Practice Problems

Work through the odd-numbered problems 1-17. Once you have completed the problem set, check your answers.


In problem 1, each quotation is a statement about a quantitity of something changing over time. Let \mathrm{f}(\mathrm{t}) represent the quantity at time \mathrm{t}. For each quotation, tell what \mathrm{f} represents and whether the first and second derivatives of \mathrm{f} are positive or negative.

1. (a) "Unemployment rose again, but the rate of increase is smaller than last month".
(b) "Our profits declined again, but at a slower rate than last month".
(c) "The population is still rising and at a faster rate than last year".

3. Sketch the graphs of functions which are defined and concave up everywhere and which have
(a) no roots.
(b) exactly 1 root.
(c) exactly 2 roots.
(d) exactly 3 roots.

5. On which intervals is the function in Fig. 12
(a) concave up?
(b) concave down?

Fig. 12

In problems 7 and 9, a function and values of x so that f^{\prime}(x)=0 are given. Use the Second Derivative Test to determine whether each point (x, f(x)) is a local maximum, a local minimum or neither

7. \mathrm{g}(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}-9 \mathrm{x}+7, \mathrm{x}=-1,3

9. \mathrm{f}(\mathrm{x})=\sin ^{5}(\mathrm{x}), \quad \mathrm{x}=\pi / 2, \pi, 3 \pi / 2

11. At which labeled values of \mathrm{x} in Fig. 13 is the point (\mathrm{x}, \mathrm{f}(\mathrm{x})) an inflection point?

Fig. 13

13. How many inflection points can a
(a) quadratic polynomial have?
(b) cubic polynomial have?
(c) polynomial of degree n have?

15. Fill in the table with "+", "-", or "0" for the function in Fig. 16

\begin{array}{l|l|l|l}x & g(x) & g^{\prime}(x) & g^{\prime \prime}(x) \\\hline 0 & & & \\1 & & & \\2 & & & \\3 & & &\end{array}

Fig. 16

17. Some people like to think of a concave up graph as one which will "hold water" and of a concave down graph as one which will "spill water". That description is accurate for a concave down graph, but it can fail for a concave up graph. Sketch the graph of a function which is concave up on an interval, but which will not "hold water".

Source: Dale Hoffman,
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