Chapter 11, Problem 6CT

(a)

To determine

To calculate:

The slope value of the following functions:

  $f\left(x\right)=3x^2-5x-2\left(2,0\right)$

Answer to Problem 6CT

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

Slope formula: $6x-5$

Slope: $7$

Explanation of Solution

Given information:

  $\begin{array}{l}f\left(x\right)=3x^2-5x-2\\ \left(2,0\right)\end{array}$

Formula Used:

The slope of the function is given by the following expressions:

  $m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Calculations:

The slope formula of $f\left(x\right)=3x^2-5x-2$ at the point $\left(2,0\right)$ :

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{3{\left(x+h\right)}^2-5\left(x+h\right)-2-\left(3x^2-5x-2\right)}{h}\\ m=\underset{h\to 0}{\lim }\frac{3x^2+6xh+3h^2-5x-5h-2-3x^2+5x+2}{h}\\ m=\underset{h\to 0}{\lim }\frac{6xh+3h^2-5h}{h}\\ m=\underset{h\to 0}{\lim }\frac{h\left(6x+3h-5\right)}{h}\\ m=\underset{h\to 0}{\lim }\frac{h\left(6x+3h-5\right)}{h}\\ m=\underset{h\to 0}{\lim }6x+3h-5\\ m=6x+3\left(0\right)-5\left[\because h=0\right]\\ m=6x-5\end{array}$

Now, the slope value at $\left(2,0\right)$ :

  $\begin{array}{l}m=6x-5\\ m=6\left(2\right)-5\\ m=7\end{array}$

(b)

To determine

To calculate:

The slope value of the following functions:

  $f\left(x\right)=2x^3+6x\left(-1,-8\right)$

Answer to Problem 6CT

The function $f\left(x\right)=2x^3+6x$ has:

Slope formula: $6\left(x^2+1\right)$

Slope: $12$

Explanation of Solution

Given information:

  $\begin{array}{l}f\left(x\right)=2x^3+6x\\ \left(-1,-8\right)\end{array}$

Formula Used:

The slope of the function is given by the following expressions:

  $m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Calculations:

The slope formula of $f\left(x\right)=2x^3+6x$ at the point $\left(-1,-8\right)$ :

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}\\ m=\underset{h\to 0}{\lim }\frac{2{\left(x+h\right)}^3+6\left(x+h\right)-2x^3+6x}{h}\\ m=\underset{h\to 0}{\lim }\frac{2x^3+6x^{2}h+6x{h}^2+h^3+6x+6h-2x^3-6x}{h}\\ m=\underset{h\to 0}{\lim }\frac{6x^{2}h+6x{h}^2+h^3+6h}{h}\\ m=\underset{h\to 0}{\lim }\frac{h\left(6x^2+6xh+h^2+6\right)}{h}\\ m=\underset{h\to 0}{\lim }6x^2+6xh+h^2+6\\ m=6x^2+6x\left(0\right)+{\left(0\right)}^2+6\left[\because h=0\right]\\ m=6x^2+6\\ m=6\left(x^2+1\right)\end{array}$

Now, the slope value at $\left(-1,-8\right)$ :

  $\begin{array}{l}m=6\left(x^2+1\right)\\ m=6\left({\left(-1\right)}^2+1\right)\\ m=12\end{array}$