3.1E: Exercises
- Page ID
- 116563
This page is a draft and is under active development.
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In exercises 1 - 10, find \(f^{\prime}(x)\) for each function.
1) \(f(x)=x^7+10\)
2) \(f(x)=5x^3−x+1\)
- Answer
- \(f^{\prime}(x)=15x^2−1\)
3) \(f(x)=4x^2−7x\)
4) \(f(x)=8x^4+9x^2−1\)
- Answer
- \(f^{\prime}(x) = 32x^3+18x\)
5) \(f(x)=x^4+2x\)
7) \(f(x)=(x+2)(2x^2−3)\)
8) \(f(x)=x^2\left(\frac{2}{x^2}+\frac{5}{x^3}\right)\)
- Answer
- \(f^{\prime}(x) = \frac{−5}{x^2}\)
9) \(f(x)=\frac{x^3+2x^2−4}{3}\)
10) \(f(x)=\frac{4x^3−2x+1}{x^2}\)
- Answer
- \(f^{\prime}(x) = \dfrac{4x^4+2x^2−2x}{x^4}\)
[Technology Required] In exercises 11 - 13, find the equation of the tangent line \(T(x)\) to the graph of the given function at the indicated point. Use graphing technology to graph the function and the tangent line.
11) \(y=3x^2+4x+1\) at \((0,1)\)
12) \(y=2\sqrt{x}+1\) at \((4,5)\)
- Answer
-
\(T(x)=\frac{1}{2}x+3\)
13) \(y=\frac{2}{x}−\frac{3}{x^2}\) at \((1,−1)\)
- Answer
-
\(T(x)=4x−5\)
In exercise 14, assume that \(f(x)\) and \(g(x)\) are both differentiable functions for all \(x\). Find the derivative of the function \(h(x)\).
14) \(h(x)=4f(x)+\frac{g(x)}{7}\)
In exercise 15, use the following figure to find the indicated derivatives, if they exist.
15) Let \(h(x)=f(x)+g(x)\). Find
a) \(h^{\prime}(1)\),
b) \(h^{\prime}(3)\), and
c) \(h^{\prime}(4)\).
[Technology Required] In exercises 16 - 19,
a) evaluate \(f^{\prime}(a)\), and
b) graph the function \(f(x)\) and the tangent line at \(x=a\).
16) \(f(x)=2x^3+3x−x^2, \quad a=2\)
- Answer
-
a. 23
b. \(y=23x−28\)
17) \(f(x)=\frac{1}{x}−x^2, \quad a=1\)
18) \(f(x)=x^2−x^{12}+3x+2, \quad a=0\)
- Answer
-
a. \(3\)
b. \(y=3x+2\)
19) \(f(x)=\frac{1}{x}−x^{2/3}, \quad a=−1\)
In exercises 20 - 22, find the equation of the tangent line to the graph of the function at the given point.
20) \(f(x)=2x^3+4x^2−5x−3\) at \(x=−1.\)
- Answer
- \(y=−7x−3\)
21) \(f(x)=x^2+\frac{4}{x}−10\) at \(x=8\).
22) \(f(x)=(3x−x^2)(3−x−x^2)\) at \(x=1\).
- Answer
- \(y=−5x+7\)
23) Find the point on the graph of \(f(x)=x^3\) such that the tangent line at that point has an \(x\)-intercept of \((6,0)\).
24) Determine all points on the graph of \(f(x)=x^3+x^2−x−1\) for which the slope of the tangent line is
a. horizontal
b. −1.
25) Find a quadratic polynomial such that \(f(1)=5,\; f^{\prime}(1)=3\) and \(f^{\prime \prime}(1)=−6.\)
- Answer
- \(y=−3x^2+9x−1\)
Applications of the Derivative: Physics
- Velocity is the change in position with respect to time. That is, \( v(t) = \frac{ d s}{dt} \), where \( s(t) \) is the position function.
- Acceleration is the change in velocity with respect to time. That is, \( a(t) = \frac{dv}{dt} = \frac{ds^2}{dt^2} \).
