91Ó°ÊÓ

State the Pythagorean theorem and give an example of a triangle that illustrates the theorem.

If $f\text{'}\left(x\right)=g\text{'}\left(x\right)$for all $x∈[a,b]$, then for some constant $C$, $f\left(x\right)=\_\_\_$ for all $x∈[a,b]$.

Each of the limits in Exercises 7–12 is of the indeterminate form $0·∞$or $∞·0$. Rewrite each limit so that it is (a) in the form $\frac{0}{0}$and then (b) in the form $\frac{∞}{∞}$. Then (c) determine which of these indeterminate forms would be easier to work with when applying L’Hopital’s rule.

7. role="math" localid="1648564249473" $\underset{x→∞}{\lim }{2}^{-x}x$

Suppose $f$is defined and continuous everywhere. Why is testing the sign of the derivative $f$ at just one point sufficient to determine the sign of $f$on the whole interval between critical points of $f$?

If a continuous, differentiable function f has zeroes at x = −4, x = 1, and x = 2, what can you say about f ' on [−4, 2]?

Sign analyses for second derivatives: Repeat the instructions of the previous block of problems, except find sign intervals for the second derivative $f\text{'}\text{'}$instead of the first derivative.

$f\left(x\right)=\frac{\sin x}{e^x}$

Suppose f is a function that is discontinuous somewhere on an interval I. Explain why comparing the values of any local extrema of f on I and the values or limits of f at the endpoints of I is not in general sufficient to determine the global extrema of f on I.

Write a formula for: A cone of height y and circular base of area A.

Part (a): The volume.

Part (b): The surface area of each solid described below.

Sketch the graph of a function f that has an inflection point at $x=c$in such a way that the derivative $f\text{'}$has a local maximum at$x=c$. Add tangent lines to your sketch to illustrate that$f\text{'}$does have a local maximum at$x=c$.

Intervals of behavior: For each of the following functions $f$, determine the intervals on which$f$ is positive, negative, increasing, decreasing, concave up, and concave down.