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

Homework

Homework Assignment #12 Read Section 3.4 48, Exercises: 1 – 45 (EOO

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

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


21. 1f x x x

2 2

2 2 2 2

2

1 , 1; 1; 2

2 1 1 2 1 3 1

3 1

f x x x uv u x u v x v x

f x uv vu x x x x x x

f x x

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


2

3

5. , 1 9t

dyy t t t

dt

2 2

2 2 2

2

2

3

1 9 1, 2 ; 9, 1

1 1 9 2 1 2 18

3 18 1

3 3 3 18 3 1 27 54 1 82 82t

y t t t uv u t u t v t v

y t uv vu t t t t t t

t t

dyy

dt

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


2

22

19. ,

1t

dg tg t

dt t

22 2

2

2 2 3 3

2 22 2 2

2 2 222

2

11, 2 ; 1, 2

1

1 2 1 2 2 2 2 2

1 1

4 24 8 82

931 2 1

8

9t

t ug t u t u t v t v t

t v

t t t tvu uv t t t tg t

v t t

tg t g

t

dg

dt

Homework, Page 148Calculate the derivative in two ways. First use the Product or Quotient Rule, then rewrite and apply the Power Rule.


213. 2 1 2f t t t

2 2

2 2 2

2

2

2 3 2 3 2

2 2 2

2 1 2 2 2, 2; 2, 2

2 1 2 2 2 4 2 2 4

6 2 4

6 2 4

2 1 2 2 4 2 2 4 2

6 2 4 , 6 2 4 6 2 4

f t t t uv u t u v t v t

f t uv vu t t t t t t

t t

f t t t

f t t t t t t t t t

f t t t t t t t

Homework, Page 148Calculate the derivative using the appropriate rule(s).


4 217. 4 1f x x x x

4 2

4 3 2

4 2 3

5 4 5 4 3 5 4 3

5 4 3

4 1

4, 4 ; 1, 2 1

4 2 1 1 4

2 8 4 4 4 4 6 5 4 8 4

6 5 4 8 4

f x x x x uv

u x u x v x x v x

f x uv vu x x x x x

x x x x x x x x x x

f x x x x x



21. 1 1f x x x

1 1

1 11, ; 1,

2 2

1 11 1

2 2

1 1 1 11

2 22 2

1

f x x x uv

u x u v x vx x

f x uv vu x xx x

x x

f x



31

125. ,

1x

dzz x

dx x

3

3 2

3 2 2

2 22 3 3

2

231

1

1

1, 0; 1, 3

1 0 1 3 3

1 1

3 1 3 31

4 41 1 x

uz x

x v

u u v x v x

x xvu uv xz x

v x x

dzz

dx



1 1

2 229. 3 5f t

1 12 2

1 12 2

1 12 2

3 5

3 , 0; 5 , 0

3 0 5 0 0

0

f t uv

u u v v

f t uv vu

f t



33.

1 1

x

x

ef x

e x

22

2 2 2 2 2

2 22 2

2 2

11 1

, ; 1, 1 1

1 1 1

1 1

2

1 1 1 1

1 1

x x

x xx

x x x x x x x

x x x x x x x

x

x x x x x x x x x

x x

x x

x

e e uf x

xe e x ve x

u e u e v xe e x v xe e e

xe e x e e xe e evu uvz x

v e x

xe e xe e xe e e e xe

e x e x

e x ez x

e x



2

437.

d xt

dt t x

2

2

2 2 2 2

2 22 2 2

2 2 2 2

2 22 2

4

4, ; , 2

4 2 2 8

8 8

d xt d u

dt t x dt v

u xt u x v t x v t

t x x xt tvu uv xt x xt tz x

v t x t x

xt t x xt t xz x

t x t x

Homework, Page 148


2

241. Let . Calculate assuming that is a variable

and is a constant.

V R dPP r

drR r

R

2

2

2 2 2

2 2 2 2

2 42 2

2 2 2

4 3 3

=

, 0; 2 , 2 2

2 0 2 2 2 2

2 2 2

V R uP

vR r

u V R u v R rR r v R r

R rR r V R R r V R R rdP vu uv

dr v R rR r

V R R r V R dP V R

drR r R r R r

Homework, Page 148


2

145. The curve is called the witch of Agnesi after the Italian

1mathematician Maria Agnesi (1718-1799) who wrote one of the first

books on calculus. Find equations of the tangent lines at

yx

x

1.

2 22 22

2 22

2 11 2 1 1

1 21 1 1

2 1 1 1 11 1

2 21 11 1

1 1 1 11 , 1

2 2 2 2

xy y y

x x

y y

y x y x


Chapter 3: DifferentiationSection 3.4: Rates of Change

Jon Rogawski

Calculus, ET First Edition


The average rate of change (ROC) of a function over an interval is:

The instantaneous ROC of a function at a point is:

We must realize that the average ROC of a function is the slope of the secant line and the instantaneous ROC is the slope of the tangent line.


Figure 1 illustrates measuring the average ROC of f (x) over (x0, x1),while Figure 2 illustrates measuring the instantaneous ROC at x = x0.

Example, Page 1582. Find the ROC of the volume of a cube with respect to the length of its side s when s = 3 and s = 5.



Table 1 illustrates some of the data used to construct the graph in Figure 3. The slope of the secant line in Figure 3 gives the average ROC of the temperature from midnight to 12:28PM on July 6, 1997.


Notice that as r increases, the area of the blue band increases, and the slope of the tangent to the area curve increases.

Example, Page 1588. Calculate the ROC dA/dD, where A is the surface area of a sphere of diameter D.



For some purposes, using the difference quotient with a small

provides a sufficiently accurate estimate of the derivative. In a few

cases, letting 1 gives a su

h

f x h f xf x

h

h

fficiently accurate estimate.


Marginal cost, in economics, gives the cost of delivering one more unit. Figure 5 shows how the cost of a flight is less per passenger,the more passengers there are. Thus, the marginal cost of flyingeach additional passenger decreases as the number of passengers increases.


From the graph, when is the car moving toward its destination?

When is the car standing still?

When is the car going towards is origin?


A truck enters a freeway off-ramp at t = 0. Its position after t seconds is s(t) = 84t – t3 ft for 0 ≤ t ≤ 5. (a) How fast is the truck traveling at the moment it enters the off-ramp?

(b) Is the truck speeding up or slowing down?


Galileo used a device such as in Figure 9 to investigate the motion of objects moving solely under the influence of gravity. From his experiments, he deduced that, in the absence of air resistance, the velocity of a falling object is proportional to the time it has been falling.


The equations below may be used to determine the vertical position and velocity of an object, absent air resistance, under the influence of gravity alone. For the equations, s0 is initial height, v0 is initial velocity, and t is time.

ExampleA baseball is tossed vertically upward with an initial velocity of 85 ft/s from the top of a 65-ft high building.

a. What is the height of the ball after 0.25 s?

b. Find the velocity of the ball after 1 s.

c. When does the ball hit the ground?


Example, Page 15818. The escape velocity at a distance r meters from the center of the Earth is vesc = (2.82 x 107)r –½ m/s. Assuming the radius of the Earth is 6.77 x 106 m, calculate the rate at which vesc changes with respect to distance at the Earth’s surface.


Homework

Homework Assignment #13 Read Section 3.5 Page 158, Exercises: 1 – 45 (EOO)


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

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