Question

graph the function (f(x) = -2x^2) and draw the tangent line to the graph at (x = -2).

a. simplify the difference quotient.

(\frac{f(x+h) - f(x)}{h} =)

b. find the derivative of (f(x)) by applying the definition.

(lim_{h \to 0} \frac{f(x+h) - f(x)}{h} =)

c. find the slope of the tangent line at (x = -2). this slope is the same as the line drawn on the grid.

Explanation:

Simplify the difference quotient

We evaluate the difference quotient for \(f(x) = -2x^2\):

$$
LATEXBLOCK0
$$

Find the derivative using definition

We take the limit as \(h \to 0\) of the simplified difference quotient:

$$ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} (-4x - 2h) = -4x $$

Find the slope at \(x = -2\)

We substitute \(x = -2\) into the derivative function \(f'(x) = -4x\):

$$ f'(-2) = -4(-2) = 8 $$

Answer:

Question A

The simplified difference quotient is:
<blank>\(-4x - 2h\)</blank>

Question B

The derivative of \(f(x)\) by applying the definition is:
<blank>\(-4x\)</blank>

Question C

The slope of the tangent line at \(x = -2\) is:
<blank>\(8\)</blank>