Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Question

Chapter 3.4, Problem 7E

To determine

To graph: the derivative of the fruit fly population. The graph of the population is reproduced below and what units should be used on the horizontal and vertical axes for the derivative’s graph.

Expert Solution

Answer to Problem 7E

The units for derivative graph are days for horizontal axis (x-axis) and flies per day for the vertical axis (y-axis).

Explanation of Solution

Given information: Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individual increases and resources are still abundant , then slowly again as the population reaches the carrying capacity of the environment. The graph of the population is reproduced is given below.

Calculus: Graphical, Numerical, Algebraic, Chapter 3.4, Problem 7E , additional homework tip 1

Calculation:

Estimate the slopes at x =0, 10, and 20,30,40,50 by drawing tangent lines at each value:

Calculus: Graphical, Numerical, Algebraic, Chapter 3.4, Problem 7E , additional homework tip 2

The estimated slopes of the tangent lines are about 0.5 for t =0, 3 for t =10, 13 for t =20, 14 for t =30, 3.5 for t =40, and 0.5 for t =50. The points on the graph of the derivative are then about (0,0.5),(10,3),(20,13),(30,14),(40,3.5), and (50,0.5). Plot these points and connect them with a smooth curve to make the graph of the derivative :

Calculus: Graphical, Numerical, Algebraic, Chapter 3.4, Problem 7E , additional homework tip 3

Since the dependent variable for the original graph was flies and the independent variable was days, the units for derivative graph are days for horizontal axis(x-axis) and flies per day for the vertical axis (y-axis).

To determine

To find: during what days does the population seem to be increasing fastest, slowest.

Expert Solution

Answer to Problem 7E

Fastest: Day 25 and Slowest: Day 0 or Day 50.

Explanation of Solution

Calculus: Graphical, Numerical, Algebraic, Chapter 3.4, Problem 7E , additional homework tip 4

Calculation:

The derivative graph is shown below.

Calculus: Graphical, Numerical, Algebraic, Chapter 3.4, Problem 7E , additional homework tip 5

From the derivative graph, the maximum of the derivative graph was around t =25.

So, Increasing the fastest: Around the 25^t^h day.

From the derivative graph, the minimum of the derivative graph was around t =0 and t =50.

So, increasing the slowest: Day 0 or Day 50.

Chapter 3.4, Problem 6E

Chapter 3.4, Problem 8E

Chapter 3 Solutions

Calculus: Graphical, Numerical, Algebraic

Ch. 3.1 - Prob. 1QR Ch. 3.1 - Prob. 2QR Ch. 3.1 - Prob. 3QR Ch. 3.1 - Prob. 4QR Ch. 3.1 - Prob. 5QR Ch. 3.1 - Prob. 6QR Ch. 3.1 - Prob. 7QR Ch. 3.1 - Prob. 8QR Ch. 3.1 - Prob. 9QR Ch. 3.1 - Prob. 10QR

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