Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
Modeling the Dynamics of Life: Calculus and Probability for Life Scientists
3rd Edition
ISBN: 9780840064189
Author: Frederick R. Adler
Publisher: Cengage Learning
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Chapter 2.2, Problem 30E
To determine

To find : the average rate of change of the function in terms of Δx and limit as Δx→0 .

Expert Solution & Answer
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Answer to Problem 30E

The average rate of change of the function in terms of Δx is 10x+7+5(Δx)

The average rate of change as Δx→0 is 10x+7

Explanation of Solution

Given information: f(x)=5x2+7x+3 near x=2 .

Formula used:

Average rate of change = f(x+Δx)−f(x)Δx

Calculation:

Finding an expression for f(x+Δx)

  5(x+Δx)2+7(x+Δx)+3=5(x2+2xΔx+(Δx)2)+7(x+Δx)+3                                                          =5x2+10xΔx+5(Δx)2+7x+7Δx+3                                                          =5x2+7x+3+(10x+7)Δx+5(Δx)2

Substituting the above in formula to find the average rate of change and simplifying further

  (5x2+7x+3+(10x+7)Δx+5(Δx)2)−(5x2+7x+3)Δx =(10x+7)Δx+5(Δx)2Δx                                                                                                                      =10x+7+5(Δx)

Finding the value of the limit by substituting 0 for Δx

  limΔx→ 0f(x+Δx)−f(x)Δx=limΔx→ 0(10x+7+5(Δx))                                                 =10x+7+5(0)                                                 =10x+7

Finding the rate of change when x=2

  10(2)+7=20+7                     =27

Finding f(2) by substituting 2 for x in the function f(x)=5x2+7x+3

  f(2)=5(2)2+7(2)+3           =20+14+3           =37

Equation of tangent line at x=2  found by substituting the point (2,37) and slope m=27 in the point slope form y−y1=m(x−x1) , then

  • Distributing 27 in right side of equation
  • Adding 37 on both sides

y−37=27(x−2)y−37=27x−54y−37+37=27x−54+37y=27x−17

Graph:

  Modeling the Dynamics of Life: Calculus and Probability for Life Scientists, Chapter 2.2, Problem 30E

Interpretation:

The rate of change of the function f(x)=5x2+7x+3 near x=2 is 27

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Chapter 2 Solutions

Modeling the Dynamics of Life: Calculus and Probability for Life Scientists

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