]> No Title

Calculus Problems

Math 504-505 Jerry L. Kazdan

1. Sketch the points $\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ in the plane ${\mathbb{R}}^{\mathrm{2}}$ that satisfy $\mathrm{|}\mathit{y}\mathrm{-}\mathit{x}\mathrm{|}\mathrm{\le }\mathrm{2}\mathrm{.}$

2. A certain function $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ has the property that ${\int }_{\mathrm{0}}^{\mathit{x}}\mathit{f}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathit{d}\mathit{t}\mathrm{=}{\mathit{e}}^{\mathit{x}}\mathrm{cos}\mathit{x}\mathrm{+}\mathit{C}$. Find both $\mathit{f}$ and the constant $\mathit{C}\mathrm{.}$

3. Compute $\underset{\mathit{x}\mathrm{\to }\mathrm{0}}{\mathrm{lim}}\mathrm{\left(}\frac{\mathrm{cos}\mathit{x}}{\mathrm{cos}\mathrm{2}\mathit{x}}{\mathrm{\right)}}^{\mathrm{1}\mathrm{/}{\mathit{x}}^{\mathrm{2}}}$

4. Sketch the curve that is defined implicitly by ${\mathit{x}}^{\mathrm{3}}\mathrm{+}{\mathit{y}}^{\mathrm{3}}\mathrm{-}\mathrm{3}\mathit{x}\mathit{y}\mathrm{=}\mathrm{0}$. Calculate $\mathit{y}\mathrm{\prime }\mathrm{\left(}\mathrm{O}\mathrm{\right)}$ .

5. Calculate $\sum _{\mathit{n}\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}\frac{\mathrm{1}}{\mathrm{4}{\mathit{n}}^{\mathrm{2}}\mathrm{-}\mathrm{1}}\mathrm{.}$

6. Determine the indefinite integral $\int \mathrm{log}\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathit{x}}^{\mathrm{2}}\mathrm{\right)}\mathit{d}\mathit{x}\mathrm{.}$

7. Let $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ be a smooth function for $\mathrm{0}\mathrm{\le }\mathit{x}\mathrm{\le }\mathrm{1}$. If $\mathit{f}\mathrm{\prime }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all $\mathrm{0}\mathrm{\le }\mathit{x}\mathrm{\le }\mathrm{1}$, what can you conclude? Prove all your assertions.

8. Solve the initial value problem $\mathrm{\left(}\mathrm{1}\mathrm{+}{\mathit{e}}^{\mathit{x}}\mathrm{\right)}\mathit{y}\mathit{y}\mathrm{\prime }\mathrm{=}{\mathit{e}}^{\mathit{x}}$ with $\mathit{y}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{=}\mathrm{1}\mathrm{.}$

9. Let the continuous function $\mathit{f}\mathrm{\left(}\mathrm{\Theta }\mathrm{\right)}\mathrm{,}$ $\mathrm{0}\mathrm{\le }\mathrm{\Theta }\mathrm{\le }\mathrm{2}\mathit{\pi }$ represent the temperature along the equator at a certain moment, say measured from the longitude at Greenwich.. Show there are antipodal points with the same temperature.

10. a) Let $\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{:}\mathrm{=}{\mathit{x}}^{\mathrm{3}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathit{x}\mathrm{\right)}$ . Without computation, show that $\mathit{g}\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}\mathit{c}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for some $\mathrm{0}\mathrm{<}$

$\mathit{c}\mathrm{<}\mathrm{1}\mathrm{.}$

b) Let $\mathit{h}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}{\mathit{x}}^{\mathrm{3}}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathit{x}{\mathrm{\right)}}^{\mathrm{3}}$. Show that $\mathit{h}\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ has exactly three distinct roots in the interval $\mathrm{0}\mathrm{<}\mathit{x}\mathrm{<}\mathrm{1}\mathrm{.}$

c) Let $\mathit{p}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{:}\mathrm{=}\mathrm{\left(}\frac{\mathit{d}}{\mathit{d}\mathit{x}}{\mathrm{\right)}}^{\mathrm{4}}\mathrm{\left(}\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathrm{2}}{\mathrm{\right)}}^{\mathrm{4}}$. Show that $\mathit{p}$ is a polynomial of degree 4 and that it has 4 real distinct zeroes, all lying in the interval $\mathrm{-}\mathrm{1}\mathrm{<}\mathit{x}\mathrm{<}\mathrm{1}\mathrm{.}$

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11. In ${\mathbb{R}}^{\mathrm{2}}$, let ${\mathit{Q}}_{\mathrm{1}}\mathrm{=}\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}\mathrm{,}{\mathit{y}}_{\mathrm{1}}\mathrm{\right)}\mathrm{,}$ ${\mathit{Q}}_{\mathrm{2}}\mathrm{=}\mathrm{\left(}{\mathit{x}}_{\mathrm{2}}\mathrm{,}{\mathit{y}}_{\mathrm{2}}\mathrm{\right)}$ , and ${\mathit{Q}}_{\mathrm{3}}\mathrm{=}\mathrm{\left(}{\mathit{x}}_{\mathrm{3}}\mathrm{,}{\mathit{y}}_{\mathrm{3}}\mathrm{\right)}$ , where ${\mathit{x}}_{\mathrm{1}}\mathrm{<}{\mathit{x}}_{\mathrm{2}}\mathrm{<}{\mathit{x}}_{\mathrm{3}}\mathrm{.}$

a) Show there is a unique quadratic polynomial $\mathit{p}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ that passes through these points:

$\mathit{p}\mathrm{\left(}{\mathit{x}}_{\mathit{j}}\mathrm{\right)}\mathrm{=}{\mathit{y}}_{\mathit{j}}\mathrm{,}\mathrm{}\mathit{j}\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{3}\mathrm{.}$

b) If ${\mathit{y}}_{\mathrm{1}}\mathrm{>}{\mathit{y}}_{\mathrm{2}}$ and ${\mathit{y}}_{\mathrm{3}}\mathrm{>}{\mathit{y}}_{\mathrm{2}}$ and $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ is any smooth function that passes through these three points, show there is some point $\mathit{c}\mathrm{\in }\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}\mathrm{,}{\mathit{x}}_{\mathrm{3}}\mathrm{\right)}$ where $\mathit{f}\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}\mathit{c}\mathrm{\right)}\mathrm{>}\mathrm{0}$. Even better, for some $\mathit{c}\mathrm{,}$ $\mathit{f}\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}\mathit{c}\mathrm{\right)}\mathrm{\ge }\mathit{p}\mathrm{\prime }\mathrm{\prime }$, so $\mathit{p}\mathrm{\prime }\mathrm{\prime }$ is the optimal constant. [Remark: It is enough to consider the special case where ${\mathit{x}}_{\mathrm{2}}\mathrm{=}\mathrm{0}$ and ${\mathit{y}}_{\mathrm{2}}\mathrm{=}\mathrm{0}$. Then write ${\mathit{x}}_{\mathrm{1}}\mathrm{=}\mathrm{-}\mathit{a}\mathrm{<}\mathrm{0}\mathrm{,}$

