3.2E: Exercises
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Exercise 3.2E.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).
1) h(x)=4f(x)+g(x)7
2) h(x)=x3f(x)
3) h(x)=f(x)g(x)2
4) h(x)=3f(x)g(x)+2
- Answers to even numbered questions
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2. hโฒ(x)=3x2f(x)+x3fโฒ(x)
4. hโฒ(x)=3fโฒ(x)(g(x)+2)โ3f(x)gโฒ(x)(g(x)+2)2
Exercise 3.2E.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 | โ2 | 0 |
g(x) | 2 | 3 | โ4 | 6 |
fโฒ(x) | โ1 | 7 | 8 | โ3 |
gโฒ(x) | 4 | 1 | 2 | 9 |
1) Find hโฒ(1) if h(x)=xf(x)+4g(x).
2) Find hโฒ(2) if h(x)=f(x)g(x).
3) Find hโฒ(3) if h(x)=2x+f(x)g(x).
4) Find hโฒ(4) if h(x)=1x+g(x)f(x).
- Answers to even numbered questions
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2. 169
4. Undefined
Exercise 3.2E.3
For the following exercises, use the following figure to find the indicated derivatives, if they exist.
1) Let h(x)=f(x)+g(x). Find
a. hโฒ(1),
b. hโฒ(3), and
c. hโฒ(4).
2) Let h(x)=f(x)g(x). Find
a. hโฒ(1),
b. hโฒ(3), and
c. hโฒ(4).
3) Let h(x)=f(x)g(x). Find
a. hโฒ(1),
b. hโฒ(3), and
c. hโฒ(4).
- Solution to even numbered question:
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Let h(x)=f(x)g(x). Notice that from the graph,
a) f(1)=3,f(3)=1, and f(4)=2
b)g(1)=3,g(3)=2.5, and g(4)=2.5
c) The rate of change of f(x) (slope),fโฒ(x), is โ1 when x=1, is DNE when x=3 and is 1 when x=4.
d) Also, the rate of change of g(x) (slope) is 1 when x=1, is 0 when x=3 and is 0 when x=4.
Now we can solve the problem:
a. hโฒ(1)=fโฒ(1)g(1)+f(1)gโฒ(1)=(โ1)(1)+(3)(1)=2,
b. hโฒ(3)=fโฒ(3)g(3)+f(3)gโฒ(3)=DNE, and
c. hโฒ(4)=fโฒ(4)g(4)+f(4)gโฒ(4)=(1)(2.5)+(2)(0)=2.5.
- Answers to even numbered questions
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2a. 2
b. does not exist
c. 2.5
Exercise 3.2E.4
For the following exercises,
a. evaluate fโฒ(a), and
b. graph the function f(x) and the tangent line at x=a.
1) f(x)=2x3+3xโx2,a=2
2) f(x)=1xโx2,a=1
3) f(x)=x2โx12+3x+2,a=0
4) f(x)=1xโx2/3,a=โ1
- Answers to odd numbered questions
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1a. 23
b. y=23xโ28
3a. 3
b. y=3x+2
Exercise 3.2E.5
Find the derivative of the following functions and simplify your answer:
1) f(x)=x7+10
2) f(x)=5x3โx+1
3) f(x)=4x2โ7x
4) f(x)=8x4+9x2โ1
5) f(x)=x4+2x
6) f(x)=3x(18x4+13x+1)
7) f(x)=(x+2)(2x2โ3)
8) f(x)=x2(2x2+5x3)
9) f(x)=x3+2x2โ43
10) f(x)=4x3โ2x+1x2
11) f(x)=x2+4x2โ4
12) f(x)=x+9x2โ7x+1
13) y(x)=3x2+52x2+xโ3.
14) y(x)=3x2+52x2+xโ3
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2. fโฒ(x)=15x2โ1
4. fโฒ(x)=32x3+18x
6.fโฒ(x)=270x4+39(x+1)2
8. fโฒ(x)=โ5x2
10. fโฒ(x)=4x4+2x2โ2xx4
12. fโฒ(x)=โx2โ18x+64(x2โ7x+1)2
Exercise 3.2E.6
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.
1) y=3x2+4x+1 at (0,1)
2) y=2โx+1 at (4,5)
3) y=2xxโ1 at (โ1,1)
4) y=2xโ3x2 at (1,โ1)
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2. T(x)=12x+3,
4. T(x)=4xโ5
Exercise 3.2E.7
1) Find the equation of the tangent line to the graph of f(x)=2x3+4x2โ5xโ3 at x=โ1.
2) Find the equation of the tangent line to the graph of f(x)=x2+4xโ10 at x=8.
3) Find the equation of the tangent line to the graph of f(x)=(3xโx2)(3โxโx2) at x=1.
4) Find the point on the graph of f(x)=x3 such that the tangent line at that point has an x intercept of 6.
5) Find the equation of the line passing through the point P(3,3) and tangent to the graph of f(x)=6xโ1.
- Answers to odd numbered questions
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1. y=โ7xโ3
3. y=โ5x+7
5. y=โ32x+152
Exercise 3.2E.8
Determine all points on the graph of f(x)=x3+x2โxโ1 for which the slope of the tangent line is
a. horizontal
b. โ1.
- Answer
- Under Construction
Exercise 3.2E.9
Find a quadratic polynomial such that f(1)=5,fโฒ(1)=3 and fโณ
- Answer
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y=โ3x^2+9xโ1
Exercise \PageIndex{10}
A car driving along a freeway with traffic has traveled s(t)=t^3โ6t^2+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.
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Under Construction
Exercise \PageIndex{11}
A herring swimming along a straight line has traveled s(t)=\frac{t^2}{t^2+2} feet in t seconds.
Determine the velocity of the herring when it has traveled 3 seconds.
- Answer
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\frac{12}{121} or 0.0992 ft/s
Exercise \PageIndex{12}
The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function P(t)=\frac{8t+3}{0.2t^2+1}, where t is measured in years.
a. Determine the initial flounder population.
b. Determine Pโฒ(10) and briefly interpret the result.
- Answer
- Under Construction
Exercise \PageIndex{12}
The concentration of antibiotic in the bloodstream t hours after being injected is given by the function C(t)=\frac{2t^2+t}{t^3+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.
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a. \frac{โ2t^4โ2t^3+200t+50}{(t^3+50)^2} b. โ0.02395 mg/L-hr, โ0.01344 mg/L-hr, โ0.003566 mg/L-hr, โ0.001579 mg/L-hr c. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.mplate active on the page.
Exercise \PageIndex{13}
A book publisher has a cost function given by C(x)=\frac{x^3+2x+3}{x^2}, where x is the number of copies of a book in thousands and C is the cost per book, measured in dollars. Evaluate Cโฒ(2) and explain its meaning.
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Under Construction
Exercise \PageIndex{14}
According to Newtonโs law of universal gravitation, the force F between two bodies of constant mass m_1 and m_2 is given by the formula F=\frac{Gm_1m_2}{d^2}, where G is the gravitational constant and d is the distance between the bodies.
a. Suppose that G,m_1, and m_2 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=6.67ร10^{โ11} Nm^2/kg^2, on two bodies 10 meters apart, each with a mass of 1000 kilograms.
- Answer
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a. F'(d)=\frac{โ2Gm_1m_2}{d_3} '
b. โ1.33ร10^{โ7} N/m
Contributors and Attributions
Gilbert Strang (MIT) and Edwin โJedโ Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.