Question

multiple choice 2 points find the difference quotient: (\frac{f(x + h) - f(x)}{h}) for (f(x) = 3x^2 + 4x) (\bigcirc) (-6x - 3 + 4h) (\bigcirc) none of the given choices (\bigcirc) (6x - 3h - 4) (\bigcirc) (-6x - 3h - 4) (\bigcirc) (-6x + 3h - 4)

Explanation:

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

Substitute \( x + h \) into \( f(x) = 3x^2 + 4x \):
\( f(x + h) = 3(x + h)^2 + 4(x + h) \)
Expand \( (x + h)^2 \): \( (x + h)^2 = x^2 + 2xh + h^2 \)
So, \( f(x + h) = 3(x^2 + 2xh + h^2) + 4x + 4h = 3x^2 + 6xh + 3h^2 + 4x + 4h \)

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

Subtract \( f(x) = 3x^2 + 4x \) from \( f(x + h) \):
\( f(x + h) - f(x) = (3x^2 + 6xh + 3h^2 + 4x + 4h) - (3x^2 + 4x) \)
Simplify: \( 3x^2 + 6xh + 3h^2 + 4x + 4h - 3x^2 - 4x = 6xh + 3h^2 + 4h \)

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

eq 0 \) )
\( \frac{f(x + h) - f(x)}{h} = \frac{6xh + 3h^2 + 4h}{h} \)
Factor out \( h \) in the numerator: \( \frac{h(6x + 3h + 4)}{h} \)
Cancel \( h \): \( 6x + 3h + 4 \)
Looking at the options, none match \( 6x + 3h + 4 \), so the answer is "None of the given choices".

Answer:

None of the given choices