미분 전체 문제

Sketch the graph of a function $f$ where $f$ has a relative maximum at $x=1$ and $f\,'(1)$ is undefined.

T/F: If $c$ is a critical value of a function $f$, then $f$ has either a relative maximum or relative minimum at $x=c$.

Fill in the blanks: The critical points of a function $f$ are found where $f\,'(x)$ is equal to _____ or where $f\,'(x)$ is _____.

Explain in your own words what the Mean Value Theorem states.

Explain in your own words what Rolle's Theorem states.

In your own words describe what it means for a function to be increasing.

What does a decreasing function “look like”?

Sketch a graph of a function on $[0,2]$ that is increasing, where it is increasing “quickly” near $x=0$ and increasing “slowly” near $x=2$.

Give an example of a function describing a situation where it is “bad” to be increasing and “good” to be decreasing.

T/F: Functions always switch from increasing to decreasing, or decreasing to increasing, at critical points.

A function $f$ has derivative $f\,'(x) = (\sin x+2)e^{x^2+1}$, where $f\,'(x) >1$ for all $x$. Is $f$ increasing, decreasing, or can we not tell from the given information?

Why is sketching curves by hand beneficial even though technology is ubiquitous?

What does “ubiquitous” mean?

T/F: When sketching graphs of functions, it is useful to find the critical points.

T/F: When sketching graphs of functions, it is useful to find the possible points of inflection.

T/F: When sketching graphs of functions, it is useful to find the horizontal and vertical asymptotes.

T/F: When sketching graphs of functions, one need not plot any points at all.

T/F: An “optimization problem” is essentially an “extreme values” problem in a “story problem” setting.

T/F: This section teaches one to find the extreme values of a function that has more than one variable.

True or false: $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\{f(x)+g(x)\}=f'(x)+g'(x)$ when $f$ and $g$ are differentiable functions.