${\mathit{x}}_{\mathrm{3}}\mathrm{=}\mathit{b}\mathrm{>}\mathrm{0}\mathrm{\right]}\mathrm{.}$

12. a) If $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{>}\mathrm{0}$ is continuous for all $\mathit{x}\mathrm{\ge }\mathrm{0}$ and the improper integral ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{d}\mathit{x}$ exists, then ${\mathrm{lim}}_{\mathit{x}\mathrm{\to }\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{0}$. Proof or counterexample.

b) If $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{>}\mathrm{0}$ is continuous for all $\mathit{x}\mathrm{\ge }\mathrm{0}$ and the improper integral ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{d}\mathit{x}$ exists, then $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ is bounded. Proof or counterexample.

13. Find explicit rational functions $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ and $\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ with the following Taylor series: $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}$ ${\sum }_{\mathrm{1}}^{\mathrm{\infty }}\mathit{n}{\mathit{x}}^{\mathit{n}}\mathrm{,}$ $\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}{\sum }_{\mathrm{1}}^{\mathrm{\infty }}{\mathit{n}}^{\mathrm{2}}{\mathit{x}}^{\mathit{n}}\mathrm{.}$

14. a) Let $\mathit{x}\mathrm{=}\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}\mathrm{,}{\mathit{x}}_{\mathrm{2}}\mathrm{\right)}$ be a point in ${\mathbb{R}}^{\mathrm{2}}$ and consider ${\int }_{{\mathbb{R}}^{\mathrm{2}}}\frac{\mathrm{1}}{\mathrm{\left(}\mathrm{1}\mathrm{+}\mathrm{|}\mathrm{|}\mathit{x}{\mathrm{‖}}^{\mathrm{2}}{\mathrm{\right)}}^{\mathit{p}}}\mathit{d}\mathit{x}$. For which $\mathit{p}$ does this improper integral converge?

b) This integral can be computed explicitly. Do so.

c) Repeat part a) where $\mathit{x}\mathrm{\in }{\mathbb{R}}^{\mathrm{3}}$ and the integral is over ${\mathbb{R}}^{\mathrm{3}}$ instead of ${\mathbb{R}}^{\mathrm{2}}\mathrm{.}$

15. Compute ${\iint }_{{\mathbb{R}}^{\mathrm{2}}}\frac{\mathrm{1}}{\mathrm{\left[}\mathrm{1}\mathrm{+}\mathrm{\left(}\mathrm{2}\mathit{x}\mathrm{+}\mathit{y}\mathrm{+}\mathrm{1}{\mathrm{\right)}}^{\mathrm{2}}\mathrm{+}\mathrm{\left(}\mathit{x}\mathrm{-}\mathit{y}\mathrm{+}\mathrm{3}{\mathrm{\right)}}^{\mathrm{2}}{\mathrm{\right]}}^{\mathrm{2}}}\mathit{d}\mathit{x}\mathit{d}\mathit{y}\mathrm{.}$

16. Let $\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}{\int }_{\mathit{x}\mathrm{-}\mathrm{2}\mathit{t}}^{\mathit{x}\mathrm{+}\mathrm{2}\mathit{t}}\mathit{g}\mathrm{\left(}\mathit{s}\mathrm{\right)}\mathit{d}\mathit{s}$, where $\mathit{g}$ is a continuous function. Compute $\mathrm{\partial }\mathit{v}\mathrm{/}\mathrm{\partial }\mathit{t}$ and $\mathrm{\partial }\mathit{v}\mathrm{/}\mathrm{\partial }\mathit{x}\mathrm{.}$

17. Let $\mathit{H}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}{\int }_{\mathit{a}\mathrm{\left(}\mathit{t}\mathrm{\right)}}^{\mathit{b}\mathrm{\left(}\mathit{t}\mathrm{\right)}}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{\right)}\mathit{d}\mathit{x}$, where $\mathit{a}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{,}$ $\mathit{b}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ , and $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{\right)}$ are smooth functions of their variables. Compute $\mathit{d}\mathit{H}\mathrm{/}\mathit{d}\mathit{t}\mathrm{.}$

18. a) Let $\mathit{p}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{:}\mathrm{=}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathit{c}\mathit{x}\mathrm{+}\mathit{d}$, where $\mathit{c}$, and $\mathit{d}$ are real. Under what conditions on $\mathit{c}$ and $\mathit{d}$ does this has three distinct real roots? $\mathrm{\left[}$ANSWER: $\mathit{c}\mathrm{<}\mathrm{0}$ and ${\mathit{d}}^{\mathrm{2}}\mathrm{<}\mathrm{-}\mathrm{4}{\mathit{c}}^{\mathrm{3}}\mathrm{/}\mathrm{2}\mathrm{7}\mathrm{\right]}\mathrm{.}$

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b) Generalize to the real polynomial $\mathit{p}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}\mathit{a}{\mathit{x}}^{\mathrm{3}}\mathrm{+}\mathit{b}{\mathit{x}}^{\mathrm{2}}\mathrm{+}\mathit{c}\mathit{x}\mathrm{+}\mathit{d}\mathrm{\left(}\mathit{a}\mathrm{\ne }\mathrm{0}\mathrm{\right)}$ by a change of variable reducing to the above special case.

19. If $\mathit{b}\mathrm{\ge }\mathrm{0}$, show that for every real $\mathit{c}$ the equation ${\mathit{x}}^{\mathrm{5}}\mathrm{+}\mathit{b}\mathit{x}\mathrm{+}\mathit{c}\mathrm{=}\mathrm{0}$ has exactly one real root. What if $\mathit{b}\mathrm{<}\mathrm{0}$? Say as much as you can.

