Explanation Given: The function, h ( t ) = ( t 4 − 1 ) 3 ( t 3 + 1 ) 4 . Concept Used: The product rule of derivatives: d d x [ f ( x ) ⋅ g ( x ) ] = f ( x ) ⋅ d d x g ( x ) + g ( x ) ⋅ d d x f ( x ) Chain rule, d d x ( f ( g ( x ) ) ) = f ' ( g ( x ) ) ⋅ g ' ( x ) Power rule of derivatives, d d x ( x n ) = n x n − 1 Derivative of any constant C is zero, that is, d d x ( C ) = 0 Calculation: In order to find the derivative of the function,useproduct rule and chain rule of derivatives and simplify further to get, h ' ( t ) = d d t ( ( t 4 − 1 ) 3 ( t 3 + 1 ) 4 ) = ( t 4 − 1 ) 3 ⋅ d d t ( t 3 + 1 ) 4 + ( t 3 + 1 ) 4 ⋅ d d t ( t 4 − 1 ) 3 = ( t 4 − 1 ) 3 ⋅ [ 4 ( t 3 + 1 ) 4 − 1 ⋅ d d t ( t 3 + 1 ) ] + ( t 3 + 1 ) 4 ⋅ [ 3 ( t 4 − 1 ) 3 − 1 ⋅ d d t ( t 4 − 1 ) ] = ( t 4 − 1 ) 3 ⋅ [ 4 ( t 3 + 1 ) 3 ⋅ ( 3 t 3 − 1 + 0 ) ] + ( t 3 + 1 ) <

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Author: James Stewart, Troy Day

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Chapter 3.5, Problem 20E

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The derivative of the function, h(t)=(t4−1)3(t3+1)4 .

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Chapter 3.5, Problem 19E

Chapter 3.5, Problem 21E

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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

Biocalculus: Calculus For Life Sciences

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The derivative of the function, h ( t ) = ( t 4 − 1 ) 3 ( t 3 + 1 ) 4 .

Solution Summary: The author explains the product rule and chain rule of derivatives and simplifies to get the derivative of the function.

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The derivative of the function, h(t)=(t4−1)3(t3+1)4 .

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