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The Fundamental Theorem of Calculus

5.4 The Fundamental Theorem of Calculus Copyright © Cengage Learning. All rights reserved.

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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus gives the precise inverse relationship between the derivative and the integral. The first part of the Fundamental Theorem deals with functions defined by an equation of the form where f is a continuous function on [a, b] and x varies between a and b.

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The Fundamental Theorem of Calculus

Observe that g depends only on x, which appears as the variable upper limit in the integral. If x is a fixed number, then the integral is a definite number. If we then let x vary, the number also varies and defines a function of x denoted by g(x).

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The Fundamental Theorem of Calculus

If f happens to be a positive function, then g(x) can be interpreted as the area under the graph f of from a to x, where x can vary from a to b. (Think of g as the “area so far” function; see Figure 1.) Figure 1

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Example 1 If f is the function whose graph is shown in Figure 2 and

, find the values of g(0), g(1), g(2), g(3), g(4), and g(5). Then sketch a rough graph of g. Figure 2

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Example 1 – Solution First we notice that

From Figure 3 we see that g(1) is the area of a triangle: Figure 3

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Example 1 – Solution cont’d To find g(2) we add to g(1) the area of a rectangle: Figure 3

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Example 1 – Solution cont’d We estimate that the area under f from 2 to 3 is about 1.3, so Figure 3

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Example 1 – Solution cont’d For t > 3, f (t) is negative and so we start subtracting areas: Figure 3

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Example 1 – Solution cont’d Figure 3

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Example 1 – Solution cont’d We use these values to sketch the graph of g in Figure 4. Figure 4

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Example 1 – Solution cont’d Notice that, because f (t) is positive for t < 3, we keep adding area for t < 3 and so g is increasing up to x = 3, where it attains a maximum value. For x > 3, g decreases because f (t) is negative.

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The Fundamental Theorem of Calculus

If g is defined as the integral of f by Equation 1, then g turns out to be an antiderivative of f. Hence g’ = f. To see why this might be generally true we consider any continuous function f with Then can be interpreted as the area under the graph of f from a to x, as in Figure 1. Figure 1

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The Fundamental Theorem of Calculus

In order to compute g (x) from the definition of derivative we first observe that, for h > 0, g(x + h) – g(x) is obtained by subtracting areas, so it is the area under the graph of f from x to x + h (the shaded area in Figure 5). Figure 5

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The Fundamental Theorem of Calculus

For small h you can see from the figure that this area is approximately equal to the area of the rectangle with height f (x) and width h: so Intuitively, we therefore expect that

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The Fundamental Theorem of Calculus

The fact that this is true, even when f is not necessarily positive, is the first part of the Fundamental Theorem of Calculus.

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The Fundamental Theorem of Calculus

Using Leibniz notation for derivatives, we can write FTC1 as when f is continuous. Roughly speaking, this equation says that if we first integrate f and then differentiate the result, we get back to the original function f.

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Example 5 Find Solution: Here we have to be careful to use the Chain Rule in conjunction with Part 1 of the Fundamental Theorem. Let Then

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Example 5 – Solution cont’d

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Differentiation and Integration as Inverse Processes

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Differentiation and Integration as Inverse Processes

We now bring together the two parts of the Fundamental Theorem. We regard Part 1 as fundamental because it relates integration and differentiation. But the Evaluation Theorem also relates integrals and derivatives, so we rename it as Part 2 of the Fundamental Theorem.

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Average Value of a Function

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Average Value of a Function

It’s easy to calculate the average value of finitely many numbers y1, y2, , yn: In general, let’s try to compute the average value of a function We start by dividing the interval [a, b] into n equal subintervals, each with length

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Average Value of a Function

Then we choose points in successive subintervals and calculate the average of the numbers : Since we can write and the average value becomes

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Average Value of a Function

If we let n increase, we would be computing the average value of a large number of closely spaced values. The limiting value is by the definition of a definite integral.

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Average Value of a Function

Therefore, we define the average value of f on the interval [a, b] as

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Example 6 Find the average value of the function on the interval [–1, 2]. Solution: With a = –1 and b = 2 we have

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Average Value of a Function

In general, is there a number c at which the value of a function f is exactly equal to the average value of the function, that is, The following theorem says that this is true for continuous functions.

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Example 7 Since is continuous on the interval [–1, 2], the Mean Value Theorem for Integrals says there is a number c in [–1, 2] such that In this particular case we can find c explicitly. From Example 6 we know that so the value of c satisfies

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Example 7 cont’d Therefore so Thus in this case there happen to be two numbers c = 1 in the interval [–1, 2] that work in the Mean Value Theorem for Integrals.

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