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  • Calculus: Early Transcendentals
  • Integrals

Calculus: Early Transcendentals

James Stewart

Chapter 5

Integrals - all with Video Answers

Educators

+ 8 more educators

Section 3

The Fundamental Theorem of Calculus

00:32

Problem 1

Explain exactly what is meant by the statement that "differentiation and integration are inverse processes."

Amrita Bhasin
Amrita Bhasin
Numerade Educator
12:51

Problem 2

Let $ \displaystyle g(x) = \int^x_0 f(t) \,dt $, where $ f $ is the function whose graph is shown.

(a) Evaluate $ g(x) $ for $ x $ = 0, 1, 2, 3, 4, 5, and 6.
(b) Estimate $ g(7) $.
(c) Where does $ g $ have a maximum value? Where does it have a minimum value?
(d) Sketch a rough graph of $ g $.

Chris Trentman
Chris Trentman
Numerade Educator
08:18

Problem 3

Let $g(x)=\int_{0}^{x} f(t) d t,$ where $f$ is the function whose graph is shown.
(a) Evaluate $g(x)$ for $x=0,1,2,3,4,5,$ and 6 .
(b) Estimate $g(7)$
(c) Where does $g$ have a maximum value? Where does it have a minimum value?
(d) Sketch a rough graph of $g$.

Chris Trentman
Chris Trentman
Numerade Educator
12:30

Problem 4

Let $ \displaystyle g(x) = \int^x_0 f(t) \,dt $, where $ f $ is the function whose graph is shown.

(a) Evaluate $ g(0) $ and $ g(6) $
(b) Estimate $ g(x) $ for $ x $ = 1, 2, 3, 4, and 5.
(c) On what interval is $ g $ increasing?
(d) Where does $ g $ have a maximum value?
(e) Sketch a rough graph of $ g $.
(f) Use the graph in part (e) to sketch the graph of $ g'(x) $. Compare with the graph of $ f $.

Chris Trentman
Chris Trentman
Numerade Educator
08:25

Problem 5

Sketch the area represented by $ g(x) $. Then find $ g'(x) $ in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.

$ \displaystyle g(x) = \int^x_1 t^2\,dt $

Leon Druch
Leon Druch
Numerade Educator
05:30

Problem 6

Sketch the area represented by $ g(x) $. Then find $ g'(x) $ in two ways: (a) by using Part 1 of the Fundamental Theorem and (b) by evaluating the integral using Part 2 and then differentiating.

$ \displaystyle g(x) = \int^x_0 (2 + \sin t)\,dt $

Chris Trentman
Chris Trentman
Numerade Educator
02:48

Problem 7

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle g(x) = \int^x_0 \sqrt{t + t^3} \,dt $

Yuki Hotta
Yuki Hotta
Numerade Educator
01:31

Problem 8

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle g(x) = \int^x_1 \ln (1 + t^2) \,dt $

Yuki Hotta
Yuki Hotta
Numerade Educator
00:25

Problem 9

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle g(s) = \int^s_5 (t - t^2)^8 \,dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:24

Problem 10

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle h(u) = \int^u_0 \frac{\sqrt{t}}{t + 1} \,dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:25

Problem 11

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle F(x) = \int^0_x \sqrt{1 + \sec t} \,dt $

$$ \biggl[ \textit{Hint:} \int^0_x \sqrt{1 + \sec t} \,dt = - \int^x_0 \sqrt{1 + \sec t} \,dt \biggr] $$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:43

Problem 12

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle R(y) = \int^2_y t^3 \sin t \,dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:30

Problem 13

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle h(x) = \int^{e^x}_1 \ln t \,dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:23

Problem 14

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle h(x) = \int^{\sqrt{x}}_1 \frac{z^2}{z^4 + 1} \,dz $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:49

Problem 15

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle y = \int^{3x + 2}_1 \frac{t}{1 + t^3} \,dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:34

Problem 16

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle y = \int^{x^4}_0 \cos^2 \theta \,d\theta $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:34

Problem 17

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle y = \int^{\pi/4}_{\sqrt{x}} \theta \tan \theta \,d\theta $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
03:05

Problem 18

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

$ \displaystyle y = \int^1_{\sin x} \sqrt{1 + t^2} \,dt $

Chris Trentman
Chris Trentman
Numerade Educator
04:53

Problem 19

Evaluate the integral.