20. Let $\mathit{f}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ and $\mathit{g}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ be smooth increasing functions of $\mathit{t}\mathrm{\in }\mathbb{R}$. Proof or counterexam- ple:

a) $\mathit{f}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{+}\mathit{g}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is an increasing functions of $\mathit{t}\mathrm{.}$

b) $\mathit{f}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathit{g}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is an increasing functions of $\mathit{t}\mathrm{.}$

c) If $\mathit{f}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{>}\mathrm{0}$ and $\mathit{g}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{>}\mathrm{1}$ then $\mathit{f}\mathrm{\left(}\mathit{t}{\mathrm{\right)}}^{\mathit{g}\mathrm{\left(}\mathit{t}\mathrm{\right)}}$ is an increasing functions of $\mathit{t}\mathrm{.}$

21. Let a smooth function $\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ have the three properties: $\mathit{g}\mathrm{\left(}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathrm{2}\mathit{g}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{g}\mathrm{\left(}\mathrm{4}\mathrm{\right)}\mathrm{=}\mathrm{6}\mathrm{.}$ Show that at some point $\mathrm{0}\mathrm{<}\mathit{c}\mathrm{<}\mathrm{4}$ one has $\mathit{g}\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}\mathit{c}\mathrm{\right)}\mathrm{>}\mathrm{0}$. Better yet, find a number $\mathit{m}\mathrm{>}\mathrm{0}$ so that $\mathit{g}\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}\mathit{c}\mathrm{\right)}\mathrm{\ge }\mathit{m}\mathrm{>}\mathrm{0}\mathrm{.}$

Is it true that $\mathit{g}\mathrm{\prime }\mathrm{\prime }$ must be positive at at least one point $\mathrm{0}\mathrm{<}\mathit{c}\mathrm{<}\mathrm{1}$ ? Proof or counterex- ample.

22. Let $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ be a differentiable function for all real $\mathit{x}$ with the property that $\mathit{f}\mathrm{\prime }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{<}\mathrm{1}$ for all $\mathit{x}$. Show has at most one fixed point, that is, at most one point $\mathit{c}$ where $\mathit{f}\mathrm{\left(}\mathit{c}\mathrm{\right)}\mathrm{=}\mathit{c}\mathrm{.}$

23. Let $\mathit{g}$ be a differentiable function with the properties $\mathit{g}\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{,}$ $\mathit{g}\mathrm{\left(}\mathit{b}\mathrm{\right)}\mathrm{=}\mathrm{0}$, and $\mathit{g}\mathrm{\prime }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\ge }$ $\mathrm{0}$ for all $\mathit{x}\mathrm{\in }\mathrm{\left[}\mathit{a}\mathrm{,}\mathrm{}\mathit{b}\mathrm{\right]}$. What can you deduce about $\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ for $\mathit{x}\mathrm{\in }\mathrm{\left[}\mathit{a}\mathrm{,}\mathrm{}\mathit{b}\mathrm{\right]}$ ? Justify your conclusions.

24. Let $\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ be a smooth real-valued function for $\mathrm{0}\mathrm{\le }\mathit{x}\mathrm{\le }\mathrm{1}$. If $\mathit{v}\mathrm{\left(}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathit{v}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{=}\mathrm{0}$ and $\mathit{v}\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{>}\mathrm{0}$ for all $\mathrm{0}\mathrm{\le }\mathit{x}\mathrm{\le }\mathrm{1}$, show that $\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\le }\mathrm{0}$ for all $\mathrm{0}\mathrm{\le }\mathit{x}\mathrm{\le }\mathrm{1}\mathrm{.}$

25. If a smooth curve $\mathit{y}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ has the property that $\mathit{f}\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{>}\mathrm{0}$, show that the chordjoining two points of the curve lies above the curve:

$\mathit{t}\mathit{f}\mathrm{\left(}\mathit{b}\mathrm{\right)}\mathrm{+}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathit{t}\mathrm{\right)}\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{\ge }\mathit{f}\mathrm{\left(}\mathit{t}\mathit{b}\mathrm{+}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathit{t}\mathrm{\right)}\mathit{a}\mathrm{\right)}$ for all $\mathrm{0}\mathrm{\le }\mathit{t}\mathrm{\le }\mathrm{1}\mathrm{.}$

26. a) Find an integer $\mathit{N}$ so that $\mathrm{1}\mathrm{+}\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\mathrm{+}\frac{\mathrm{1}}{\sqrt{\mathrm{3}}}\mathrm{+}\mathrm{\text{...}}\mathrm{+}\frac{\mathrm{1}}{\sqrt{\mathit{N}}}\mathrm{>}\mathrm{1}\mathrm{0}\mathrm{0}\mathrm{.}$ b) Find an integer $\mathit{N}$ so that $\mathrm{1}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{+}\frac{\mathrm{1}}{\mathrm{3}}\mathrm{+}\mathrm{\text{...}}\mathrm{+}\frac{\mathrm{1}}{\mathit{N}}\mathrm{>}\mathrm{1}\mathrm{0}\mathrm{0}\mathrm{.}$

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27. Let $\mathit{c}$ be any complex number. Show that $\underset{\mathit{n}\mathrm{\to }\mathrm{\infty }}{\mathrm{lim}}\frac{{\mathit{c}}^{\mathit{n}}}{\mathit{n}\mathrm{!}}\mathrm{=}\mathrm{0}\mathrm{.}$

28. a) Show that $\mathrm{sin}\mathit{x}$ is not a polynomial.

b) Show that $\mathrm{sin}\mathit{x}$ is not a rational function, that is, it cannot be the quotient of two polynomials.

c) Let $\mathit{f}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ be periodic with period 1, so $\mathit{f}\mathrm{\left(}\mathit{t}\mathrm{+}\mathrm{1}\mathrm{\right)}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ for all real $\mathit{t}$. If $\mathit{f}$ is not a constant, show that it cannot be a rational function. that is, $\mathit{f}$ cannot be the quotient of two polynomials.

d) Show that ${\mathit{e}}^{\mathit{x}}$ is not a rational function.

29. Let $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ be a differentiable function of $\mathit{x}\mathrm{:}\mathrm{=}\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}\mathrm{,}{\mathit{x}}_{\mathrm{2}}\mathrm{,}{\mathit{x}}_{\mathrm{3}}\mathrm{\right)}$ for all $\mathit{x}\mathrm{\in }{\mathbb{R}}^{\mathrm{3}}$. If $\mathit{f}$ is ho- mogeneous of degree $\mathit{k}$ in the sense that $\mathit{f}\mathrm{\left(}\mathit{c}\mathit{x}\mathrm{\right)}\mathrm{=}{\mathit{c}}^{\mathit{k}}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ for all $\mathit{c}\mathrm{>}\mathrm{0}$, show that $\mathit{x}\mathrm{\cdot }\mathrm{\nabla }\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathit{k}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ (Euler).

