Start your trial now! First week only $4.99!*arrow_forward*

BuyFind*launch*

4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.6, Problem 4E

(a)

To determine

**To find:** The slope of the tangent line to the curve

Expert Solution

The slope of the tangent line to the curve

(i) Using Definition 1 is

(ii) Using Equation 2 is

**Given:**

The equation of the curve

The slope of the tangent line to the curve at the point

**Formula used:**

Definition 1: The slope of the tangent curve

Equation 2: The slope of the tangent line in definition 1 becomes,

**Difference of square formula:**

**Calculation:**

**Section (i)**

Obtain the slope of the tangent line to the curve at the point (1, 0) using Definition 1.

Substitute

Apply the difference of squares formula,

Since the limit *x* approaches 1 but not equal to 1, cancel the common term

Thus, the slope of the tangent line to the parabola at the point (1, 0) by using Definition 1 is

**Section (ii)**

Obtain the slope of the tangent line to the parabola at the point (1, 0) by using Equation 2.

Substitute

Simplify the terms by expanding

Since the limit *h* approaches 0 but not equal to 0, cancel the common term

Thus, the slope of the tangent line to the parabola at the point (1, 0) by using Equation 2 is

**(b)**

To determine

**To find:** The equation of the tangent line in part (a).

Expert Solution

The equation of the tangent line in part (a) is

**Formula used:**

The equation of the tangent line to the curve

**Calculation:**

Obtain the equation of the tangent line in part (a).

Since the tangent line to the curve *a*,

Substitute

Isolate *y,*

Thus, the equation of the tangent line is

(c)

To determine

**To sketch**: The graph of the tangent line and the curve. Check zoom in the graph toward the point (1, 0) until the curve and the tangent line is indistinguishable.

Expert Solution

The equation of the tangent line

The equation of the given curve is

Use the online graphing calculator to draw the graph of the functions as shown below in Figure 1.

**Graph:**

Use the online graphing calculator to zoom toward the point (1, 0) as shown below in Figure 2.

From Figure 2, it is observed that when zoom in the graph toward the point (1, 0), the graph of the tangent line and the curve looks likes almost identical.

Hence, it is verified that the graph of the tangent line and the curve zooming toward the point (1, 0) until the tangent line are indistinguishable.