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Chapter 4: Q. 56 (page 386)

For each function f and interval[a, b] in Exercises 56鈥�67, use definite integrals and the Fundamental Theorem of Calculus to find the exact average value of f from x = a to x = b. Then use a graph of f to verify that your answer is reasonable.

f(x)=x-1,-1,3

Short Answer

Expert verified

The exact average value of f is 0and it is verified from the graph of f.

The graph is

Step by step solution

01

Step 1. Given Information.

The given function and interval isf(x)=x-1,-1,3.

02

Step 2. Finding the exact average value.

To find the exact average value of f from x=atox=b,we will use the formula: 1b-a鈭�abf(x)dx.

Thus,

1b-a鈭�abf(x)dx=13-(-1)鈭�-13(x-1)dx=14x22-x-13=14322-3--122-(-1)=1492-3-12-1=149-6-1-22=140=0

03

Step 3. Verification.

The graph of f is

From the graph, we can depict that the average value is 0.Thus, the answer is right.

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Most popular questions from this chapter

Show thatF(x)=12ex2x2-1is an anti-derivative off(x)=x3ex2.

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

鈭�13(x+1)2dx

Find the sum or quantity without completely expanding or calculating any sums.

Given鈭�k=310ak=12and鈭�k=210ak=23, finda2.

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].

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

limn鈫�鈭�鈭�k=1nk+12n3-1
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