30. The Gamma function is defined by $\mathrm{\Gamma }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}{\int }_{\mathrm{0}}^{\mathrm{\infty }}{\mathit{e}}^{\mathrm{-}\mathit{t}}{\mathit{t}}^{\mathit{x}\mathrm{-}\mathrm{1}}\mathit{d}\mathit{t}\mathrm{.}$

a) For which real $\mathit{x}$ does this improper integral converge?

b) Show that $\mathrm{\Gamma }\mathrm{\left(}\mathit{x}\mathrm{+}\mathrm{1}\mathrm{\right)}\mathrm{=}\mathit{x}\mathrm{\Gamma }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ and deduce that $\mathrm{\Gamma }\mathrm{\left(}\mathit{n}\mathrm{+}\mathrm{1}\mathrm{\right)}\mathrm{=}\mathit{n}\mathrm{!}$ for any integer $\mathit{n}\mathrm{\ge }\mathrm{0}\mathrm{.}$

31. Say $\mathit{\gamma }\mathrm{\left(}\mathit{t}\mathrm{\right)}$ : $\mathbb{R}\mathrm{\to }{\mathbb{R}}^{\mathrm{2}}$ defines a smooth curve in the plane.

a) If $\mathit{\gamma }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$ and $\mathrm{‖}\mathrm{sqrt}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{‖}\mathrm{\le }\mathit{c}$, show that for any $\mathit{T}\mathrm{\ge }\mathrm{0}\mathrm{,}$ $\mathrm{‖}\mathit{\gamma }\mathrm{\left(}\mathit{T}\mathrm{\right)}\mathrm{‖}\mathrm{\le }\mathit{c}\mathit{T}$. Moreover, show that equality can occur if and only if one has $\mathit{\gamma }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{c}\mathit{v}\mathit{t}$ where $\mathit{v}$ is a unit vector that does not depend on $\mathit{t}\mathrm{.}$

b) If $\mathit{\gamma }\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{,}$ $\mathrm{sqrt}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$ and $\mathrm{‖}\mathrm{sqrt}\mathrm{\prime }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{‖}\mathrm{\le }\mathrm{1}\mathrm{2}$, give an upper bound estimate for $\mathrm{‖}\mathit{\gamma }\mathrm{\left(}\mathrm{2}\mathrm{\right)}\mathrm{‖}\mathrm{.}$ When can this upper bound be achieved?

32. Let $\mathrm{r}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ define a smooth curve that does not pass through the origin.

a) If the point $\mathrm{a}\mathrm{=}\mathrm{r}\mathrm{\left(}{\mathit{t}}_{\mathrm{0}}\mathrm{\right)}$ is a point on the curve that is closest to the origin (and not an end point of the curve), show that the position vector $\mathrm{r}\mathrm{\left(}{\mathit{t}}_{\mathrm{0}}\mathrm{\right)}$ is perpendicular to the tangent vector $\mathrm{r}\mathrm{\prime }\mathrm{\left(}{\mathit{t}}_{\mathrm{0}}\mathrm{\right)}$ .

b) What can you say about a point $\mathrm{b}\mathrm{=}\mathrm{r}\mathrm{\left(}{\mathit{t}}_{\mathrm{1}}\mathrm{\right)}$ that is furthest from the origin?

33. Consider two smooth plane curves ${\mathit{\gamma }}_{\mathrm{1}}\mathrm{,}$ ${\mathit{\gamma }}_{\mathrm{2}}\mathrm{:}\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}\mathrm{\to }{\mathbb{R}}^{\mathrm{2}}$ that do not intersect. Suppose ${\mathit{P}}_{\mathrm{1}}$ and ${\mathit{P}}_{\mathrm{2}}$ are points on ${\mathit{\gamma }}_{\mathrm{1}}$ and ${\mathit{\gamma }}_{\mathrm{2}}$, respectively, such that the distance $\mathrm{|}{\mathit{P}}_{\mathrm{1}}{\mathit{P}}_{\mathrm{2}}\mathrm{|}$ is mini- mal. Prove that the straight line ${\mathit{P}}_{\mathrm{1}}{\mathit{P}}_{\mathrm{2}}$ is normal to both curves.

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34. Let $\mathit{h}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{z}\mathrm{\right)}\mathrm{=}\mathrm{0}$ define a smooth surface in ${\mathbb{R}}^{\mathrm{3}}$ and let $\mathit{P}\mathrm{:}\mathrm{=}\mathrm{\left(}\mathit{a}\mathrm{,}\mathit{b}\mathrm{,}\mathrm{}\mathit{c}\mathrm{\right)}$ be a point not on the surface. If $\mathit{Q}\mathrm{:}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{z}\mathrm{\right)}$ is a point on the surface that is closest to $\mathit{P}$, show that the line $\mathit{P}\mathit{Q}$ is perpendicular to the tangent plane to the surface at $\mathit{Q}\mathrm{.}$

35. Let $\mathrm{r}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ describe a smooth curve and let V be a fixed vector. If $\mathrm{r}\mathrm{\prime }\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is perpendicular to V for all $\mathit{t}$ and if $\mathrm{r}\mathrm{\left(}\mathrm{O}\mathrm{\right)}$ is perpendicular to V, show that $\mathrm{r}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is perpendicular to V for all $\mathit{t}\mathrm{.}$

36. Let $\mathit{f}\mathrm{\left(}\mathit{s}\mathrm{\right)}$ be any differentiable function of the real variable $\mathit{s}$. Show that $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}$ $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{+}\mathrm{3}\mathit{t}\mathrm{\right)}$ has the property that ${\mathit{u}}_{\mathit{t}}\mathrm{=}\mathrm{3}{\mathit{u}}_{\mathit{x}}$. Show that $\mathit{u}$ also satisfies the wave equation ${\mathit{u}}_{\mathit{t}\mathit{t}}\mathrm{=}\mathrm{9}{\mathit{u}}_{\mathit{x}\mathit{x}}\mathrm{.}$

