The expression for slope of tangent line to the curve y = f ( x ) at the point ( a , f ( a ) ) .

Question

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Answer 1

Question

Chapter 3, Problem 1CC

To determine

To define: The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ .

Expert Solution & Answer

Answer to Problem 1CC

The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $limx→af(x)−f(a)x−a$ .

Explanation of Solution

Given information:

The curve $y=f(x)$ and the point $(a,f(a))$ .

The expression for slope mof tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limx→af(x)−f(a)x−a$ , provided that the limit exists.

Let $x−a=h$ then $x=a+h$ . When limit of x tends to a, the limit of h tends to 0 so another expression for slope m of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limh→0f(a+h)−f(a)h$ , provided that the limit exists.

For example: Slope of the curve $y=9x$ at the point $(1,9)$ is,

$m=limh→0f(a+h)−f(a)h=limh→09(a+h)−9ah=limh→09a+9h−9ah=limh→09hh$

Simplify it further as,

$m=9$

Therefore, slope of the curve $y=9x$ at the point $(1,9)$ is 9.

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Answer 2

Question

Chapter 3, Problem 1CC

To determine

To define: The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ .

Expert Solution & Answer

Answer to Problem 1CC

The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $limx→af(x)−f(a)x−a$ .

Explanation of Solution

Given information:

The curve $y=f(x)$ and the point $(a,f(a))$ .

The expression for slope mof tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limx→af(x)−f(a)x−a$ , provided that the limit exists.

Let $x−a=h$ then $x=a+h$ . When limit of x tends to a, the limit of h tends to 0 so another expression for slope m of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limh→0f(a+h)−f(a)h$ , provided that the limit exists.

For example: Slope of the curve $y=9x$ at the point $(1,9)$ is,

$m=limh→0f(a+h)−f(a)h=limh→09(a+h)−9ah=limh→09a+9h−9ah=limh→09hh$

Simplify it further as,

$m=9$

Therefore, slope of the curve $y=9x$ at the point $(1,9)$ is 9.

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Answer 3

Question

Chapter 3, Problem 1CC

To determine

To define: The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ .

Expert Solution & Answer

Answer to Problem 1CC

The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $limx→af(x)−f(a)x−a$ .

Explanation of Solution

Given information:

The curve $y=f(x)$ and the point $(a,f(a))$ .

The expression for slope mof tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limx→af(x)−f(a)x−a$ , provided that the limit exists.

Let $x−a=h$ then $x=a+h$ . When limit of x tends to a, the limit of h tends to 0 so another expression for slope m of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limh→0f(a+h)−f(a)h$ , provided that the limit exists.

For example: Slope of the curve $y=9x$ at the point $(1,9)$ is,

$m=limh→0f(a+h)−f(a)h=limh→09(a+h)−9ah=limh→09a+9h−9ah=limh→09hh$

Simplify it further as,

$m=9$

Therefore, slope of the curve $y=9x$ at the point $(1,9)$ is 9.

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Answer 4

Question

Chapter 3, Problem 1CC

To determine

To define: The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ .

Expert Solution & Answer

Answer to Problem 1CC

The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $limx→af(x)−f(a)x−a$ .

Explanation of Solution

Given information:

The curve $y=f(x)$ and the point $(a,f(a))$ .

The expression for slope mof tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limx→af(x)−f(a)x−a$ , provided that the limit exists.

Let $x−a=h$ then $x=a+h$ . When limit of x tends to a, the limit of h tends to 0 so another expression for slope m of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limh→0f(a+h)−f(a)h$ , provided that the limit exists.

For example: Slope of the curve $y=9x$ at the point $(1,9)$ is,

$m=limh→0f(a+h)−f(a)h=limh→09(a+h)−9ah=limh→09a+9h−9ah=limh→09hh$

Simplify it further as,

$m=9$

Therefore, slope of the curve $y=9x$ at the point $(1,9)$ is 9.

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Answer 5

Chapter 3, Problem 1CC

To determine

To define: The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ .

Expert Solution & Answer

Answer to Problem 1CC

The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $limx→af(x)−f(a)x−a$ .

Explanation of Solution

Given information:

The curve $y=f(x)$ and the point $(a,f(a))$ .

The expression for slope mof tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limx→af(x)−f(a)x−a$ , provided that the limit exists.

Let $x−a=h$ then $x=a+h$ . When limit of x tends to a, the limit of h tends to 0 so another expression for slope m of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limh→0f(a+h)−f(a)h$ , provided that the limit exists.

For example: Slope of the curve $y=9x$ at the point $(1,9)$ is,

$m=limh→0f(a+h)−f(a)h=limh→09(a+h)−9ah=limh→09a+9h−9ah=limh→09hh$

Simplify it further as,

$m=9$

Therefore, slope of the curve $y=9x$ at the point $(1,9)$ is 9.

Answer 6

Chapter 3, Problem 1CC

To determine

To define: The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ .

Expert Solution & Answer

Answer to Problem 1CC

The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $limx→af(x)−f(a)x−a$ .

Explanation of Solution

Given information:

The curve $y=f(x)$ and the point $(a,f(a))$ .

The expression for slope mof tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limx→af(x)−f(a)x−a$ , provided that the limit exists.

Let $x−a=h$ then $x=a+h$ . When limit of x tends to a, the limit of h tends to 0 so another expression for slope m of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limh→0f(a+h)−f(a)h$ , provided that the limit exists.

For example: Slope of the curve $y=9x$ at the point $(1,9)$ is,

$m=limh→0f(a+h)−f(a)h=limh→09(a+h)−9ah=limh→09a+9h−9ah=limh→09hh$

Simplify it further as,

$m=9$

Therefore, slope of the curve $y=9x$ at the point $(1,9)$ is 9.

Answer 7

Chapter 3, Problem 1CC

To determine

To define: The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ .

Answer 8

Expert Solution & Answer

Answer to Problem 1CC

The expression for slope of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $limx→af(x)−f(a)x−a$ .

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Given information:

The curve $y=f(x)$ and the point $(a,f(a))$ .

The expression for slope mof tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limx→af(x)−f(a)x−a$ , provided that the limit exists.

Let $x−a=h$ then $x=a+h$ . When limit of x tends to a, the limit of h tends to 0 so another expression for slope m of tangent line to the curve $y=f(x)$ at the point $(a,f(a))$ is $m=limh→0f(a+h)−f(a)h$ , provided that the limit exists.