In exercises 26 - 27, the given functions represent the position of a particle traveling along a horizontal line.
a. Find the velocity and acceleration functions.
b. Determine the time intervals when the object is slowing down or speeding up.
26) \(s(t)=2t^3−3t^2−12t+8\)
27) \(s(t)=2t^3−15t^2+36t−10\)
- Answer
- a. \(v(t)=6t^2−30t+36,\quad a(t)=12t−30\);
b. speeds up for \( (2,2.5)∪(3,∞)\), slows down for \((0,2)∪(2.5,3)\)
28) A rocket is fired vertically upward from the ground. The distance \(s\) in feet that the rocket travels from the ground after \(t\) seconds is given by \(s(t)=−16t^2+560t\).
a. Find the velocity of the rocket 3 seconds after being fired.
b. Find the acceleration of the rocket 3 seconds after being fired.
29) A ball is thrown downward with a speed of 8 ft/s from the top of a 64-foot-tall building. After \(t\) seconds, its height above the ground is given by \(s(t)=−16t^2−8t+64.\)
a. Determine how long it takes for the ball to hit the ground.
b. Determine the velocity of the ball when it hits the ground.
30) The position function \(s(t)=t^2−3t−4\) represents the position of the back of a car backing out of a driveway and then driving in a straight line, where \(s\) is in feet and \(t\) is in seconds. In this case, \(s(t)=0\) represents the time at which the back of the car is at the garage door, so \(s(0)=−4\) is the starting position of the car, 4 feet inside the garage.
a. Determine the velocity of the car when \(s(t)=0\).
b. Determine the velocity of the car when \(s(t)=14\).
31) The position of a hummingbird flying along a straight line in \(t\) seconds is given by \(s(t)=3t^3−7t\) meters.
a. Determine the velocity of the bird at \(t=1\) sec.
b. Determine the acceleration of the bird at \(t=1\) sec.
c. Determine the acceleration of the bird when the velocity equals 0.
32) A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot-tall building. The distance in feet that the potato travels from the ground after \(t\) seconds is given by \(s(t)=−16t^2+100t+85\).
a. Find the velocity of the potato after \(0.5\) s and \(5.75\) s.
b. Find the speed of the potato at \(0.5\) s and \(5.75\) s.
c. Determine when the potato reaches its maximum height.
d. Find the acceleration of the potato at \(0.5\) s and \(1.5\) s.
e. Determine how long the potato is in the air.
f. Determine the velocity of the potato upon hitting the ground.
- Answer
- a. 84 ft/s, −84 ft/s
b. 84 ft/s
c. \(\frac{25}{8}\) s
d. \(−32 \; \text{ft/s}^2\) in both cases
e. \(\frac{1}{8}(25+\sqrt{965})\) s
f. \(−4\sqrt{965}\) ft/s
33) The position function \(s(t)=t^3−8t\) gives the position in miles of a freight train where east is the positive direction and \(t\) is measured in hours.
a. Determine the direction the train is traveling when \(s(t)=0\).
b. Determine the direction the train is traveling when \(a(t)=0\).
c. Determine the time intervals when the train is slowing down or speeding up.
34) The following graph shows the position \(y=s(t)\) of an object moving along a straight line.
a. Use the graph of the position function to determine the time intervals when the velocity is positive, negative, or zero.
b. Sketch the graph of the velocity function.
c. Use the graph of the velocity function to determine the time intervals when the acceleration is positive, negative, or zero.
d. Determine the time intervals when the object is speeding up or slowing down.
- Answer
- a. Velocity is positive on \((0,1.5)∪(6,7)\), negative on \((1.5,2)∪(5,6)\), and zero on \((2,5)\).
b.
c. Acceleration is positive on \((5,7)\), negative on \((0,2)\), and zero on \((2,5)\).
d. The object is speeding up on \((6,7)∪(1.5,2)\) and slowing down on \((0,1.5)∪(5,6)\).
