Question

find the limit as x approaches c of the average rate of change of each function from c to x.

$c=-1;f\left(x\right)=2x^2-3x$

Short answer

The limit is$-7$

Step-by-step solution

Step 1. The given information is

$c=-1;f\left(x\right)=2x^2-3x$

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

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

Step 3. Finding limits we get,

$$\begin{gathered} \underset{x\to c}{\lim }\frac{∆y}{∆x}=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c} \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=\underset{x\to -1}{\lim }(2x-5) \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=\underset{x\to -1}{\lim }2(-1)-5 \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=-2-5 \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=-7 \end{gathered}$$