$ \displaystyle \int^3_1 (x^2 + 2x - 4) \,dx $

MB
Michelle Bartolo
Numerade Educator
00:31

Problem 20

Evaluate the integral.

$ \displaystyle \int^1_{-1} x^{100} \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:26

Problem 21

Evaluate the integral.

$ \displaystyle \int^2_0 \biggl(\frac{4}{5}t^3 - \frac{3}{4}t^2 + \frac{2}{5}t \biggr) \,dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:28

Problem 22

Evaluate the integral.

$ \displaystyle \int^1_0 (1 - 8v^3 + 16v^7) \,dv $

Yuki Hotta
Yuki Hotta
Numerade Educator
00:28

Problem 23

Evaluate the integral.

$ \displaystyle \int^9_1 \sqrt{x} \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:28

Problem 24

Evaluate the integral.

$ \displaystyle \int^8_1 x^{-2/3} \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:29

Problem 25

Evaluate the integral.

$ \displaystyle \int^{\pi}_{\pi/6} \sin \theta \,d\theta $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:31

Problem 26

Evaluate the integral.

$ \displaystyle \int^5_{-5} e \,dx $

Yuki Hotta
Yuki Hotta
Numerade Educator
00:40

Problem 27

Evaluate the integral.

$ \displaystyle \int^1_0 (u + 2) (u - 3) \,du $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:53

Problem 28

Evaluate the integral.
$\int_{0}^{4}(4-t) \sqrt{t} d t$

Mary Wakumoto
Mary Wakumoto
Numerade Educator
00:39

Problem 29

Evaluate the integral.

$ \displaystyle \int^4_1 \frac{2 + x^2}{\sqrt{x}} \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:32

Problem 30

Evaluate the integral.

$ \displaystyle \int^2_{-1} (3u - 2)(u + 1) \,du $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:35

Problem 31

Evaluate the integral.

$ \displaystyle \int^{\pi/2}_{\pi/6} \csc t \cot t \,dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:28

Problem 32

Evaluate the integral.

$ \displaystyle \int^{\pi/3}_{\pi/4} \csc^2 \theta \,d\theta $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:26

Problem 33

Evaluate the integral.

$ \displaystyle \int^1_0 (1 + r)^3 \,dr $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:53

Problem 34

Evaluate the integral.

$ \displaystyle \int^3_0 (2\sin x - e^x) \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:29

Problem 35

Evaluate the integral.

$ \displaystyle \int^2_1 \frac{v^3 + 3v^6}{v^4} \,dv $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:50

Problem 36

Evaluate the integral.

$ \displaystyle \int^{18}_1 \sqrt{\frac{3}{z}} \,dz $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:47

Problem 37

Evaluate the integral.

$ \displaystyle \int^1_0 (x^e + e^x) \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:38

Problem 38

Evaluate the integral.

$ \displaystyle \int^1_0 \cosh t \,dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:04

Problem 39

Evaluate the integral.

$ \displaystyle \int^{\sqrt{3}}_{1/\sqrt{3}} \frac{8}{1 + x^2} \,dx $

Mary Wakumoto
Mary Wakumoto
Numerade Educator
00:45

Problem 40

Evaluate the integral.

$ \displaystyle \int^3_1 \frac{y^3 - 2y^2 - y}{y^2} \,dy $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:25

Problem 41

Evaluate the integral.

$ \displaystyle \int^4_0 2^s \,ds $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:48

Problem 42

Evaluate the integral.

$ \displaystyle \int^{1/\sqrt{2}}_{1/2} \frac{4}{\sqrt{1 - x^2}} \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:03

Problem 43

Evaluate the integral.

$ \displaystyle \int^{\pi}_0 f(x) \,dx $ where $ f(x) = \left\{
\begin{array}{ll}
\sin x & \mbox{if $ 0 \le x < \pi/2 $}\\
\cos x & \mbox{if $ \pi/2 \le x \le \pi $}
\end{array} \right.$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:45

Problem 44

Evaluate the integral.

