Question

given the function $f(x)=-3x^{2}$. find the difference quotient $\frac{f(x + h)-f(x)}{h}$. find $f(x)$ by determining $lim_{h
ightarrow0}\frac{f(x + h)-f(x)}{h}$. use the previous answer to find the values indicated below. the values should match the slopes of the tangent lines you will draw on the graph at the bottom. find $f(-2)$. find $f(0)$. find $f(1)$. graph the function $f(x)=-3x^{2}$ and draw the tangent lines to the graph at points whose $x$ - coordinates are -2, 0, and 1.

Explanation:

Step1: Find the difference - quotient

First, find \(f(x + h)\):

$$
LATEXBLOCK0
$$

Then, calculate \(\frac{f(x + h)-f(x)}{h}\):

$$
LATEXBLOCK1
$$

Step2: Find the derivative \(f^{\prime}(x)\)

$$
LATEXBLOCK2
$$

Step3: Find \(f^{\prime}(-2)\)

Substitute \(x=-2\) into \(f^{\prime}(x)\):

$$ f^{\prime}(-2)=-6\times(-2)=12 $$

Step4: Find \(f^{\prime}(0)\)

Substitute \(x = 0\) into \(f^{\prime}(x)\):

$$ f^{\prime}(0)=-6\times0 = 0 $$

Step5: Find \(f^{\prime}(1)\)

Substitute \(x = 1\) into \(f^{\prime}(x)\):

$$ f^{\prime}(1)=-6\times1=-6 $$

Answer:

1. Difference - quotient: \(-6x - 3h\)
2. \(f^{\prime}(x)\): \(-6x\)
3. \(f^{\prime}(-2)\): \(12\)
4. \(f^{\prime}(0)\): \(0\)
5. \(f^{\prime}(1)\): \(-6\)