Question

Find the value of $f\left(a\right),f\left(a+h\right),\frac{f\left(a+h\right)-f\left(a\right)}{h}$for the function $f\left(x\right)=x^3$.

Short answer

The value of $f\left(a\right)=a^3,f\left(a+h\right)=a^3+h^3+3a^{2}h+3a{h}^2,\frac{f\left(a+h\right)-f\left(a\right)}{h}=h^2+3a^2+3ah$.

Step-by-step solution

Step 1. Evaluate the values of fa and fa+h.

From the given function,

$f\left(x\right)=x^3\mathrm{....}\left(1\right)$

Evaluate $f\left(a\right)\text{and}f\left(a+h\right)$gives,

role="math" localid="1644571715795" $$\begin{gathered} f\left(a\right)={\left(a\right)}^3\mathrm{....}\left(\text{Substitute}x=a\text{in}\left(1\right)\right) \\ =a^3 \\ f\left(a+h\right)={\left(a+h\right)}^3\mathrm{....}\left(\text{Substitute}x=a+h\text{in}\left(1\right)\right) \\ =a^3+h^3+3a^{2}h+3a{h}^2 \end{gathered}$$

Step 2. Subtract fa from fa+h and divide by h where h≠0.

Evaluate $\frac{f\left(a+h\right)-f\left(a\right)}{h}$gives,

role="math" localid="1644571869107" $$\begin{gathered} \frac{f\left(a+h\right)-f\left(a\right)}{h}=\frac{\left(a^3+h^3+3a^{2}h+3a{h}^2\right)-\left(a^3\right)}{h} \\ =\frac{h^3+3a^{2}h+3a{h}^2}{h} \\ =h^2+3a^2+3ah \end{gathered}$$

Step 3. Description of steps.

The value of $f\left(a\right)=a^3,f\left(a+h\right)=a^3+h^3+3a^{2}h+3a{h}^2,\frac{f\left(a+h\right)-f\left(a\right)}{h}=h^2+3a^2+3ah$.