Calculus_Volume_1_-_WEB_68M1Z5W-55

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3.3 EXERCISES
For the following exercises, find f ′(x) for each function.
106. f (x) = x7 + 10
107. f (x) = 5x3 − x + 1
108. f (x) = 4x2 − 7x
109. f (x) = 8x4 + 9x2 − 1
110. f (x) = x4 + 2
x
111. f (x) = 3x⎛⎝18x4 + 13
x + 1
⎞
⎠
112. f (x) = (x + 2)⎛⎝2x2 − 3⎞⎠
113. f (x) = x2⎛⎝ 2
x2 + 5
x3
⎞
⎠
114. f (x) = x3 + 2x2 − 4
3
115. f (x) = 4x3 − 2x + 1
x2
116. f (x) = x2 + 4
x2 − 4
117. f (x) = x + 9
x2 − 7x + 1
For the following exercises, find the equation of the tangent
line T(x) to the graph of the given function at the indicated
point. Use a graphing calculator to graph the function and
the tangent line.
118. [T] y = 3x2 + 4x + 1 at (0, 1)
119. [T] y = 2 x + 1 at (4, 5)
120. [T] y = 2x
x − 1 at (−1, 1)
121. [T] y = 2
x − 3
x2 at (1, −1)
For the following exercises, assume that f (x) and g(x)
are both differentiable functions for all x. Find the
derivative of each of the functions h(x).
122. h(x) = 4 f (x) + g(x)
7
123. h(x) = x3 f (x)
124. h(x) = f (x)g(x)
2
125. h(x) = 3 f (x)
g(x) + 2
For the following exercises, assume that f (x) and g(x)
are both differentiable functions with values as given in
the following table. Use the following table to calculate the
following derivatives.
x 1 2 3 4
f(x) 3 5 −2 0
g(x) 2 3 −4 6
f′(x) −1 7 8 −3
g′(x) 4 1 2 9
126. Find h′(1) if h(x) = x f (x) + 4g(x).
127. Find h′ (2) if h(x) = f (x)
g(x) .
128. Find h′ (3) if h(x) = 2x + f (x)g(x).
129. Find h′ (4) if h(x) = 1
x + g(x)
f (x).
For the following exercises, use the following figure to find
the indicated derivatives, if they exist.
Chapter 3 | Derivatives 263
130. Let h(x) = f (x) + g(x). Find
a. h′ (1),
b. h′ (3), and
c. h′ (4).
131. Let h(x) = f (x)g(x). Find
a. h′ (1),
b. h′ (3), and
c. h′ (4).
132. Let h(x) = f (x)
g(x) . Find
a. h′ (1),
b. h′ (3), and
c. h′ (4).
For the following exercises,
a. evaluate f ′ (a), and
b. graph the function f (x) and the tangent line at
x = a.
133. [T] f (x) = 2x3 + 3x − x2, a = 2
134. [T] f (x) = 1
x − x2, a = 1
135. [T] f (x) = x2 − x12 + 3x + 2, a = 0
136. [T] f (x) = 1
x − x2/3, a = −1
137. Find the equation of the tangent line to the graph of
f (x) = 2x3 + 4x2 − 5x − 3 at x = −1.
138. Find the equation of the tangent line to the graph of
f (x) = x2 + 4
x − 10 at x = 8.
139. Find the equation of the tangent line to the graph of
f (x) = (3x − x2)(3 − x − x2) at x = 1.
140. Find the point on the graph of f (x) = x3 such that
the tangent line at that point has an x intercept of 6.
141. Find the equation of the line passing through the
point P(3, 3) and tangent to the graph of f (x) = 6
x − 1.
142. Determine all points on the graph of
f (x) = x3 + x2 − x − 1 for which
a. the tangent line is horizontal
b. the tangent line has a slope of −1.
143. Find a quadratic polynomial such that
f (1) = 5, f ′ (1) = 3 and f ″(1) = −6.
144. A car driving along a freeway with traffic has
traveled s(t) = t3 − 6t2 + 9t meters in t seconds.
a. Determine the time in seconds when the velocity of
the car is 0.
b. Determine the acceleration of the car when the
velocity is 0.
145. [T] A herring swimming along a straight line has
traveled s(t) = t2
t2 + 2
feet in t seconds. Determine the
velocity of the herring when it has traveled 3 seconds.
146. The population in millions of arctic flounder in the
Atlantic Ocean is modeled by the function
P(t) = 8t + 3
0.2t2 + 1
, where t is measured in years.
a. Determine the initial flounder population.
b. Determine P′ (10) and briefly interpret the result.
