0 0 0

4 3 4 3 lim lim

4 4 3 4 3 4lim lim lim 4 4

4

h h

h h h

f x h f x x h xf x

h hx h x h

h h

f x

Homework, Page 139Compute f ′ (x) using the limit definition.


15. f x x

1 1

0 0

0 0 0

20 0

2

lim lim

1 1

lim lim lim

1 1lim lim

1

h h

h h h

h h

x h xf x h f xf x

h h

x x h x x h x x hx h xh x x h h x x h h x x h

h

h x x h x x h x

f xx

Homework, Page 139Use the Power Rule to compute the derivative.


4

2

9. x

dx

dx

34 3

22

4

2

4 4 2 4 8 32

32

xx

x

dx x

dx

dx

dx

Homework, Page 139Use the Power Rule to compute the derivative.


0.3513. d

xdx

0.35 0.35 1 0.65

0.35 0.65

0.35 0.35

0.35

dx x x

dx

dx x

dx

Homework, Page 139Compute f ′ (a) and find an equation of the tangent line to the graph at x = a.


517. , 1f x x x

5 441 1 1 1,1 5 1 5 1 5

1 5 1 1 5 5 1 1

f f x x f

y x f y x

Homework, Page 13921. Calculate:


29 3 4 x t td d da e b t e c e

dx dt dt

0

2 2 2 2 2 2 2

9 9 9 9

3 4 3 4 3 1 4 3 4

3 4 3 4

x x x x

t t t t

t t

t t t t t t t

d da e e e e

dx dx

d d db t e t e t e e

dt dt dt

dt e e

dx

d d d dc e e e e e e e e e e

dt dt dt dt

Homework, Page 139Calculate the derivative of the function.


3 125. 2 10f x x x

2 2 2 2

2 2

2 3 10 1 6 10

6 10

f x x x x x

f x x x



3429. f s s s

1 1 3 21 1 1134 3 3 34 4 4

3 234

1 1 1 1

4 3 4 3

1 1

4 3

f s s s s s f s s s s s

f s s s



33. 3 1 2 1P z z z

23 1 2 1 6 1 6 2 1 12 1

12 1

P z z z z z P z z z

P z z



337. 5 th t e

3 3 3

3

5 5 5

5

t t t

t

h t e e e h t e e

h t e

Homework, Page 139Calculate the derivative indicated.


23

8

41. , 3C

dTT C

dC

2 113 33 3

8

8

2 2 2 23 2 1

3 28

1

C

C

dT dTC C

dC dCC

dT

dC

Homework, 9Calculate the derivative indicated.


2

4

45. , 7 h

h

dpp e

dh

2 2

2 2 4 4 2 2

4

2

4

7 7

7 7 7 7

7

h h

h

h

h

p e e e

dp dpe e e e e e

dh dh

dpe

dh

Homework, Page 13949. Sketch the graph of f ′(x) for f (x) omitting points where f (x) is not differentiable.


Homework, Page 13953. Find the points on the curve y = x2 + 3x – 7 at which the slope of the tangent line is equal to 4..


2

2

13 7 2 3 2 3 4 2 1

2

1 1 1 1 3 1 6 28 213 7 7

2 2 2 4 2 4 4

1 21 ,

2 4

y x x y x x x x

y

Homework, Page 13957. Determine the coefficients a and b such that p(x) = x2 + ax + b satisfies p(1) = 0 and p′(1) =4.


2

2

2

2 1 4 2 1 2

1 0 1 2 1 3

2, 3, 2 3

p x x ax b p x x a p a a

p b b

a b p x x x

Homework, Page 13961. Compute the derivative of f (x) = x–2 using the limit definition.


2 2 2 22 2

220 0

22 2 2 2

2 22 20 0

2

2 2 4 32 20 0

3

1 1

lim lim

1 2lim lim

2 2 2 2lim lim

2

h h

h h

h h

xx h x x h x x hf x

h h x x h

x x h x x xh h

h x x h hx x h

xh h x h x

x xhx x h x x h

f xx

Homework, Page 13965.


2

The average speed (m/s) of a gas molecule is 8 / ,

where is the temperature (K), M is the molar mass (kg/mol), and R = 8.31.

Calulate / at = 300K for O , which has a molar mass

avg

avg

v RT M

T

dv dT T

of

0.032 kg/mol.

12

12

300

300

8 / 8 / 8 /

1 1 8 1 1 88 /

2 2 2

8 8.3110.742

2 0.032 300

0.742

avg

avg

avg

T K

avg

T K

v RT M R M T R M T

dv R RR M T

dT M T MT

dv msdT

dv msdT

Homework, Page 13969. Let L be a tangent line to the hyperbola xy = 1 at x = a, where a > 0. Show that the area of the triangle bounded by L and the coordinate axes does not depend upon a.


1 22

2 2

2 2

1 11 1

1 1 2 20

2 2 1 1 20 2 2 2

2 2

Area is independent of .

xy y x y x y a y aa a

xy x a y y h

a a aa ax x

x a b A bh aa a aa a

a

Homework, Page 139


71. Match the functions (A – C)with their derivatives (I – III) inFigure 18.

A – III

B – I

C – II

Homework, Page 13973. Make a rough sketch of the graph of the derivative of the function shown in Figure 19(A).


x

y

1 2 3 4x

y

1 2 3 4


Chapter 3: DifferentiationSection 3.3: Product and Quotient Rules

Jon Rogawski

Calculus, ET First Edition


As noted above, the derivative of a product is NOT the productof the derivatives.

Example, Page 148Use the Product Rule to find the derivative.


24. 1 4 xf x x e

Example, Page 148Use the Product Rule to find the derivative.


1 12

4

6. , 2 7x

dhh x x x x

dx


As noted above, the derivative of a quotient is NOT the quotientof the derivatives.

Example, Page 148Use the Quotient Rule to find the derivative.


2

48.

1

xf x

x x

Example, Page 148Use the Quotient Rule to find the derivative, then evaluate.


2

9

10. , z

dw zw z

dz z z


The power P delivered to load R by the battery (V and r are constants) is:

(a) Calculate dP/dR; (b) Find the value of R where the tangent to the graph of P is horizontal.

2

2

V RP

R r

Homework

Homework Assignment #12 Read Section 3.4 Page 148, Exercises: 1 – 45 (EOO


homework homework assignment #11 read section 3.3 page 139, exercises: 1 – 73 (eoo), 71 rogawski...

Documents