Chapter 5.4, Problem 88E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

# Linear and Quadratic Approximation In Exercises 87 and 88, use a graphing utility to graph the function. Then graph P 1 ( x ) = f ( 0 ) + f ' ( 0 ) ( x − 0 ) and P 2 ( x ) = f ( 0 ) + f ' ( 0 ) ( x − 0 ) + 1 2 f ″ ( 0 ) ( x − 0 ) 2 in the same viewing window. Compare the values of f , P 1 , P 2 , and their first derivatives at x = 0 . f ( x ) = e x / 2

To determine

To graph: The graph of function f(x)=ex2 and the functions P1(x)=f(0)+f(0)(x0) and P2(x)=f(0)+f(0)(x0)+12f(0)(x0)2 further compare these three graphs.

Explanation

Graph:

The graph of f(x)=ex, P1=1+12x and P2=1+12x+18x2 is:

Formula used:

deaxdx=aeax where a is a constant value.

dcf(x)dx=cf'(x), where c is a constant value.

Calculation:

Use Ti-83 to draw the graph of the function f(x)=ex2. The steps are:

Step 1: Press ON button.

Step 2: Press 2nd then press Y=. Enter the function Y1=ex2.

Step 3: Press TRACE button.

Since,

deaxdx=aeax where a is a constant value.

So,

f'(x)=12ex2f''(x)=ddx(12ex2)=12(12ex2)=14ex2

Now, calculate the value of P1(x)=f(0)+f(0)(x0):

P1(x)=f(0)+f(0)(x0)=e02+12e02×x=

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