$ \displaystyle \int^2_{-2} f(x) \,dx $ where $ f(x) = \left\{
\begin{array}{ll}
2 & \mbox{if $ -2 \le x \le 0 $}\\
4 - x^2 & \mbox{if $ 0 < x \le 2 $}
\end{array} \right.$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:46

Problem 45

Sketch the region enclosed by the given curves and calculate its area.

$ y = \sqrt{x} $, $ y = 0 $, $ x = 4 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:36

Problem 46

Sketch the region enclosed by the given curves and calculate its area.

$ y = x^3 $, $ y = 0 $, $ x = 1 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
04:06

Problem 47

Sketch the region enclosed by the given curves and calculate its area.

$ y = 4 - x^2 $, $ y = 0 $

Leon Druch
Leon Druch
Numerade Educator
00:40

Problem 48

Sketch the region enclosed by the given curves and calculate its area.

$ y = 2x - x^2 $, $ y = 0 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:50

Problem 49

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = \sqrt[3]{x} $, $ 0 \le x \le 27 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:50

Problem 50

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = x^{-4} $, $ 1 \le x \le 6 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:31

Problem 51

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = \sin x $, $ 0 \le x \le \pi $

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
01:01

Problem 52

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area.

$ y = \sec^2 x $, $ 0 \le x \le \pi/3 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:48

Problem 53

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.

$ \displaystyle \int^2_{-1} x^3 \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:44

Problem 54

Evaluate the integral and interpret it as a difference of areas. Illustrate with a sketch.

$ \displaystyle \int^{2\pi}_{\pi/6} \cos x \,dx $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:41

Problem 55

What is wrong with the equation?

$ \displaystyle \int^1_{-2} x^{-4} \, dx = \frac{x^{-3}}{-3} \Bigg]^1_{-2} = -\frac{3}{8} $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
03:18

Problem 56

What is wrong with the equation?

$ \displaystyle \int^2_{-1} \frac{4}{x^3} \, dx = -\frac{2}{x^2} \Bigg]^2_{-1} = -\frac{3}{2} $

Yuki Hotta
Yuki Hotta
Numerade Educator
02:10

Problem 57

What is wrong with the equation?

$ \displaystyle \int^{\pi}_{\pi/3} \sec \theta \tan \theta \, d\theta = \sec \theta \Bigg]^{\pi}_{\pi/3} = -3 $

Bobby Barnes
Bobby Barnes
University of North Texas
00:31

Problem 58

What is wrong with the equation?

$ \displaystyle \int^{\pi}_{0} \sec^2 x \, dx = \tan x \Bigg]^{\pi}_{0} = 0 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:56

Problem 59

Find the derivative of the function.

$ g(x) = \displaystyle \int^{3x}_{2x} \frac{u^2 - 1}{u^2 + 1} \, du $

$ \displaystyle \Biggl[ Hint: \int^{3x}_{2x} f(u) \, du = \int^0_{2x} f(u) \, du + \int^{3x}_0 f(u) \, du \Biggr] $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:00

Problem 60

Find the derivative of the function.

$ g(x) = \displaystyle \int^{1 + 2x}_{1 - 2x} t \sin t \, dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:28

Problem 61

Find the derivative of the function.

$ F(x) = \displaystyle \int^{x^2}_{x} e^{t^2} \, dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:05

Problem 62

Find the derivative of the function.

$ F(x) = \displaystyle \int^{2x}_{\sqrt{x}} \arctan t \, dt $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
View

Problem 63

Find the derivative of the function.

$ y = \displaystyle \int^{\sin x}_{\cos x} \ln (1 + 2v) \, dv $

Ma. Theresa  Alin
Ma. Theresa Alin
Numerade Educator
00:44

Problem 64

If $ \displaystyle f(x) = \int^x_0 (1 - t^2) e^{t^2} \,dt $, on what interval is $ f $ increasing?

Amrita Bhasin
Amrita Bhasin
Numerade Educator
01:01

Problem 65

On what interval is the curve $$ y = \displaystyle \int^x_0 \frac{t^2}{t^2 + t + 2} \, dt $$ concave downward?

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:46

Problem 66

Let $ \displaystyle F(x) = \int^x_1 f(t) \, dt $, where $ f $ is the function whose graph is shown. Where is $ F $ concave downward?

