Chapter 10, Problem 1CR

To determine

The value of the expression $\frac{f\left(x+h\right)-f\left(x\right)}{h},h\ne 0$, where $f\left(x\right)=-3x^2+5x-2$

Answer to Problem 1CR

Solution:

The value of the expression $\frac{f\left(x+h\right)-f\left(x\right)}{h},h\ne 0$, where $f\left(x\right)=-3x^2+5x-2$ is $-3h-6x+5$

Explanation of Solution

Given information:

The function $f\left(x\right)=-3x^2+5x-2$

Consider the function $f\left(x\right)=-3x^2+5x-2$ .

Replace $x$ by $x+h$ to find $f(x+h)$

  $\Rightarrow f\left(x+h\right)=-3{\left(x+h\right)}^2+5\left(x+h\right)-2$ .

Substitute the values of $f(x+h)$ and $f(x)$ in difference quotient formula.

Therefore, $\frac{f\left(x+h\right)-f\left(x\right)}{h}=\frac{-3{\left(x+h\right)}^2+5\left(x+h\right)-2-\left(-3x^2+5x-2\right)}{h}$ .

  $=\frac{-3{\left(x+h\right)}^2+5\left(x+h\right)-2+3x^2-5x+2}{h}$

  $=\frac{-3\left(x^2+h^2+2xh\right)+5\left(x+h\right)-2+3x^2-5x+2}{h}$

  $=\frac{-3x^2-3h^2-6xh+5x+5h-2+3x^2-5x+2}{h}$

  $=\frac{-3h^2-6xh+5h}{h}$

  $\Rightarrow \frac{f\left(x+h\right)-f\left(x\right)}{h}=-3h-6x+5$

Therefore, the value of the expression $\frac{f\left(x+h\right)-f\left(x\right)}{h},h\ne 0$ where, $f\left(x\right)=-3x^2+5x-2$ is $-3h-6x+5$ .