Question

Evaluate \(f(2)\) and \(f(2.1)\) and use the results to approximate \(f^{\prime}(2)\) \(f(x)=\frac{1}{4} x^{3}\)

Short answer

The approximate value of \(f^{\prime}(2)\) is 3.0725.

Step-by-step solution

Evaluate the function

Evaluate \(f(x)\) at \(x=2\) and \(x=2.1\). Using \(f(x)=\frac{1}{4} x^{3}\), we obtain \(f(2)=\frac{1}{4} 2^{3}=2\) and \(f(2.1)=\frac{1}{4} (2.1)^{3} \approx 2.30725\).

Approximate the derivative

Using the results from step 1, we can estimate the derivative at the point \(x=2\) using the formula \[f^{\prime}(2) \approx \frac{f(2.1)-f(2)}{2.1-2} \rightarrow f^{\prime}(2) \approx \frac{2.30725-2}{2.1-2}=3.0725\].