Explanation Given information: f ( x ) = ( x 3 − x 2 + 2 x − 1 ) 4 Formula used: The equation of the tangent line is given by y − y 1 = f ' ( x ) ( x − x 1 ) The equation of the normal line is given by y − y 1 = − 1 f ' ( x ) ( x − x 1 ) Calculation: First, differentiate with respect to x , we get f ' ( x ) = 4 ( x 3 − x 2 + 2 x − 1 ) 3 × ( 3 x 2 − 2 x + 2 − 0 ) f ' ( x ) = 4 ( 3 x 2 − 2 x + 2 ) ( x 3 − x 2 + 2 x − 1 ) 3 At x = 1, f ' ( 1 ) = 4 ( 3 ( 1 ) 2 − 2 ( 1 ) + 2 ) ( ( 1 ) 3 − ( 1 ) 2 + 2

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

Author: STRAUSS MONTY J, TODA MAGDALENA DANIELE, SMITH KARL J

Publisher: Kendall Hunt Publishing

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 3, Problem 46SP

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To find: The equations of the tangent line and normal line to the graph o f(x)=(x3−x2+2x−1)4 at x=1

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Chapter 3, Problem 45SP

Chapter 3, Problem 47SP

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Evaluate the limit along the stated paths, or type "DNE" if the limit Does Not Exist: lim xy+y³ (x,y)(0,0) x²+ y² Along the path = = 0: Along the path y = = 0: Along the path y = 2x:

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A graph of the function f is given below: Study the graph of ƒ at the value given below. Select each of the following that applies for the value a = 1 Of is defined at a. If is not defined at x = a. Of is continuous at x = a. If is discontinuous at x = a. Of is smooth at x = a. Of is not smooth at = a. If has a horizontal tangent line at = a. f has a vertical tangent line at x = a. Of has a oblique/slanted tangent line at x = a. If has no tangent line at x = a. f(a + h) - f(a) lim is finite. h→0 h f(a + h) - f(a) lim h->0+ and lim h h->0- f(a + h) - f(a) h are infinite. lim does not exist. h→0 f(a+h) - f(a) h f'(a) is defined. f'(a) is undefined. If is differentiable at x = a. If is not differentiable at x = a.

Chapter 3 Solutions

Student's Solution and Survival Manual for Calculus

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The equations of the tangent line and normal line to the graph o f ( x ) = ( x 3 − x 2 + 2 x − 1 ) 4 at x = 1

Solution Summary: The author compares the equations of the tangent line and the normal line to the graph.

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To find: The equations of the tangent line and normal line to the graph o f(x)=(x3−x2+2x−1)4 at x=1

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Chapter 3 Solutions