3.3E: Both 3.3 and 3.4 Exercises
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3.3 and 3.4:Differentiation Rules
Exercise:
For the following exercises, find f′(x) for each function.
106) f(x)=x7+10
107) f(x)=5x3−x+1
- Answer:
- f′(x)=15x2−1
108) f(x)=4x2−7x
109) f(x)=8x4+9x2−1
- Answer:
- f′(x)=32x3+18x
110) f(x)=x4+2x
111) f(x)=3x(18x4+13x+1)
- Answer:
- f′(x)=270x4+39(x+1)2
112) f(x)=(x+2)(2x2−3)
113) f(x)=x2(2x2+5x3)
- Answer:
- f′(x)=−5x2
114) f(x)=x3+2x2−43
115) f(x)=4x3−2x+1x2
- Answer:
- f′(x)=4x4+2x2−2xx4
116) f(x)=x2+4x2−4
117) f(x)=x+9x2−7x+1
- Answer:
- f′(x)=−x2−18x+64(x2−7x+1)2
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)
- Answer:
-
T(x)=12x+3
120) [T] y=2xx−1 at (−1,1)
121) [T] y=2x−3x2 at (1,−1)
- Answer:
-
T(x)=4x−5
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)=4f(x)+g(x)7
123) h(x)=x3f(x)
- Answer:
- h′(x)=3x2f(x)+x3f′(x)
124) h(x)=f(x)g(x)2
125) h(x)=3f(x)g(x)+2
- Answer:
- h′(x)=3f′(x)(g(x)+2)−3f(x)g′(x)(g(x)+2)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)=xf(x)+4g(x).
127) Find h′(2) if h(x)=f(x)g(x).
- Answer:
-
169
128) Find h′(3) if h(x)=2x+f(x)g(x).
129) Find h′(4) if h(x)=1x+g(x)f(x).
- Answer:
-
Undefined
For the following exercises, use the following figure to find the indicated derivatives, if they exist.For the following exercises, use the following figure to find the indicated derivatives, if they exist.
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).
- Answer:
-
a. 2
b. does not exist
c. 2.5
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
- Answer:
-
a. 23
b. y=23x−28
134) [T] f(x)=1x−x2,a=1
135) [T] f(x)=x2−x12+3x+2,a=0
- Answer:
-
a. 3
b. y=3x+2
136) [T] f(x)=1x−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.
- Answer:
-
y=−7x−3
138) Find the equation of the tangent line to the graph of f(x)=x2+4x−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.
- Answer:
-
y=−5x+7
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)=6x−1.
- Answer:
-
y=−32x+152
J141) Find f′(x) for f(x)=5ex
- Answer:
-
5ex
J142) Find the equation of the line passing through x=5 and tangent to the graph of f(x)=3x+ex.
J143) Find the equation of the line passing through x=ln4 and tangent to the graph of f(x)=3x+ex.
- Answer:
-
y−(ln64+4)=7(x−ln4)
J144) Determine all points on the graph of f(x)=3x+ex for which the slope of the tangent line is 10
J145) Determine all points on the graph of f(x)=3x+ex for which the slope of the tangent line is horizontal.
- Answer:
-
none
J146) Determine all points on the graph of f(x)=3x+ex for which the slope of the tangent line is 10
J147) For the following functions, find f″(x).
a. f(x)=3x+ex b. f(x)=2x3+7ex
- Answer:
-
a. f″(x)=ex.
b. f″(x)=6+7ex.
Solution:
a. f″(x)=ex.
b. f″(x)=6+7ex.
For the following exercises, find f′(x) for each function.
J148) f(x)=xex
J149) f(x)=x2ex
- Answer:
-
2xex+x2ex
J150) f(x)=6x3−2ex
J151) f(x)=5ex
- Answer:
-
−5ex
J152) f(x)=e−xx
J153) f(x)=5x+13x−2
- Answer:
-
−13(3x−2)2
J154) f(x)=7xex.
142) 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.
143) Find a quadratic polynomial such that f(1)=5,f′(1)=3 and f″(1)=−6.
- Answer:
-
y=−3x2+9x−1
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) 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
146) 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.
147) [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.
Solution: 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.
148) 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.
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=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.−1.33×10−7N/m