Problem 18E

18. Let $f(x)=x^{3}$.

(a) Estimate the values of $f^{\prime}(0), f^{\prime}\left(\frac{1}{2}\right), f^{\prime}(1), f^{\prime}(2)$, and $f^{\prime}(3)$ by using a graphing device to zoom in on the graph of $f$.

(b) Use symmetry to deduce the values of $f^{\prime}\left(-\frac{1}{2}\right), f^{\prime}(-1)$, $f^{\prime}(-2)$, and $f^{\prime}(-3)$

(c) Use the values from parts (a) and (b) to graph $f$ !

(d) Guess a formula for $f^{\prime}(x)$.

(e) Use the definition of derivative to prove that your guess in part (d) is correct.

Step-by-Step Solution