BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 4, Problem 4RCC

(a)

To determine

To state: The Rolle ’s Theorem.

Expert Solution

Explanation of Solution

The Rolle’s theorem states that, “if function f satisfies the following three hypotheses,

(1) Function f is continuous on [a,b] .

(2) Function f is differentiable on (a,b)

(3) f(a)=f(b)

Then, there is a number c in (a,b) such that f(c)=0 ”.

(b)

To determine

To state: The mean value theorem and interpret its geometrical meaning.

Expert Solution

Explanation of Solution

The mean value theorem states that, “if a function f satisfies the following hypotheses,

(1)Function f is continuous on [a,b] .

(2) Function f is differentiable on (a,b)

Then, there is a number c in (a,b) such that f(c)=f(b)f(a)ba ”.

Geometrically mean value theorem can be explained as follow.

The slope of the secant line joining the points (a,f(a)) and (b,f(b)) to the curve is, f(b)f(a)ba .

Therefore, the equation f(b)f(a)ba=f(c) represents the tangent to the function y=f(x) .

Diagrammatic representation of the function is shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4, Problem 4RCC

From Figure 1, it is noticed that secant line becomes tangent with slope f(b)f(a)ba .

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