35) A car driving along a freeway with traffic has traveled \(s(t)=t^3−6t^2+9t\) 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.
36) [Technology Required] According to Newton’s law of universal gravitation, the force \(F\) between two bodies of constant mass \(m_1\) and \(m_2\) is given by the formula \(F=\dfrac{Gm_1m_2}{d^2}\), 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 \(G=6.67×10^{−11} \text{Nm}^2/\text{kg}^2\), on two bodies 10 meters apart, each with a mass of 1000 kilograms.
- Answer
- a. \(F^{\prime}(d)=\dfrac{−2Gm_1m_2}{d_3}\)
b. \(−1.33×10^{−7}\) N/m
37) The centripetal force of an object of mass m is given by \(F(r)=\frac{mv^2}{r}\), where \(v\) is the speed of rotation and \(r\) is the distance from the center of rotation.
a. Find the rate of change of centripetal force with respect to the distance from the center of rotation.
b. Find the rate of change of centripetal force of an object with mass 1000 kilograms, velocity of 13.89 m/s, and a distance from the center of rotation of 200 meters.
38) [Technology Required] Consider an astronaut on a large planet in another galaxy. To learn more about the composition of this planet, the astronaut drops an electronic sensor into a deep trench. The sensor transmits its vertical position every second in relation to the astronaut’s position. The summary of the falling sensor data is displayed in the following table.
| Time after dropping (s) | Position (m) |
| 0 | 0 |
| 1 | −1 |
| 2 | −2 |
| 3 | −5 |
| 4 | −7 |
| 5 | −14 |
a. Using a calculator or computer program, find the best-fit quadratic curve to the data.
b. Find the derivative of the position function and explain its physical meaning.
c. Find the second derivative of the position function and explain its physical meaning.
- Answer
- a. \(p(t)=−0.6071x^2+0.4357x−0.3571\)
b. \(p^{\prime}(t)=−1.214x+0.4357\). This is the velocity of the sensor.
c. \(p^{\prime \prime}(t)=−1.214\). This is the acceleration of the sensor; it is a constant acceleration downward.
Applications of the Derivative: Business
- Revenue is the product of the price of the item and the number of items sold. Mathematically, \( R(n) = p(n) \cdot n \), where \( p(n) \) is the price-demand function (the price the company charges when the market demands \( n \) items) and \( n \) is the number of items sold.
- Profit is the difference between the revenue from sales and the cost of producing an item. Mathematically, \( P(n) = R(n) - C(n) \), where \( R(n) \) is the revenue function and \( C(n) \) is the cost function.
- Marginal cost is the change in the total cost that arises when the quantity produced is incremented. That is, it is the change in cost with respect to the number of items produced. Mathematically, \( M_C(n) = \frac{dC}{dn} \), where \( C(n) \) is the cost function (i.e., the cost of producing \( n \) items).
- Marginal profit is the change in the total profit that arises when the quantity produced is incremented. That is, it is the change in profit with respect to the number of items produced. Mathematically, \( M_P(n) = \frac{dP}{dn} \), where \( P(n) \) is the profit function.
- Marginal revenue is the change in the total revenue that arises when the quantity produced is incremented. That is, it is the change in revenue with respect to the number of items produced. Mathematically, \( M_R(n) = \frac{dR}{dn} \), where \( R(n) \) is the revenue function.
39) 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 \(C^{\prime}(2)\) and explain its meaning.
40) The cost function, in dollars, of a company that manufactures food processors is given by \(C(x)=200+\frac{7}{x}+\frac{x}{27}\), where \(x\) is the number of food processors manufactured.
a. Find the marginal cost function.
b. Find the marginal cost of manufacturing 12 food processors.
c. Find the actual cost of manufacturing the thirteenth food processor.
41) The price p (in dollars) and the demand \(x\) for a certain digital clock radio is given by the price–demand function \(p=10−0.001x\).
a. Find the revenue function \(R(x)\)
b. Find the marginal revenue function.
c. Find the marginal revenue at \(x=2000\) and \(5000\).
