1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such...

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1 3.2 The Mean Value Theorem

Transcript of 1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such...

Page 1: 1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

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3.2 The Mean Value Theorem

Page 2: 1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

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Rolle’s Theorem

Page 3: 1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

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Figure 1 shows the graphs of four such functions.

Figure 1

(c)

(b)

(d)

(a)

Examples:

Page 4: 1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

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The Mean Value Theorem

This theorem is an extension of Rolle’s Theorem

Page 5: 1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

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Example

Show that f(x) satisfies the Mean Value Theorem on [a,b]

f (x) = x3 – x, Interval: a = 0, b = 2.

Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 2] and differentiable on (0, 2).

Therefore, by the Mean Value Theorem, there is a number c in (0, 2) such that

f (2) – f (0) = f (c)(2 – 0)

Page 6: 1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

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Example - proof

f (2) = 6,

f (0) = 0, and

f (x) = 3x2 – 1,

so this equation becomes:

6 = (3c2 – 1)2

= 6c2 – 2

which gives that is, c = But c must lie in

(0, 2), so

Page 7: 1 3.2 The Mean Value Theorem. 2 Rolle’s Theorem 3 Figure 1 shows the graphs of four such functions. Figure 1 (c) (b) (d) (a) Examples:

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The Mean Value TheoremThe Mean Value Theorem can be used to establish some of the basic facts of differential calculus.


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