Question

chapter 1 quiz (1.1,1.2, 1.4, 1.5, 1.7)
14 points possible answered: 2/14
question 3
for the function $f(x) = 3x^2 + x$, evaluate and simplify.
$\frac{f(x + h) - f(x)}{h} = $

Explanation:

Step1: Find \( f(x + h) \)

Substitute \( x + h \) into \( f(x) = 3x^2 + x \):
\( f(x + h) = 3(x + h)^2 + (x + h) \)
Expand \( (x + h)^2 \): \( (x + h)^2 = x^2 + 2xh + h^2 \)
So, \( f(x + h) = 3(x^2 + 2xh + h^2) + x + h = 3x^2 + 6xh + 3h^2 + x + h \)

Step2: Calculate \( f(x + h) - f(x) \)

Subtract \( f(x) = 3x^2 + x \) from \( f(x + h) \):
\( f(x + h) - f(x) = (3x^2 + 6xh + 3h^2 + x + h) - (3x^2 + x) \)
Simplify: \( 3x^2 + 6xh + 3h^2 + x + h - 3x^2 - x = 6xh + 3h^2 + h \)

Step3: Divide by \( h \) ( \( h

eq 0 \) )
\( \frac{f(x + h) - f(x)}{h} = \frac{6xh + 3h^2 + h}{h} \)
Factor out \( h \) in the numerator: \( \frac{h(6x + 3h + 1)}{h} \)
Cancel \( h \): \( 6x + 3h + 1 \)

Answer:

\( 6x + 3h + 1 \)