37. Let $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ be a smooth function.

a) If ${\mathit{u}}_{\mathit{x}}\mathrm{=}\mathrm{0}$ with $\mathit{u}\mathrm{\left(}\mathrm{O}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{sin}\mathrm{\left(}\mathrm{3}\mathit{y}\mathrm{\right)}$ , find $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ .

b) If ${\mathit{u}}_{\mathit{x}}\mathrm{=}\mathrm{2}\mathit{x}\mathit{y}$ with $\mathit{u}\mathrm{\left(}\mathrm{O}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{sin}\mathrm{\left(}\mathrm{3}\mathit{y}\mathrm{\right)}$ , find $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ .

c) If ${\mathit{u}}_{\mathit{x}}\mathrm{+}{\mathit{u}}_{\mathit{y}}\mathrm{=}\mathrm{0}$ with $\mathit{u}\mathrm{\left(}\mathrm{O}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{sin}\mathrm{\left(}\mathrm{3}\mathit{y}\mathrm{\right)}$ , find $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ . Is there more than one such function?

d) If ${\mathit{u}}_{\mathit{x}}\mathrm{+}{\mathit{u}}_{\mathit{y}}\mathrm{=}\mathrm{3}\mathrm{-}\mathrm{2}\mathit{x}\mathit{y}$ with $\mathit{u}\mathrm{\left(}\mathrm{O}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{sin}\mathrm{\left(}\mathrm{3}\mathit{y}\mathrm{\right)}$ , find $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ . Is

e) If ${\mathit{u}}_{\mathit{x}}\mathrm{-}\mathrm{2}{\mathit{u}}_{\mathit{y}}\mathrm{=}\mathrm{0}$ with $\mathit{u}\mathrm{\left(}\mathrm{O}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{sin}\mathrm{\left(}\mathrm{3}\mathit{y}\mathrm{\right)}$ , find $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ . Is there more than one such function?

38. Let $\mathrm{r}\mathrm{:}\mathrm{=}\mathit{x}\mathrm{i}\mathrm{+}\mathit{y}\mathrm{j}$ and $\mathrm{V}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}\mathit{p}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{i}\mathrm{+}\mathit{q}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{j}$ be (smooth) vector fields and $\mathit{C}$ a smooth curve in the plane. In this problem $\mathit{I}$ is the line integral $\mathit{I}\mathrm{=}{\int }_{\mathit{C}}\mathrm{V}$. dr. For each of the following, either give a proof or give a counterexample.

a) If $\mathit{C}$ is a vertical line segment and $\mathit{q}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{0}$, then $\mathit{I}\mathrm{=}\mathrm{0}\mathrm{.}$

b) If $\mathit{C}$ is a circle and $\mathit{q}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{0}$, then $\mathit{I}\mathrm{=}\mathrm{0}\mathrm{.}$

c) If $\mathit{C}$ is a circle centered at the origin and $\mathit{p}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{-}\mathit{q}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ , then $\mathit{I}\mathrm{=}\mathrm{0}\mathrm{.}$

d) If $\mathit{p}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{>}\mathrm{0}$ and $\mathit{q}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{>}\mathrm{0}$, then $\mathit{I}\mathrm{>}\mathrm{0}\mathrm{.}$

39. Let $\mathit{C}$ denote the unit circle centered at the origin of the plane, and $\mathit{D}$ denote the circle of radius 5 centered at $\mathrm{\left(}\mathrm{2}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{\right)}$ , both oriented counterclockwise. Let $\mathit{Q}$ denote the ring region between these curves. If a vector field V satisfies $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{V}\mathrm{=}\mathrm{0}$, show that the line integral ${\int }_{\mathit{C}}\mathrm{V}\mathrm{\cdot }\mathrm{N}\mathit{d}\mathit{s}\mathrm{=}{\int }_{\mathit{D}}\mathrm{V}\mathrm{\cdot }\mathrm{N}\mathit{d}\mathit{s}\mathrm{=}$ [This extends immediately to the situation where $\mathit{C}$ and $\mathit{D}$ are more general curves and $\mathit{Q}$ is the region between them. For fluid flow it is an expression of conservation of mass, since $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{V}\mathrm{=}\mathrm{0}$ means there are no sources or sinks in the region $\mathit{Q}\mathrm{.}$]

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40. (Integration by Parts for Multiple Integrals) Let $\mathrm{F}$ be a smooth vector field and $\mathit{u}$ a smooth scalar-valued function.

a) Prove the identity $\mathrm{\nabla }\mathrm{\cdot }\mathrm{\left(}\mathit{u}\mathrm{F}\mathrm{\right)}\mathrm{=}\mathrm{\nabla }\mathit{u}\mathrm{\cdot }\mathrm{F}\mathrm{+}\mathit{u}\mathrm{\nabla }$. F. Compare this with the special case of a function of one variable.

b) Let q) be a bounded region in the plane whose boundary is the curve $\mathit{C}$ with unit outer normal N. Also, let $\mathit{u}$ be a scalar-valued function, and $\mathrm{F}$ a vector field. Prove the identity

$\int {\int }_{\mathit{q}\mathrm{\right)}}\mathit{u}\mathrm{\nabla }\mathrm{\cdot }\mathrm{F}\mathit{d}\mathit{A}\mathrm{=}{\oint }_{\mathit{C}}\mathit{u}\mathrm{F}\mathrm{\cdot }\mathrm{N}\mathit{d}\mathit{s}\mathrm{-}\int {\int }_{\mathit{q}\mathrm{\right)}}\mathrm{\nabla }\mathit{u}\mathrm{\cdot }\mathrm{F}\mathit{d}\mathit{A}\mathrm{.}$

Notice that for a function of one variable with q) being the interval $\mathrm{\left\{}\mathit{a}\mathrm{<}\mathit{x}\mathrm{<}\mathit{b}\mathrm{\right\}}\mathrm{,}$ this reduces precisely to the usual formula for integration by parts.

c) Generalize this formula to the case where q) is a bounded (solid) region in three dimensional space.

d) One frequently uses this with $\mathrm{F}\mathrm{=}\mathrm{\nabla }\mathit{v}$. Show the above formula for integration by parts becomes (say in two dimensions)

