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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 2.6, Problem 3E

(a)

To determine

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

Expert Solution

The slope of the tangent line to the parabola

(i) Using Definition 1 is

(ii) Using Equation 2 is

**Given:**

The slope of the tangent line to the parabola

**Formula used:**

Definition 1: The slope of the tangent curve

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

**Section (i)**

Obtain the slope of the tangent line to the parabola at the point (1, 3) by using Definition 1.

Substitute

The factors of

Therefore, the slope of the tangent line to the parabola becomes,

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, 3) is

**Section (ii)**

Obtain the slope of the tangent line to the parabola at the point (1, 3) by using Equation 2.

Substitute

Simplify further and obtain the value of *m*.

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

Thus, the slope of the tangent line becomes,

Thus, the slope of the tangent line to the parabola at the point (1, 3) 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

**Equation of the tangent line:**

The equation of the tangent line to the curve

Since the tangent line to the curve *a*, the value of

Substitute

Isolate *y* as shown below.

Thus, the equation of the tangent line is

(c)

To determine

**To sketch**: The graph of the function is the tangent line,

Expert Solution

**Calculation:**

The equation of the tangent line is

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 (1,3) as shown below in Figure 2.

From Figure 2, it is observed that when zoom in the graph toward the point (1, 3), 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 parabola zoom in toward the point (1, 3) until the tangent line are indistinguishable.