(a) Explanation Given Information: The provided function is f ( x ) = 4 x − x 2 and the provided point is x = 5 . Formula Used: According to sum rule of derivatives, If f ( x ) = u ( x ) + v ( x ) then f ′ ( x ) = u ′ ( x ) + v ′ ( x ) . According to power rule, If f ( x ) = x n then f ′ ( x ) = n x n − 1 . The equation of tangent at any point ( x 1 , y 1 ) is given by y − y 1 = y ′ ( x − x 1 ) Calculation: The provided function is f ( x ) = 4 x − x 2 . The slope of the tangent at any point of the curve is given by derivative of the function. Therefore, to calculate the slope of the tangent, calculate the derivative of the function. Apply the sum rule of derivatives, f ( x ) = 4 x − x 2 f ′ ( x ) = d d x ( 4 x − x 2 ) y ′ = d d x ( 4 x ) − d d x ( x 2 ) Use the power rule to find the derivative of the function, y ′ = d d x ( 4 x ) − d d x ( x 2 ) = ( 4 ( 1 ) x 1 − 1 ) + ( 2 x 2 − 1 ) = 4 − 2 x To determine the slope of the tangent at x = 5 (b) To determine To graph: The function f ( x ) = 4 x − x 2 and the tangent line y = − 6 x + 25 . (c) To determine If both the curves f ( x ) = 4 x − x 2 and y = − 6 x + 25 become indistinguishable at x = 5 .

12th Edition

ISBN: 9780840058225

Author: HARSHBARGER

Publisher: CENGAGE L

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Chapter 9.4, Problem 42E

(a)

To determine

To calculate: The equation of the tangent line to curve f(x)=4x−x2 at x=5.

(b)

To determine

To graph: The function f(x)=4x−x2 and the tangent line y=−6x+25.

(c)

To determine

If both the curves f(x)=4x−x2 and y=−6x+25 become indistinguishable at x=5.

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Chapter 9.4, Problem 41E

Chapter 9.4, Problem 43E

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The equation of the tangent line to curve f ( x ) = 4 x − x 2 at x = 5 .

Solution Summary: The author calculates the equation of the tangent line to curve f(x)=4x-x

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To calculate: The equation of the tangent line to curve f(x)=4x−x2 at x=5.

To graph: The function f(x)=4x−x2 and the tangent line y=−6x+25.

If both the curves f(x)=4x−x2 and y=−6x+25 become indistinguishable at x=5.

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