3.3E: Exercises for Section 3.3
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In exercises 1 - 12, find f′(x) for each function.
1) f(x)=x7+10
2) f(x)=5x3−x+1
- Answer
- f′(x)=15x2−1
3) f(x)=4x2−7x
4) f(x)=8x4+9x2−1
- Answer
- f′(x)=32x3+18x
5) f(x)=x4+2x
6) f(x)=3x(18x4+13x+1)
- Answer
- f′(x)=270x4+39(x+1)2
7) f(x)=(x+2)(2x2−3)
8) f(x)=x2(2x2+5x3)
- Answer
- f′(x)=−5x2
9) f(x)=x3+2x2−43
10) f(x)=4x3−2x+1x2
- Answer
- f′(x)=4x4+2x2−2xx4
11) f(x)=x2+4x2−4
12) f(x)=x+9x2−7x+1
- Answer
- f′(x)=−x2−18x+64(x2−7x+1)2
In exercises 13 - 16, 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.
13) [T] y=3x2+4x+1 at (0,1)
14) [T] y=2√x+1 at (4,5)
- Answer
-
T(x)=12x+3
15) [T] y=2xx−1 at (−1,1)
16) [T] y=2x−3x2 at (1,−1)
- Answer
-
T(x)=4x−5
In exercises 17 - 20, assume that f(x) and g(x) are both differentiable functions for all x. Find the derivative of each of the functions h(x).
17) h(x)=4f(x)+g(x)7
18) h(x)=x3f(x)
- Answer
- h′(x)=3x2f(x)+x3f′(x)
19) h(x)=f(x)g(x)2
20) h(x)=3f(x)g(x)+2
- Answer
- h′(x)=3f′(x)(g(x)+2)−3f(x)g′(x)(g(x)+2)2
For exercises 21 - 24, 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 |
21) Find h′(1) if h(x)=xf(x)+4g(x).
22) Find h′(2) if h(x)=f(x)g(x).
- Answer
- h′(2)=169
23) Find h′(3) if h(x)=2x+f(x)g(x).
24) Find h′(4) if h(x)=1x+g(x)f(x).
- Answer
- h′(4) is undefined.
In exercises 25 - 27, use the following figure to find the indicated derivatives, if they exist.
25) Let h(x)=f(x)+g(x). Find
a) h′(1),
b) h′(3), and
c) h′(4).
26) Let h(x)=f(x)g(x). Find
a) h′(1),
b) h′(3), and
c) h′(4).
- Answer
- a. h′(1)=2,
b. h′(3) does not exist,
c. h′(4)=2.5
27) Let h(x)=f(x)g(x). Find
a) h′(1),
b) h′(3), and
c) h′(4).
In exercises 28 - 31,
a) evaluate f′(a), and
b) graph the function f(x) and the tangent line at x=a.
28) [T] f(x)=2x3+3x−x2,a=2
- Answer
-
a. 23
b. y=23x−28
29) [T] f(x)=1x−x2,a=1
30) [T] f(x)=x2−x12+3x+2,a=0
- Answer
-
a. 3
b. y=3x+2
31) [T] f(x)=1x−x2/3,a=−1
32) Find the equation of the tangent line to the graph of f(x)=2x3+4x2−5x−3 at x=−1.
- Answer
- y=−7x−3
33) Find the equation of the tangent line to the graph of f(x)=x2+4x−10 at x=8.
34) Find the equation of the tangent line to the graph of f(x)=(3x−x2)(3−x−x2) at x=1.
- Answer
- y=−5x+7
35) Find the point on the graph of f(x)=x3 such that the tangent line at that point has an x-intercept of (6,0).
36) Find the equation of the line passing through the point P(3,3) and tangent to the graph of f(x)=6x−1.
- Answer
- y=−32x+152
37) Determine all points on the graph of f(x)=x3+x2−x−1 for which the slope of the tangent line is
a. horizontal
b. −1.
38) Find a quadratic polynomial such that f(1)=5,f′(1)=3 and f″(1)=−6.
- Answer
- y=−3x2+9x−1
39) 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.
40) [T] A herring swimming along a straight line has traveled s(t)=t2t2+2 feet in t
seconds. Determine the velocity of the herring when it has traveled 3 seconds.
- Answer
- 12121 or 0.0992 ft/s
41) The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function P(t)=8t+30.2t2+1, where t is measured in years.
a. Determine the initial flounder population.
b. Determine P′(10) and briefly interpret the result.
42) [T] The concentration of antibiotic in the bloodstream t hours after being injected is given by the function C(t)=2t2+tt3+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.
- Answer
- a. −2t4−2t3+200t+50(t3+50)2
b. −0.02395 mg/L-hr, −0.01344 mg/L-hr, −0.003566 mg/L-hr, −0.001579 mg/L-hr
c. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.
43) A book publisher has a cost function given by C(x)=x3+2x+3x2, 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.
44) [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=Gm1m2d2, 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=6.67×10−11Nm2/kg2, on two bodies 10 meters apart, each with a mass of 1000 kilograms.
- Answer
- a. F′(d)=−2Gm1m2d3
b. −1.33×10−7 N/m