Explanation Given information: f ( x ) = x 2 sin x 2 Concept and Formula Used: We use the product rule and chain rule of differentiation. The following are the rules: Chain rule: d d x u ( v ( x ) ) = u ' ( v ( x ) ) v ' ( x ) Product rule: d d x ( u ( x ) v ( x ) ) = u ' ( x ) v ( x ) + u ( x ) v ' ( x ) Also, we use the following differentiation formula d d x ( x n ) = n x n − 1 d d x ( sin x ) = cos x d d x ( cos x ) = − sin x Calculation: As the given function is f ( x ) = x 2 sin x 2 , applying the product rule we have f ( x ) = x 2 sin x 2 d d x ( f ( x ) ) = d d x ( x 2 sin x 2 ) f ' ( x ) = sin x 2 d d x ( x 2 ) + x 2 d d x ( sin x 2 ) Now, applying chain rule and the formula d d x ( x n ) = n x n − 1 and d d x ( x n ) = n x n − 1 , we get f ' ( x ) = 2 x sin x 2 + x 2 cos x 2 d d x ( x 2 ) </

7th Edition

ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 3, Problem 47SP

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To Calculate: The double derivative f''(x) of the function f(x) .

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Chapter 3, Problem 46SP

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5:38 Video Message instructor Submit Question ||| D

8:38 *** TEMU TEMU -3 -2 7 B 2 1 & 5G. 61% 1 2 -1 Based on the graph above, determine the amplitude, period, midline, and equation of the function. Use f(x) as the output. Amplitude: 2 Period: 2 Midline: 2 ☑ syntax error: this is not an equation. Function: f(x) = −2 cos(πx + 2.5π) +2× Question Help: Worked Example 1 ☑ Message instructor Submit Question ||| <

8:39 *** TEMU 5G 60% A ferris wheel is 28 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. What is the amplitude? 14 meters What is the equation of the Midline? y = 16 What is the period? 4 meters minutes The equation that models the height of the ferris wheel after t minutes is: f(t): = ƒ (3) = ·−14(0) + 16 syntax error: you gave an equation, not an expression. syntax error. Check your variables - you might be using an incorrect one. How high are you off of the ground after 3 minutes? Round your answe the nearest meter. ||| <

Chapter 3 Solutions

Calculus

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The double derivative f ' ' ( x ) of the function f ( x ) .

Solution Summary: The author explains the product rule and chain rule of differentiation.

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To Calculate: The double derivative f''(x) of the function f(x) .

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Chapter 3 Solutions