Q. 7 What is the relationship between... [FREE SOLUTION]

Chapter 2: Q. 7 (page 165)

What is the relationship between the derivative of a function $f$ at a point $x = c$ , the slope of the tangent line to the graph of $f$ at $x = c$ , and the instantaneous rate of change of $f$ at $x = c$ ?

Short Answer

Expert verified

The derivative of $f$ at $x = c$ , the slope of the tangent line at $x = c$ and instantaneous rate of change at $x = c$ all represent the same value i.e. $\frac{f (c + h) - f (c)}{h}$ .

Step by step solution

Step 1. Given information

A function $f (x)$ .

Step 2. Explanation

The derivative of a function at any two instantaneous points $x = c$ and $x = c + h$ is given by $\frac{f (c + h) - f (c)}{h}$ .

While the slope of the tangent line at $x = c$ is also the rate of change of the function $f$ for two instantaneous points $x = c$ and $x = c + h$ .

Therefore, the slope of the tangent line is $\frac{f (c + h) - f (c)}{h}$ .

Thus the derivative of $f$ at $x = c$ , the slope of the tangent line at $x = c$ and instantaneous rate of change at $x = c$ all represent the same value.

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What is the relationship between the derivative of a function $f$ at a point $x = c$ , the slope of the tangent line to the graph of $f$ at $x = c$ , and the instantaneous rate of change of $f$ at $x = c$ ?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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91影视

What is the relationship between the derivative of a function fat a point x=c, the slope of the tangent line to the graph of fat x=c, and the instantaneous rate of change of fat x=c?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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What is the relationship between the derivative of a function $f$ at a point $x = c$ , the slope of the tangent line to the graph of $f$ at $x = c$ , and the instantaneous rate of change of $f$ at $x = c$ ?