Homework Homework Assignment #12 Read Section 3.4 Page 148, Exercises: 1 – 45 (EOO Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
Jan 04, 2016
Homework
Homework Assignment #12 Read Section 3.4 Page 148, Exercises: 1 – 45 (EOO
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 148Use the Product Rule to find the derivative.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Homework, Page 148Calculate the derivative using the appropriate rule(s).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Homework, Page 148Calculate the derivative using the appropriate rule(s).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Homework, Page 148Calculate the derivative using the appropriate rule(s).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Homework, Page 148Calculate the derivative using the appropriate rule(s).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Homework, Page 148Calculate the derivative using the appropriate rule(s).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Chapter 3: DifferentiationSection 3.4: Rates of Change
Jon Rogawski
Calculus, ET First Edition
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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?
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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?
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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?
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework
Homework Assignment #13 Read Section 3.5 Page 158, Exercises: 1 – 45 (EOO)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company