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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

15-16 Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant fine through the endpoints, and the tangent line at (c, f(c)). Are the secant line and the tangent line parallel?

f ( x ) = x ,    [ 0 , 4 ]

To determine

To:

(i) Find the number c that satisfies Mean value theorem on given interval.

(ii) Graph the functions, the secant line through the end points and tangent lines at c, fc.

(iii) Check whether secant line and tangent line parallel.

Explanation

1) Concept:

i. Using the Mean Value Theorem verify the result

ii. If slopes of secant and tangent are same then they are parallel.

2) Theorem:

Mean Value Theorem: Let f be a function that satisfies the following hypotheses:(i) f  is continuous on the closed interval [a, b].(ii) f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that f'c=fb -f(a)b - a or equivalently, f(b)  f(a) = f(c) (b  a).

3) Formula:

(i) Slope of tangent

= mtanα=dydx

(ii) Equation of tangent through a, b with slope m is y-b=m (x-a)

(iii) Slope of line through points (a, f(a)) and (b, f(b)) is given by,

m=fb-f(a)b - a

4) Given:

fx=x ,   [0, 4]

5) Calculations:

(i) As fx=x is continuous and differentiable on for all x0, therefore

fx is continuous on [0,4] and differentiable at (0,1).

Therefore, by mean value theorem,

f'c=f4 -f(0) 4 - 0

f'x=12x

12c=4-04-0

1c=22

c=1

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