$\int {\int }_{\mathit{q}\mathrm{\right)}}\mathit{u}\mathrm{\nabla }\mathrm{\cdot }\mathrm{\nabla }\mathit{v}\mathit{d}\mathit{A}\mathrm{=}{\oint }_{\mathit{C}}\mathit{u}\mathrm{\nabla }\mathit{v}\mathrm{\cdot }\mathrm{N}\mathit{d}\mathit{s}\mathrm{-}\int {\int }_{\mathit{q}\mathrm{\right)}}\mathrm{\nabla }\mathit{u}\mathrm{\cdot }\mathrm{\nabla }\mathit{v}\mathit{d}\mathit{A}\mathrm{.}$

This is Green's theorem. To what does this reduce for functions on one variable? e) As a short application using this, say $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ is a harmonic function in a bounded region $\mathit{q}$), so $\mathrm{\Delta }\mathit{u}\mathrm{:}\mathrm{=}\mathrm{\nabla }\mathrm{\cdot }\mathrm{\nabla }\mathit{u}\mathrm{=}{\mathit{u}}_{\mathit{x}\mathit{x}}\mathrm{+}{\mathit{u}}_{\mathit{y}\mathit{y}}\mathrm{=}\mathrm{0}$. One can think of $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ as being the equilibrium temperature of $\mathit{q}$). Let $\mathit{C}$ is the boundary of $\mathit{q}$). If $\mathit{u}\mathrm{=}\mathrm{0}$ on $\mathit{C}$, it is plausible that one must have $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{0}$ throughout $\mathit{q}$). Show how this follows from the above formula. What is the analogous assertion for functions of one variable, where a harmonic function is just a solution of $\mathit{u}\mathrm{\prime }\mathrm{\prime }\mathrm{=}\mathrm{0}$?

41. Let q) be a bounded region in the plane, and let $\mathit{B}$ be its boundary.

a) Use the divergence theorem (or any related formula you know) to show that for any smooth function $\mathit{v}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$

$\int \mathit{b}\mathrm{\Delta }\mathit{v}\mathit{d}\mathit{x}\mathit{d}\mathit{y}\mathrm{=}{\int }_{\mathit{B}}\frac{\mathrm{\partial }\mathit{v}}{\mathrm{\partial }\mathit{N}}\mathit{d}\mathit{s}$

where $\mathrm{\partial }\mathit{v}\mathrm{/}\mathrm{\partial }\mathit{N}\mathrm{:}\mathrm{=}\mathrm{\nabla }\mathit{v}\mathrm{\cdot }\mathrm{n}$ is the outer normal directional derivative on $\mathit{B}\mathrm{.}$

b) Let $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{t}\mathrm{\right)}$ be a solution of the heat equation ${\mathit{u}}_{\mathit{t}}\mathrm{=}\mathrm{\Delta }\mathit{u}$ for $\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ in $\mathit{q}$). Assume that the boundary, $\mathit{B}$, is insulated, so the outer normal derivative there is zero: $\frac{{\mathrm{\partial }}_{{U}}}{\mathrm{\partial }\mathit{N}}\mathrm{=}\mathrm{0}$ on $\mathit{B}\mathrm{.}$

Show that $\mathit{Q}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}\int \mathit{b}\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{t}\mathrm{\right)}\mathit{d}\mathit{x}\mathit{d}\mathit{y}$ is a constant.

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42. Continuing the notation of the previous problem, say that instead the temperature $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all points $\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ on the boundary $\mathit{B}\mathrm{.}$

a) Show that the function $\mathit{E}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathit{q}\mathrm{\right)}}{\mathit{u}}^{\mathrm{2}}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{t}\mathrm{\right)}\mathit{d}\mathit{x}\mathit{d}\mathit{y}$ has the property that $\mathit{d}\mathit{E}\mathrm{/}\mathit{d}\mathit{t}\mathrm{\le }$ $\mathrm{0}\mathrm{.}$

b) Use this to show that with these zero boundary conditions, if the initial temperature is zero, $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathrm{0}$, then $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all $\mathit{t}\mathrm{\ge }\mathrm{0}\mathrm{.}$

43. Let $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{t}\mathrm{\right)}$ describe the motion of a vibrating drumhead $\mathit{q}$). A reasonable mathe- matical model shows that $\mathit{u}$ satisfies the wave equation ${\mathit{u}}_{\mathit{t}\mathit{t}}\mathrm{=}\mathrm{\Delta }\mathit{u}$ in q) with boundary condition $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all $\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ on the boundary $\mathrm{\partial }\mathit{q}$).

Physical reasoning leads one to define the energy as

$\mathit{E}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{}\mathrm{:}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{\iint }_{\mathit{q}\mathrm{\right)}}\mathrm{\left(}{\mathit{u}}_{\mathit{t}}^{\mathrm{2}}\mathrm{+}\mathrm{|}\mathrm{\nabla }\mathit{u}{\mathrm{|}}^{\mathrm{2}}\mathrm{\right)}\mathit{d}\mathit{A}\mathrm{.}$

a) Show that energy is conserved: $\mathit{E}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{E}\mathrm{\left(}\mathrm{0}\mathrm{\right)}$ . [HINT: Show $\mathit{d}\mathit{E}\mathrm{/}\mathit{d}\mathit{t}\mathrm{=}\mathrm{0}\mathrm{.}$]

b) If in addition one knows that the initial position $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathrm{0}$ and that the initial velocity ${\mathit{u}}_{\mathit{t}}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathrm{0}$, show that $\mathit{E}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all $\mathit{t}$ and deduce that $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{t}\mathrm{\right)}\mathrm{\equiv }$ $\mathrm{0}$. [This is hardly a surprise on physical grounds, but it should be interpreted as reassuring us that this mathematical model is indeed reasonably correct.]