147. [T] The concentration of antibiotic in the
bloodstream t hours after being injected is given by the
function C(t) = 2t2 + t
t3 + 50
, where C is measured in
milligrams per liter of blood.
a. Find the rate of change of C(t).
b. Determine the rate of change for t = 8, 12, 24,
and 36.
c. Briefly describe what seems to be occurring as the
number of hours increases.
148. A book publisher has a cost function given by
C(x) = x3 + 2x + 3
x2 , where x is the number of copies of
a book in thousands and C is the cost, per book, measured
in dollars. Evaluate C′ (2) and explain its meaning.
264 Chapter 3 | Derivatives
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149. [T] According to Newton’s law of universal
gravitation, the force F between two bodies of constant
mass m1 and m2 is given by the formula F = Gm1m2
d2 ,
where G is the gravitational constant and d is the distance
between the bodies.
a. Suppose that G, m1, and m2 are constants. Find
the rate of change of force F with respect to
distance d.
b. Find the rate of change of force F with
gravitational constant G = 6.67 × 10−11
Nm2 /kg2, on two bodies 10 meters apart, each
with a mass of 1000 kilograms.
Chapter 3 | Derivatives 265
3.4 | Derivatives as Rates of Change
Learning Objectives
3.4.1 Determine a new value of a quantity from the old value and the amount of change.
3.4.2 Calculate the average rate of change and explain how it differs from the instantaneous rate
of change.
3.4.3 Apply rates of change to displacement, velocity, and acceleration of an object moving along
a straight line.
3.4.4 Predict the future population from the present value and the population growth rate.
3.4.5 Use derivatives to calculate marginal cost and revenue in a business situation.
In this section we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of
change of a function. These applications include acceleration and velocity in physics, population growth rates in biology,
and marginal functions in economics.
Amount of Change Formula
One application for derivatives is to estimate an unknown value of a function at a point by using a known value of a
function at some given point together with its rate of change at the given point. If f (x) is a function defined on an interval
⎡
⎣a, a + h⎤⎦, then the amount of change of f (x) over the interval is the change in the y values of the function over that
interval and is given by
f (a + h) − f (a).
The average rate of change of the function f over that same interval is the ratio of the amount of change over that interval
to the corresponding change in the x values. It is given by
f (a + h) − f (a)
h .
As we already know, the instantaneous rate of change of f (x) at a is its derivative
f ′ (a) = lim
h → 0
f (a + h) − f (a)
h .
For small enough values of h, f ′ (a) ≈ f (a + h) − f (a)
h . We can then solve for f (a + h) to get the amount of change
formula:
(3.10)f (a + h) ≈ f (a) + f ′(a)h.
We can use this formula if we know only f (a) and f ′(a) and wish to estimate the value of f (a + h). For example, we
may use the current population of a city and the rate at which it is growing to estimate its population in the near future. As
we can see in Figure 3.22, we are approximating f (a + h) by the y coordinate at a + h on the line tangent to f (x) at
x = a. Observe that the accuracy of this estimate depends on the value of h as well as the value of f ′ (a).
266 Chapter 3 | Derivatives
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3.21
Figure 3.22 The new value of a changed quantity equals the
original value plus the rate of change times the interval of
change: f (a + h) ≈ f (a) + f ′ (a)h.
Here is an interesting demonstration (http://www.openstax.org/l/20_chainrule) of rate of change.
Example 3.33
Estimating the Value of a Function
If f (3) = 2 and f ′ (3) = 5, estimate f (3.2).
Solution
Begin by finding h. We have h = 3.2 − 3 = 0.2. Thus,
f (3.2) = f (3 + 0.2) ≈ f (3) + (0.2) f ′ (3) = 2 + 0.2(5) = 3.
Given f (10) = −5 and f ′ (10) = 6, estimate f (10.1).
Motion along a Line
Another use for the derivative is to analyze motion along a line. We have described velocity as the rate of change of position.
If we take the derivative of the velocity, we can find the acceleration, or the rate of change of velocity. It is also important to
introduce the idea of speed, which is the magnitudeof velocity. Thus, we can state the following mathematical definitions.
Definition
Let s(t) be a function giving the position of an object at time t.
The velocity of the object at time t is given by v(t) = s′ (t).
The speed of the object at time t is given by |v(t)|.
The acceleration of the object at t is given by a(t) = v′ (t) = s″(t).
Chapter 3 | Derivatives 267
http://www.openstax.org/l/20_chainrule
	Chapter 3. Derivatives
	3.4. Derivatives as Rates of Change*