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:44

Problem 67

Let $ \displaystyle F(x) = \int^x_2 e^{t^2} \, dt $. Find an equation of the tangent line to the curve $ y = F(x) $ at the point with $ x $-coordinate 2.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
View

Problem 68

If $ \displaystyle f(x) = \int^{\sin x}_0 \sqrt{1 + t^2} \, dt $ and $ \displaystyle g(y) = \int^y_3 f(x) \, dx $, find $ g''(\pi/6) $.

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator
00:34

Problem 69

If $ f(1) = 12 $, $ f' $ is continuous, and $ \displaystyle \int^4_1 f'(x) \, dx = 17 $, what is the value of $ f(4) $?

Amrita Bhasin
Amrita Bhasin
Numerade Educator
06:04

Problem 70

The error function $$ \displaystyle \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int^x_0 e^{-t^2} \, dt $$ is used in probability, statistics, and engineering.
(a) Show that $ \displaystyle \int^b_a e^{-t^2} \, dt = \frac{1}{2} \sqrt{\pi} [ \text{erf}(b) - \text{erf}(a) ] $.
(b) Show that the function $ y = e^{x^2} \text{erf}(x) $ satisfies the differential equation $ y' = 2xy + 2/\sqrt{\pi} $.

Chris Trentman
Chris Trentman
Numerade Educator
06:52

Problem 71

The Fresnel function $ S $ was defined in Example 3 and graphed in Figures 7 and 8.

(a) At what values of $ x $ does this function have local maximum values?
(b) On what intervals is the function concave upward?
(c) Use a graph to solve the following equation correct to two decimal places:
$$ \displaystyle \int^x_0 \sin (\pi t^2/2) \, dt = 0.2 $$

Frank Lin
Frank Lin
Numerade Educator
06:57

Problem 72

The sine integral function $$ \displaystyle \text{Si}(x) = \int^x_0 \frac{\sin t}{t} \, dt $$ is important in electrical engineering. [The integrand $ f(t) = (\sin t)/t $ is not defined when $ t = 0 $, but we know that its limit is 1 when $ t \to 0 $. So we defined $ f(0) = 1 $ and this makes $ f $ a continuous function everywhere.]

(a) Draw the graph of $ \text{Si} $.
(b) At what values of $ x $ does this function have local maximum values?
(c) Find the coordinates of the first inflection point to the right of the origin.
(d) Does this function have horizontal asymptotes?
(e) Solve the following equation correct to one decimal place:
$$ \displaystyle \int^x_0 \frac{\sin t}{t} \, dt = 1 $$

Frank Lin
Frank Lin
Numerade Educator
11:34

Problem 73

Let $ \displaystyle g(x) = \int^x_0 f(t) \, dt $, where $ f $ is the function whose graph is shown.

(a) At what values of $ x $ do the local maximum and minimum values of $ g $ occur?
(b) Where does $ g $ attain its absolute maximum value?
(c) On what intervals is $ g $ concave downward?
(d) Sketch the graph of $ g $.

Chris Trentman
Chris Trentman
Numerade Educator
09:37

Problem 74

Let $ \displaystyle g(x) = \int^x_0 f(t) \, dt $, where $ f $ is the function whose graph is shown.

(a) At what values of $ x $ do the local maximum and minimum values of $ g $ occur?
(b) Where does $ g $ attain its absolute maximum value?
(c) On what intervals is $ g $ concave downward?
(d) Sketch the graph of $ g $.

Chris Trentman
Chris Trentman
Numerade Educator
00:43

Problem 75

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on $ [0, 1] $.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n \biggl( \frac{i^4}{n^5} + \frac{i}{n^2} \biggr) $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:51

Problem 76

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on $ [0, 1] $.

