a. Explanation Given information: The function f ( x ) = ( x 3 − 2 x 2 ) ( x + 2 ) . Equation of tangent: Let the line passes through the point ( x 1 , y 1 ) and the slope of the line is m . So, the equation of the tangent is y − y 1 = m ( x − x 1 ) Derivative power formula: If function f ( x ) = x n , then the derivative of function is d d x ( f ( x ) ) = f ' ( x ) = n x n − 1 . Calculation: Simplify the function f ( x ) = ( x 3 − 2 x 2 ) ( x + 2 ) f ( x ) = x ( x 3 − 2 x 2 ) + 2 ( x 3 − 2 x 2 ) f ( x ) = ( x 4 − 2 x 3 + 2 x 3 − 4 x 2 ) f ( x ) = ( x 4 − 4 x 2 ) ....................... ( 1 ) The slope of the line is the derivative of the function at a point. So, differentiate to the above equation with respect to x. f ( x ) = ( x 4 − 4 x 2 ) f ' ( x ) = d d x ( x 4 − 4 x 2 ) f ' ( x ) = ( 4 x 3 − 4 × 2 x ) f ' ( x ) = ( 4 x 3 − 8 x ) The derivative at point x = 1 b. To determine To find the equation of the normal line to the graph of function f at the point where x = 0 .

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Concept explainers

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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Question

Chapter 3.2, Problem 43PS

To determine

To find the equation of the tangent line to the graph of function f at the point where x=1 .

To determine

To find the equation of the normal line to the graph of function f at the point where x=0 .

Expert Solution & Answer

Chapter 3 Solutions

Calculus

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To find the equation of the tangent line to the graph of function f at the point where x = 1 .

Solution Summary: The author explains how to find the equation of the tangent line to the graph of function f at the point where x=1.

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To find the equation of the tangent line to the graph of function f at the point where x=1 .

To find the equation of the normal line to the graph of function f at the point where x=0 .

Chapter 3 Solutions