BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2.2, Problem 26E

(a)

To determine

To estimate: The value of limx0sinxsinπx by graphing the function f(x)=sinxsinπx.

Expert Solution

Answer to Problem 26E

The value of limx0sinxsinπx0.32.

Explanation of Solution

Using the graphing calculator, the graph of the function f(x)=sinxsinπx is drawn and shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.2, Problem 26E , additional homework tip  1

Zoom the graph towards the point where the graph crosses the y-axis as shown below in Figure 2.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 2.2, Problem 26E , additional homework tip  2

From Figure 2, it is observed that graph of f(x)=sinxsinπx approaches 0.32 as x approaches 0 from either side.

Since the right hand and the left hand limits are equal, the value of limx0sinxsinπx exists.

That is, limx0sinxsinπx=limx0+sinxsinπx0.32.

Thus, the value of limx0sinxsinπx0.32.

(b)

To determine

To check: The correctness of the answer obtained in part (a) by evaluating the value of f(x)=sinxsinπx as x approaches 0.

Expert Solution

Explanation of Solution

As x approaches 0, evaluate the function f(x)=sinxsinπx for the x values 0.5,0.1,0.01,0.001,0.0001,0.5,0.1,0.01,0.001 and 0.0001.

Evaluate the function (correct to 6 decimal places) for the values of x and get the following table of values.

xsinxsinπxf(x)=sinxsinπx
−0.5−1.571428571−0.99999980.318346
−0.1−0.314285714−0.3091372520.318311
−0.01−0.031428571−0.0314233980.318228
−0.001−0.003142857−0.0031428520.318309
−0.0001−0.000314286−0.0003142860.318309
0.51.5714285710.99999980.318345
0.10.3142857140.3091372520.318311
0.010.0314285710.0314233980.318309
0.0010.0031428570.0031428520.318309
0.00010.0003142860.0003142860.318309

From the above table, it is observed that the value of f(x) is approximately closer to 0.32 as x approaches to 0 from either side. That is, limx0sinxsinπx0.32.

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