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Chapter 4: Q. 15 (page 399)

Let fbe the function shown earlier at the right, and define A(x)=鈭�0xf(t)dt. On which interval(s) is Apositive? Negative? Increasing? Decreasing? Sketch a rough graph of A.

Short Answer

Expert verified

The graph of fis positive from 0,1and 5,6so the graph of Ais increasing from 0,1and 5,6.

The graph of fis negative from 1,5so the graph of Ais decreasing from 1,5.

The graph of Ais,

Step by step solution

01

Step 1. Given information

A(x)=鈭�0xf(t)dt.

The given graph is,

02

Step 2.The objective is to determine the intervals the graph of A is positive, negative increasing and decreasing.

The graph of Ais positive on [0,2]and negative on 2,6since the slope of the graph shown is negative on 0,2and positive on 2,6.

The graph of fis positive from 0,1and 5,6so the graph of Ais increasing from 0,1and 5,6.

The graph of fis negative from 1,5so the graph of Ais decreasing from 1,5.

03

Step 3. The graph of A is shown below:

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Most popular questions from this chapter

Properties of addition: State the associative law for addition, the commutative law for addition, and the distributive law for multiplication over addition of real numbers. (You may have to think back to a previous algebra course.)

Without using absolute values, how many definite integrals would we need in order to calculate the area between the graphs of f(x) = sin x and g(x) = 12on [-蟺2,2蟺] ?

Verify that鈭�lnxdx=x(lnx-1)+C(Do not try to solve the integral from scratch.

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: tanx=sinxcosx).

鈭�2xlnx-xlnx2dx.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The absolute area between the graph of f and the x-axis on [a, b] is equal to|鈭�abf(x)dx|.

(b) True or False: The area of the region between f(x) = x 鈭� 4 and g(x) = -x2on the interval [鈭�3, 3] is negative.

(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.

(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by 鈭�ab(f(x)-g(x))dx.

(e) True or False: The average value of the function f(x) = x2-3 on [2, 6] is

f(6)+f(2)2= 33+12= 17.

(f) True or False: The average value of the function f(x) = x2-3on [2, 6] is f(6)-f(2)4= 33-14= 8.

(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].

(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].
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