### Home > PC3 > Chapter 11 > Lesson 11.2.2 > Problem11-105

11-105.

Let $f(x) = x^3$.

1. Calculate the average rate of change from $x = 3$ to $x = 5$.

$\frac{f(5)-f(3)}{5-3}$

2. Show that the average rate of change from $x = 4$ to $x = 4 + h$ is $48 + 12h + h^{2}$.

$\frac{f(4+h)-f(4)}{(4+h)-4}$

3. Evaluate the expression from part (b) as $h → 0$ to determine the instantaneous rate of change when $x = 4$.

$\lim\limits_{ h \to 0} \frac{f(4+h)-f(4)}{(4+h)-4}$

4. Show that the average rate of change from $x$ to $x + h$ is $3x^{2} + 3xh + h^{2}$.

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

5. Evaluate the expression from part (d) as $h → 0$ to write an expression for the instantaneous rate of change for any value of $x$.

$\lim\limits_{ h \to 0}\frac{f(x+h)-f(x)}{(x+h)-x}$