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High School Math Calculus EBK PRECALCULUS W/LIMITS

Verify that the tangent line is horizontal.

Verify that the tangent line is horizontal.

EBK PRECALCULUS W/LIMITS

EBK PRECALCULUS W/LIMITS

3rd Edition

ISBN: 9781285607160

Author: Larson

Publisher: CENGAGE CO

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Question

Chapter 12.3, Problem 68E

To determine

Verify that the tangent line is horizontal.

Expert Solution & Answer

Answer to Problem 68E

Tangents drawn on the function $f(x)=xe−x$ are horizontal.

Explanation of Solution

Given information:

Use the function and its derivative to determine any points on the graph of $f$ at which the tangent line is horizontal. Use a graphing utility to verify the results.

$f(x)=xe−x$ , $f'(x)=e−x−xe−x$

Calculation:

Consider the function, $f(x)=xe−x$

The slope of the function or the derivative is $f'(x)=e−x−xe−x$

The slope of the tangent drawn on the function $f(x)=xe−x$ should be horizontal.

So, the slope will be $0$

$−xe−x−e−x=0e−x(−x+1)=0$

$x=$

$0$ , $1$

Now, $x=$

$0$ , $1$ should satisfy the function.

When, $x=$

$0$

$f(0)=0×e−0$

$=0$

When, $x=$

$1$

$f(1)=e−1$

So, the points at which the function $f(x)=xe−x$ and tangents drawn it is horiontal are $(0,0)$ and $(1,e−1)$ .

Now to check the above point by calculator, proceed as follows.

Press button $"=Y"$ and then enter the expression in the window as given below.

EBK PRECALCULUS W/LIMITS, Chapter 12.3, Problem 68E , additional homework tip 1

Now, press the $"WINDOW"$ button to set the range.

EBK PRECALCULUS W/LIMITS, Chapter 12.3, Problem 68E , additional homework tip 2

Now, press the $"GRAPH"$ button to plot the graph as for different point .

For point $(0,0)$

EBK PRECALCULUS W/LIMITS, Chapter 12.3, Problem 68E , additional homework tip 3

For point $(1,e−1)$

EBK PRECALCULUS W/LIMITS, Chapter 12.3, Problem 68E , additional homework tip 4

Hence, it is verified that the tangents drawn on the function $f(x)=xe−x$ are horizontal.

Chapter 12.3, Problem 67E

Chapter 12.3, Problem 69E

Chapter 12 Solutions

EBK PRECALCULUS W/LIMITS

Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E Ch. 12.1 - Prob. 5E Ch. 12.1 - Prob. 6E Ch. 12.1 - Prob. 7E Ch. 12.1 - Prob. 8E Ch. 12.1 - Prob. 9E Ch. 12.1 - Prob. 10E

Ch. 12.1 - Prob. 11E Ch. 12.1 - Prob. 12E Ch. 12.1 - Prob. 13E Ch. 12.1 - Prob. 14E Ch. 12.1 - Prob. 15E Ch. 12.1 - Prob. 16E Ch. 12.1 - Prob. 17E Ch. 12.1 - Prob. 18E Ch. 12.1 - Prob. 19E Ch. 12.1 - Prob. 20E Ch. 12.1 - Prob. 21E Ch. 12.1 - Prob. 22E Ch. 12.1 - Prob. 23E Ch. 12.1 - Prob. 24E Ch. 12.1 - Prob. 25E Ch. 12.1 - Prob. 26E Ch. 12.1 - Prob. 27E Ch. 12.1 - Prob. 28E Ch. 12.1 - Prob. 29E Ch. 12.1 - Prob. 30E Ch. 12.1 - Prob. 31E Ch. 12.1 - Prob. 32E Ch. 12.1 - Prob. 33E Ch. 12.1 - Prob. 34E Ch. 12.1 - Prob. 35E Ch. 12.1 - Prob. 36E Ch. 12.1 - Prob. 37E Ch. 12.1 - Prob. 38E Ch. 12.1 - Prob. 39E Ch. 12.1 - Prob. 40E Ch. 12.1 - Prob. 41E Ch. 12.1 - Prob. 42E Ch. 12.1 - Prob. 43E Ch. 12.1 - Prob. 44E Ch. 12.1 - Prob. 45E Ch. 12.1 - Prob. 46E Ch. 12.1 - Prob. 47E Ch. 12.1 - Prob. 48E Ch. 12.1 - Prob. 49E Ch. 12.1 - Prob. 50E Ch. 12.1 - Prob. 51E Ch. 12.1 - Prob. 52E Ch. 12.1 - Prob. 53E Ch. 12.1 - Prob. 54E Ch. 12.1 - Prob. 55E Ch. 12.1 - Prob. 56E Ch. 12.1 - Prob. 57E Ch. 12.1 - Prob. 58E Ch. 12.1 - 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