Problem 17E

17. Let $f(x)=x^{2}$.

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

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

(c) Use the results from parts (a) and (b) to guess a formula for $f^{\prime}(x)$.

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

 

Step-by-Step Solution