- Answer
- a. \(R(x)=10x−0.001x^2\)
b. \( R^{\prime}(x)=10−0.002x\)
c. $6 per item, $0 per item
42) [Technology Required] A profit is earned when revenue exceeds cost. Suppose the profit function for a skateboard manufacturer is given by \(P(x)=30x−0.3x^2−250\), where \(x\) is the number of skateboards sold.
a. Find the exact profit from the sale of the thirtieth skateboard.
b. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard.
43) [Technology Required] In general, the profit function is the difference between the revenue and cost functions: \(P(x)=R(x)−C(x)\).
Suppose the price-demand and cost functions for the production of cordless drills is given respectively by \(p=143−0.03x\) and \(C(x)=75,000+65x\), where \(x\) is the number of cordless drills that are sold at a price of \(p\) dollars per drill and \(C(x)\) is the cost of producing \(x\) cordless drills.
a. Find the marginal cost function.
b. Find the revenue and marginal revenue functions.
c. Find \(R^{\prime}(1000)\) and \(R^{\prime}(4000)\). Interpret the results.
d. Find the profit and marginal profit functions.
e. Find \(P^{\prime}(1000)\) and \(P^{\prime}(4000)\). Interpret the results.
- Answer
- a. \(C^{\prime}(x)=65\)
b. \(R(x)=143x−0.03x^2\),\(R′(x)=143−0.06x\)
c. \(R^{\prime}(1000)=83, \quad R^{\prime}(4000) = −97\). At a production level of 1000 cordless drills, revenue is increasing at a rate of $83 per drill; at a production level of 4000 cordless drills, revenue is decreasing at a rate of $97 per drill.
d. \(P(x)=−0.03x^2+78x−75000, \quad P′(x)=−0.06x+78\)
e. \(P^{\prime}(1000)=18, \quad P^{\prime}(4000) =−162\). At a production level of 1000 cordless drills, profit is increasing at a rate of $18 per drill; at a production level of 4000 cordless drills, profit is decreasing at a rate of $162 per drill.
Applications of the Derivative: Population Models
44) A small town in Ohio commissioned an actuarial firm to conduct a study that modeled the rate of change of the town’s population. The study found that the town’s population (measured in thousands of people) can be modeled by the function \(P(t)=−\frac{1}{3}t^3+64t+3000\), where \(t\) is measured in years.
a. Find the rate of change function \(P^{\prime}(t)\) of the population function.
b. Find \(P^{\prime}(1),\; P^{\prime}(2),\; P^{\prime}(3)\), and \(P^{\prime}(4)\). Interpret what the results mean for the town.
c. Find \(P^{\prime \prime}(1),\; P^{\prime \prime}(2),\; P^{\prime \prime}(3)\), and \(P^{\prime \prime}(4)\). Interpret what the results mean for the town’s population.
[Technology Required] In Exercises 45 - 46, use the following table concerning the population (in millions) of London by decade in the 19th century.
| Year Since 1800 | Population (millions) |
| 1 | 0.8975 |
| 11 | 1.040 |
| 21 | 1.264 |
| 31 | 1.516 |
| 41 | 1.661 |
| 51 | 2.000 |
| 61 | 2.634 |
| 71 | 3.272 |
| 81 | 3.911 |
| 91 | 4.422 |
Population of LondonSource: http://en.Wikipedia.org/wiki/Demographics_of_London
45)
a. Using a calculator or a computer program, find the best-fit linear function to measure the population.
b. Find the derivative of the equation in a. and explain its physical meaning.
c. Find the second derivative of the equation and explain its physical meaning.
Answer
a. \(P(t)=0.03983t+0.4280\)
b. \(P^{\prime}(t)=0.03983\). The population is increasing.
c. \(P^{\prime \prime}(t)=0\). The rate at which the population is increasing is constant.
46)
a. Using a calculator or a computer program, find the best-fit quadratic curve through the data.
b. Find the derivative of the equation and explain its physical meaning.
c. Find the second derivative of the equation and explain its physical meaning.