Chapter 3: Applications of the Derivative

State the law of similar triangles and give an example of a pair of triangles that illustrate this law.

If a continuous, differentiable function f is equal to 2 at x = 3 and at x = 5, what can you say about f ' on [3, 5]?

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.

The first-derivative test: Suppose $x=c$is a ____ of a differentiable function $f$. If _____ , then f has a local maximum at $x=c$. If ______ , then f has a local minimum at $x=c$. If _____ , then f has neither a local maximum nor a local minimum at$x=c$.

Describe what the first-derivative test is for and how to use it. Sketch graphs and sign charts to illustrate your description.

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.

$f\left(x\right)=\ln \left(\ln x\right)$

Given the following graph of f , graphically estimate the global extrema of f on each of the six intervals listed:

$$\begin{gathered} \left(a\right)\left[-2,4\right]\left(b\right)\left(-2,4\right)\left(c\right)\left(-1,1\right) \\ \left(d\right)(0,4]\left(e\right)[0,4)\left(f\right)\left(-\infty ,\infty \right) \end{gathered}$$

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 minimum at$x=c$. Add tangent lines to your sketch to illustrate that $f\text{'}$does have a local minimum at$x=c$.

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

8.role="math" localid="1648570578299" $\underset{x\to 0}{\lim }\left(2^x-1\right)x^{-2}$

The Pythagorean Theorem: If a right triangle has legs of lengths $x$and $y$ and a hypotenuse of length $h$, then ____ .