Home > APCALC > Chapter 6 > Lesson 6.1.3 > Problem6-36

6-36.

Sketch a continuous function that satisfies all of the following conditions. Read carefully—some are limits of the derivative, not the function. Where is f non‑differentiable? Homework Help ✎

• $\lim\limits _ { x \rightarrow - \infty } f ( x ) = \infty$

• $\lim\limits _ { x \rightarrow 0 ^ { - } } f ^ { \prime } ( x ) = - 1$

• $\lim\limits _ { x \rightarrow 0 ^ { + } } f ^ { \prime } ( x ) = 1$

• $\lim\limits _ { x \rightarrow \infty } f ( x ) = 5$

Clues 1 and 4 refer to the end behavior of f(x). Notice that there f(x) approaches ∞ in one direction, but approaches a horizontal asymptote in the other.

Clues 2 and 3 refer the slopes before and after x = 0. Notice that these limits describes the derivative, f '(x), not the actual function that you are graphing. Also notice that the limit from the left does not agree with the limit from the right, so f(x) is NOT differentiable at x = 1.