Question

Evaluate the difference quotient for the given function. Simplify your answer.

\(f\left( x \right) = {x^3}\), \(\frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\)

Short answer

The value of the expression \(\frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\)is \(3{a^2} + 3ah + {h^2}\).

Step-by-step solution

Describe the given information

The given function is as follows:

\(f\left( x \right) = {x^3}\)

It is required to find\(\frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\).

Simplify the expression \(\frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\)

Solve for\(f\left( {a + h} \right)\).

\(\begin{aligned}f\left( {a + h} \right) &= {\left( {a + h} \right)^3}\\ &= {a^3} + {h^3} + 3ah\left( {a + h} \right)\\ &= {a^3} + {h^3} + 3{a^2}h + 3a{h^2}\end{aligned}\)

Solve for\(f\left( a \right)\).

\(f\left( 3 \right) = {a^3}\)

Substitute all the values in the expression\(\frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\).

\(\begin{aligned}\frac{{f\left( {a + h} \right) - f\left( a \right)}}{h} &= \frac{{{a^3} + {h^3} + 3{a^2}h + 3a{h^2} - {a^3}}}{h}\\ &= \frac{{{h^3} + 3{a^2}h + 3a{h^2}}}{h}\\ &= h\left( {\frac{{{h^2} + 3{a^2} + 3ah}}{h}} \right)\\ &= 3{a^2} + 3ah + {h^2}\end{aligned}\)

Therefore, the value of the expression \(\frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\)is\(3{a^2} + 3ah + {h^2}\).