$ \displaystyle \lim_{n \to \infty} \frac{1}{n} \biggl( \sqrt{\frac{1}{n}} + \sqrt{\frac{2}{n}} + \sqrt{\frac{3}{n}} + \cdots + \sqrt{\frac{n}{n}} \biggr) $

Bobby Barnes
Bobby Barnes
University of North Texas
01:02

Problem 77

Justify (3) for the case $ h < 0 $.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:55

Problem 78

If $ f $ is continuous and $ g $ and $ h $ are differentiable functions, find a formula for
$$ \displaystyle \frac{d}{dx} \int^{h(x)}_{g(x)} f(t) \, dt $$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
05:09

Problem 79

(a) Show that $ 1 \le \sqrt{1 + x^3} \le 1 + x^3 $ for $ x \ge 0 $.
(b) Show that $ \displaystyle 1 \le \int^1_0 \sqrt{1 + x^3} \, dx \le 1.25 $

Chris Trentman
Chris Trentman
Numerade Educator
05:18

Problem 80

(a) Show that $ \cos (x^2) \ge \cos x $ for $ 0 \le x \le 1 $.
(b) Deduce that $ \displaystyle \int^{\pi/6}_0 \cos (x^2) \, dx \ge \frac{1}{2} $.

Chris Trentman
Chris Trentman
Numerade Educator
00:57

Problem 81

Show that $$ 0 \le \int^{10}_5 \frac{x^2}{x^4 + x^2 + 1} \, dx \le 0.1 $$ by comparing the integrand to a simpler function.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
11:11

Problem 82

Let $ f(x) = \left\{
\begin{array}{ll}
0 & \mbox{if $ x < 0 $}\\
x & \mbox{if $ 0 \le x \le 1 $}\\
2 - x & \mbox{if $ 1 < x \le 2 $}\\
0 & \mbox{if $ x > 2 $}
\end{array} \right.$
and $$ g(x) = \int^x_0 f(t) \, dt $$

(a) Find an expression for $ g(x) $ similar to the one for $ f(x) $.
(b) Sketch the graphs of $ f $ and $ g $.
(c) Where is $ f $ differentiable? Where is $ g $ differentiable?

Chris Trentman
Chris Trentman
Numerade Educator
00:51

Problem 83

Find a function $ f $ and a number $ a $ such that
$ \displaystyle 6 + \int^x_a \frac{f(t)}{t^2} \, dt = 2 \sqrt{x} $ for all $ x > 0 $

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:39

Problem 84

The area labeled $ B $ is three times the area labeled $ A $. Express $ b $ in terms of $ a $.

Aparna Shakti
Aparna Shakti
Numerade Educator
10:50

Problem 85

A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate $ f = f(t) $, where $ t $ is the time measured in months since its last overhaul. Because a fixed cost $ A $ is incurred each time the machine is overhauled, the company wants to determine the optimal time $ T $ (in months) between overhauls.

(a) Explain why $ \displaystyle \int^t_0 f(s) ds $ represents the loss in value of the machine over the period of time $ t $ since the last overhaul.
(b) Let $ C = C(t) $ be given by $$ C(t) = \frac{1}{t} \Biggl[ A + \int^t_0 f(s) ds \Biggr] $$
What does $ C $ represent and why would the company want to minimize $ C $?
(c) Show that $ C $ has a minimum value at the numbers $ t = T $ where $ C(T) = f(T) $.

Chris Trentman
Chris Trentman
Numerade Educator
22:11

Problem 86

A high-tech company purchases a new computing system whose initial value is $ V $. The system will depreciate at the rate $ f = f(t) $ and will accumulate maintenance costs at the rate $ g = g(t) $, where $ t $ is the time measured in months. The company wants to determine the optimal time to replace the system.
(a) Let $$ C(t) = \frac{1}{t} \int^t_0 [f(s) + g(s)] \, ds $$
Show that the critical numbers of $ C $ occur at the numbers $ t $ where $ C(t) = f(t) + g(t) $.
(b) Suppose that
$ f(t) = \left\{
\begin{array}{ll}
\frac{V}{15} - \frac{V}{450}t & \mbox{if $ 0 < t \le 30 $}\\
0 & \mbox{if $ t > 30 $}
\end{array} \right.$
and $ g(t) = \frac{Vt^2}{12,900} $ $ t > 0 $
Determine the length of time $ T $ for the total depreciation $ \displaystyle D(t) = \int^t_0 f(s) \, ds $ to equal the initial value $ V $.
(c) Determine the absolute minimum of $ C $ on $ (0, T] $.
(d) Sketch the graphs of $ C $ and $ f + g $ in the same coordinate system, and verify the result in part (a) in this case.

Chris Trentman
Chris Trentman
Numerade Educator

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