44. If $\mathit{h}\mathrm{\prime }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\le }\mathit{c}\mathit{h}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ , where $\mathit{c}$ is a constant, show that $\mathit{h}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\le }{\mathit{e}}^{\mathit{c}\mathit{t}}\mathit{h}\mathrm{\left(}\mathrm{0}\mathrm{\right)}$ for all $\mathit{t}\mathrm{\ge }\mathrm{0}\mathrm{.}$

45. Say $\mathit{u}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ satisfies $\mathit{u}\mathrm{\prime }\mathrm{\prime }\mathrm{+}\mathit{b}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathit{u}\mathrm{\prime }\mathrm{+}\mathit{c}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathit{u}\mathrm{=}\mathrm{0}$, where $\mathit{b}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ and $\mathit{c}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ are bounded functions. Let $\mathit{E}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\left(}{\mathit{u}}^{\mathrm{\prime }\mathrm{2}}\mathrm{+}{\mathit{u}}^{\mathrm{2}}\mathrm{\right)}$ .

a) Show that $\mathit{E}\mathrm{\prime }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\le }\mathit{\gamma }\mathit{E}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ , where $\mathrm{Y}$ is a constant.

b) Deduce that if $\mathit{u}\mathrm{\left(}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathrm{0}$ and $\mathit{u}\mathrm{\prime }\mathrm{\left(}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathrm{0}$, then $\mathit{u}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{0}$ for all $\mathit{t}\mathrm{.}$

46. Let $\mathit{\nu }\mathrm{:}\mathrm{=}\mathrm{\left\{}\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\in }{\mathit{C}}^{\mathrm{2}}\mathrm{\left(}\mathbb{R}\mathrm{\right)}\mathrm{|}\mathit{u}\mathrm{\prime }\mathrm{\prime }\mathrm{+}\mathit{u}\mathrm{=}\mathrm{0}\mathrm{\right\}}$. Prove that $\mathrm{dim}\mathit{\nu }\mathrm{=}\mathrm{2}$. Prove all of your assertions in detail.

47. The solutions to the following matrix differential equation

$\mathit{X}\mathrm{\prime }\mathrm{=}\left\{\begin{array}{cc}\mathrm{3}& \mathrm{-}\mathrm{1}\\ \mathrm{1}& \mathrm{1}\end{array}\right\}\mathit{X}$

form a vector space. Find a basis for this vector space.

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48. Consider the differential equation $\mathit{X}\mathrm{\prime }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{A}\mathit{X}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ where

$\mathit{A}\mathrm{=}\left(\begin{array}{ccc}\mathrm{0}& \mathrm{0}& \mathrm{-}\mathrm{1}\\ \mathrm{0}& \mathrm{-}\mathrm{2}& \mathrm{0}\\ \mathrm{1}& \mathrm{0}& \mathrm{0}\end{array}\right)$

Which of the following assertions are correct-and why?

a) There is a solution of the form $\mathit{X}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{U}$, where $\mathit{U}$ is a real constant (non-zero) vector.

b) There is a solution of the form $\mathit{X}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{V}{\mathit{e}}^{\mathrm{2}\mathit{t}}$, where $\mathit{V}$ is a real constant (non-zero) vector.

c) There is a solution of the form $\mathit{X}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{V}{\mathit{e}}^{\mathrm{-}\mathrm{2}\mathit{t}}$, where $\mathit{V}$ is a real constant (non-zero) vector.

d) There is a complex solution of the form $\mathit{X}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{W}{\mathit{e}}^{\mathit{i}\mathit{t}}$, where $\mathit{W}$ is a constant (non- zero) vector.

e) All of the solutions of this equation remain bounded as $\mathit{t}\mathrm{\to }\mathrm{\infty }\mathrm{.}$

49. Consider the second order differential equation $\mathit{X}\mathrm{\prime }\mathrm{\prime }\mathrm{=}\mathit{A}\mathit{X}$ where $\mathit{A}$ is a symmetric $\mathrm{2}\mathrm{×}\mathrm{2}$ matrix.

a) Find the general solution if $\mathit{A}\mathrm{=}\left(\begin{array}{cc}\mathrm{5}& \mathrm{0}\\ \mathrm{0}\mathrm{-}\mathrm{3}\end{array}\right)\mathrm{.}$

b) Find the general solution if $\mathit{A}\mathrm{=}\left(\begin{array}{cc}\mathrm{1}& \mathrm{4}\\ \mathrm{4}& \mathrm{1}\end{array}\right)$. [Suggestion: First diagonalize $\mathit{A}$, so $\mathit{D}\mathrm{:}\mathrm{=}{\mathit{R}}^{\mathrm{-}\mathrm{1}}\mathit{A}\mathit{R}$ is diagonal. Then make the change of variables $\mathit{X}\mathrm{=}\mathit{R}\mathrm{Y}$ to obtain a simpler differential equation for $\mathrm{Y}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{.}$]

c) Find the general solution if $\mathit{A}\mathrm{=}\left(\begin{array}{cc}\mathrm{-}\mathrm{2}& \mathrm{1}\\ \mathrm{1}& \mathrm{-}\mathrm{2}\end{array}\right)\mathrm{.}$

50. For which complex numbers $\mathit{z}$ does the series ${\sum }_{\mathrm{1}}^{\mathrm{\infty }}\mathit{n}{\mathit{e}}^{\mathrm{-}\mathit{n}\mathit{z}}$ converge?

51. a) Let $\mathit{u}\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}{\mathit{x}}_{\mathit{n}}\mathrm{\right)}$ be a smooth function that depends only on the distance $\mathit{r}\mathrm{=}$ $\sqrt{{\mathit{x}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{+}\mathrm{+}{\mathit{x}}_{\mathit{n}}^{\mathrm{2}}}$. Show that

$\frac{{\mathrm{\partial }}^{\mathrm{2}}\mathit{u}}{\mathrm{\partial }{\mathit{x}}_{\mathit{j}}^{\mathrm{2}}}\mathrm{=}\frac{{\mathit{x}}_{\mathit{j}}^{\mathrm{2}}}{{\mathit{r}}^{\mathrm{2}}}\frac{{\mathit{d}}^{\mathrm{2}}\mathit{u}}{\mathit{d}{\mathit{r}}^{\mathrm{2}}}\mathrm{+}\frac{\mathrm{\left(}{\mathit{r}}^{\mathrm{2}}\mathrm{-}{\mathit{x}}_{\mathit{j}}^{\mathrm{2}}\mathrm{\right)}}{{\mathit{r}}^{\mathrm{3}}}\frac{\mathit{d}\mathit{u}}{\mathit{d}\mathit{r}}$ , and hence $\mathrm{\Delta }\mathit{u}\mathrm{=}{\mathit{u}}_{\mathit{r}\mathit{r}}\mathrm{+}\frac{\mathit{n}\mathrm{-}\mathrm{1}}{\mathit{r}}{\mathit{u}}_{\mathit{r}}\mathrm{.}$

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b) Find all harmonic functions (these are the solutions of $\mathrm{\Delta }\mathit{u}\mathrm{=}\mathrm{0}$) that depend only on $\mathit{r}\mathrm{.}$

52. a) Find the equation of the tangent plane to the surface ${\mathit{x}}^{\mathrm{2}}\mathrm{+}\mathit{x}\mathit{y}\mathrm{+}{\mathit{y}}^{\mathrm{3}}\mathrm{-}{\mathit{z}}^{\mathrm{2}}\mathrm{=}\mathrm{2}$ at the point $\mathrm{\left(}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{\right)}$ .

b) Say the function $\mathit{T}\mathrm{=}{\mathit{x}}^{\mathrm{2}}\mathrm{+}\mathit{x}\mathit{y}\mathrm{+}{\mathit{y}}^{\mathrm{3}}\mathrm{-}{\mathit{z}}^{\mathrm{2}}$ gives the temperature at the point $\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{z}\mathrm{\right)}$ . At the point $\mathrm{\left(}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{\right)}$ , in which direction should one move so that the temperature increases fastest?

