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Chapter 14: Q. 48 (page 883)

find the limit as x approaches c of the average rate of change of each function from c to x.
$c = - 1; f (x) = 2 x^{2} - 3 x$

Short Answer

Expert verified

The limit is $- 7$

Step by step solution

01

Step 1. The given information is

$c = - 1; f (x) = 2 x^{2} - 3 x$

02

Step 2. An average rate change of function f from point c to x

$\frac{∆ y}{∆ x} = \frac{f (x) - f (c)}{x - c} \frac{∆ y}{∆ x} = \frac{2 x^{2} - 3 x - (2 c^{2} - 3 c)}{x - (- 1)} \frac{∆ y}{∆ x} = \frac{2 x^{2} - 3 x - (2 {(- 1)}^{2} - 3 (- 1))}{x + 1} \frac{∆ y}{∆ x} = \frac{2 x^{2} - 3 x - (2 + 3)}{x + 1} \frac{∆ y}{∆ x} = \frac{2 x^{2} - 3 x - 5}{x + 1} \frac{∆ y}{∆ x} = \frac{2 x^{2} + 2 x - 5 x - 5}{x + 1} \frac{∆ y}{∆ x} = \frac{2 x (x + 1) - 5 (x + 1)}{x + 1} \frac{∆ y}{∆ x} = \frac{(2 x - 5) (x + 1)}{x + 1} \frac{∆ y}{∆ x} = 2 x - 5$

03

Step 3. Finding limits we get,

$\lim_{x \to c} \frac{∆ y}{∆ x} = \lim_{x \to c} \frac{f (x) - f (c)}{x - c} \lim_{x \to c} \frac{∆ y}{∆ x} = \lim_{x \to - 1} (2 x - 5) \lim_{x \to c} \frac{∆ y}{∆ x} = \lim_{x \to - 1} 2 (- 1) - 5 \lim_{x \to c} \frac{∆ y}{∆ x} = - 2 - 5 \lim_{x \to c} \frac{∆ y}{∆ x} = - 7$

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Most popular questions from this chapter

In Problems 13–32, use the accompanying graph of $y = f (x)$ .

Find the x-intercept(s), if any, of $f$ .

In Problems 13–32, use the accompanying graph of y = f(x).

Find $\lim_{x \to 2^{-}} f (x)$ .

In Problems 13–32, use the accompanying graph of y = f(x).

Find $\lim_{x \to - 6^{+}} f (x) .$

In Problems 7– 42, find each limit algebraically

$\lim_{x \to 1} (5 x^{4} - 3 x^{2} + 6 x - 9)$

Use the graph shown to determine if the limit exists. If it does, find its value.

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