Question

homework assignment 1.1: functions and function notat
score: 13.75/15 answered: 14/15
question 15
for the function $f(x) = 4x^3 + 5x$, evaluate and simplify.
$\frac{f(x + h) - f(x)}{h} = \square$
question help: video written example

Explanation:

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

Substitute \( x + h \) into \( f(x) = 4x^3 + 5x \):
\( f(x + h)=4(x + h)^3+5(x + h) \)
Expand \( (x + h)^3=x^3 + 3x^2h+3xh^2 + h^3 \), so:
\( f(x + h)=4(x^3 + 3x^2h + 3xh^2+h^3)+5x + 5h=4x^3+12x^2h + 12xh^2+4h^3+5x + 5h \)

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

Subtract \( f(x)=4x^3 + 5x \) from \( f(x + h) \):
\( f(x + h)-f(x)=(4x^3+12x^2h + 12xh^2+4h^3+5x + 5h)-(4x^3 + 5x) \)
Simplify by canceling \( 4x^3 \) and \( 5x \):
\( f(x + h)-f(x)=12x^2h + 12xh^2+4h^3+5h \)

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

eq0 \) )
Divide \( f(x + h)-f(x) \) by \( h \):
\( \frac{f(x + h)-f(x)}{h}=\frac{12x^2h + 12xh^2+4h^3+5h}{h} \)
Factor out \( h \) from the numerator:
\( \frac{h(12x^2 + 12xh+4h^2+5)}{h} \)
Cancel \( h \) (since \( h
eq0 \)):
\( 12x^2 + 12xh+4h^2+5 \)

Answer:

\( 12x^2 + 12xh + 4h^2 + 5 \)