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Chapter 4: Q. 4 (page 375)

Use the Fundamental Theorem of Calculus to find an equation for A(x) that does not involve an integral.

Short Answer

Expert verified

A(x)≈∑nf1x1x

Step by step solution

01

Step 1. Given information is:

Suppose A(x) is a function that for each x>0 is equal to area under the graph of f(t)=t2 from 0 to x.

02

Step 2. Determining A(x)

Theareaunderthecurveisgivenby:A(x)=f1x1x+f1x1x+.....+f1x1xA(x)≈∑nf1x1xThus,A(x)≈∑nf1x1x

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Most popular questions from this chapter

Shade in the regions between the two functions shown here on the intervals (a) [−2, 3]; (b) [−1, 2]; and (c) [1, 3]. Which of these regions has the largest area? The smallest?

Consider the sequence A(1), A(2), A(3),.....,A(n) write our the sequence up to n. What do you notice?

For each function f and interval [a, b] in Exercises 27–33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.

f(x)=sin(x),[a,b]=[0,π], n = 3 with

a) Trapezoid sim b) Upper sum

Suppose f is a function whose average value on [-3,1] is

-2 and whose average rate of change on the same in-

terval is 4. Sketch a possible graph for f . Illustrate the

average value and the average rate of change on your

graph of f .

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A function f for which the signed area between f and the x-axis on [0, 4] is zero, and a different function g for which the absolute area between g and the x-axis on [0, 4] is zero.

(b) A function f whose signed area on [0, 5] is less than its signed area on [0, 3].

(c) A function f whose average value on [−1, 6] is negative while its average rate of change on the same interval is positive.
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