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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.1, Problem 2E

(a)

To determine

**To sketch:** The graph of the function and obtain the point when the function crosses the *y*-axis.

Expert Solution

The graph of the function

The function *y* axis at

**Given:**

The function is

**Formula used:** Derivative of Exponential Function

**Calculation:**

Obtain the derivative of

Use the derivative of exponential function in equation (1).

Since the given function and its derivative are same,

The graph of the function

From the graph, it is observed that the function *x*, because the derivative of the function *x*.

Moreover, the function is closer to zero as *x* approaches minus infinity and it is closer to infinity as *x* approaches plus infinity.

That is,

Substitute 0 for *x* in

Therefore, the function is crosses the *y* axis at

(b)

To determine

**To describe:** The type of functions

Expert Solution

Both the functions

The differentiation formulas for

**Given:**

The function are

**Formula used:** Power Rule

If *n* is a real number, then

**Calculation:**

The graph of the function

From the graph, it is observed that the function *x* and *x*.

Thus, *x*.

From part (a), *x* and it is an increasing function.

The derivatives of both the functions *x* increases.

Therefore, both the functions

Obtain the derivatives of

From part (a), the derivative of

Since the derivative of

Apply the Power rule (2),

Thus, the derivative of

Therefore, the differentiation formulas for

(c)

To determine

**To identify:** The function which grows more rapidly when *x* is large.

Expert Solution

The function *x* is large.

**Given:**

The function are

The graph of the functions

From the graph, it is observed that the value of

That is,

Therefore, the function *x* is large.