53. Let $\mathit{\psi }\mathrm{\left(}\mathit{t}\mathrm{\right)}$ be a scalar-valued function with a continuous derivative for $\mathrm{0}\mathrm{<}\mathit{t}\mathrm{<}\mathrm{\infty }$ and let $\mathrm{X}\mathrm{=}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{z}\mathrm{\right)}\mathrm{\in }{\mathbb{R}}^{\mathrm{3}}$. Define the vector field $\mathrm{F}\mathrm{\left(}\mathrm{X}\mathrm{\right)}\mathrm{:}\mathrm{=}\mathit{\psi }\mathrm{\left(}\mathrm{‖}\mathrm{X}\mathrm{‖}\mathrm{\right)}\mathrm{X}$ for all $\mathrm{X}\mathrm{\ne }\mathrm{0}$. Show that this vector field is conservative by finding a scalar-valued function $\mathit{\phi }\mathrm{\left(}\mathit{r}\mathrm{\right)}$ with the property that $\mathrm{F}\mathrm{\left(}\mathrm{X}\mathrm{\right)}$ $\mathrm{:}\mathrm{=}\mathrm{\nabla }\mathit{\phi }\mathrm{\left(}\mathrm{‖}\mathrm{X}\mathrm{‖}\mathrm{\right)}$ . In particular, this shows that every centralforce field is conservative.

54. Let q) be a bounded region in the plane with smooth boundary $\mathit{B}$. Show that

Area $\mathrm{\left(}\mathit{q}\mathrm{\right)}$ ) $\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}}{\int }_{\mathit{B}}\mathit{x}\mathit{d}\mathit{y}\mathrm{-}\mathit{y}\mathit{d}\mathit{x}\mathrm{.}$

Use this to find the area inside the ellipse $\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathrm{\left(}\mathit{a}\mathrm{cos}\mathrm{6}\mathrm{,}\mathrm{}\mathit{b}\mathrm{}\mathrm{sin}\mathrm{6}\mathrm{\right)}$ for $\mathrm{0}\mathrm{\le }\mathrm{\Theta }\mathrm{\le }\mathrm{2}\mathit{\pi }\mathrm{.}$

55. If $\mathrm{\left\{}{\mathit{b}}_{\mathit{j}}\mathrm{\right\}}\mathrm{>}\mathrm{0}$, prove the arithmetic-geometric mean inequality

$\mathrm{\left(}{\mathit{b}}_{\mathrm{1}}{\mathit{b}}_{\mathrm{2}}\mathrm{\text{...}}{\mathit{b}}_{\mathit{n}}{\mathrm{\right)}}^{\mathrm{1}\mathrm{/}\mathit{n}}\mathrm{\le }\frac{{\mathit{b}}_{\mathrm{1}}\mathrm{+}{\mathit{b}}_{\mathrm{2}}\mathrm{+}\mathrm{\text{...}}\mathrm{+}{\mathit{b}}_{\mathit{n}}}{\mathit{n}}\mathrm{.}$

When is there equality?

56. Let $\mathrm{0}\mathrm{<}\mathit{c}\mathrm{<}\mathrm{1}$. Show that ${\mathit{s}}^{\mathit{c}}{\mathit{t}}^{\mathrm{1}\mathrm{-}\mathit{c}}\mathrm{<}\mathit{c}\mathit{s}\mathrm{+}\mathrm{\left(}\mathrm{1}\mathrm{-}\mathit{c}\mathrm{\right)}\mathit{t}$ for all $\mathit{s}\mathrm{,}$ $\mathit{t}\mathrm{>}\mathrm{0}\mathrm{,}$ $\mathit{s}\mathrm{\ne }\mathit{t}$ (if $\mathit{s}\mathrm{=}\mathit{t}$, then this becomes an equality).

57. Let $\mathit{p}\mathrm{,}$ $\mathit{q}\mathrm{\ge }\mathrm{1}$ with $\frac{\mathrm{1}}{\mathit{p}}\mathrm{+}\frac{\mathrm{1}}{\mathit{q}}\mathrm{=}\mathrm{1}$. Show that $\mathit{x}\mathit{y}\mathrm{\le }\frac{{\mathit{x}}^{\mathit{p}}}{\mathit{p}}\mathrm{+}\frac{{\mathit{y}}^{\mathit{q}}}{\mathit{q}}$ for all $\mathit{x}\mathrm{,}$ $\mathit{y}\mathrm{>}\mathrm{0}\mathrm{.}$

58. Let ${\mathit{P}}_{\mathrm{1}}\mathrm{,}\mathrm{\text{...}}{\mathit{P}}_{\mathit{n}}$ be $\mathit{n}\mathrm{\ge }\mathrm{3}$ points on a circle and let $\mathit{Q}$ be the polygon obtained by connect- ing these successive points. How should the points be situated to maximize the area of $\mathit{Q}$?

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59. a) Find a smooth function $\mathit{f}$ : ${\mathbb{R}}^{\mathrm{2}}\mathrm{\to }\mathbb{R}$ that has exactly three critical points, all non- degenerate, one being a local $\mathrm{max}$, one a local $\mathrm{min}$, and the third a saddle.

b) Show there is no such $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ of the form $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{+}\mathit{h}\mathrm{\left(}\mathit{y}\mathrm{\right)}$ .

60. Compute $\underset{\mathit{\lambda }\mathrm{\to }\mathrm{\infty }}{\mathrm{lim}}{\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{|}\mathrm{sin}\mathit{\lambda }\mathit{x}\mathrm{|}\mathit{d}\mathit{x}$ (part of the problem is to show that the limit exists).

61. a) State what it means for a real-valued function defined on the closed, bounded in- terval $\mathrm{\left[}\mathit{a}\mathrm{,}\mathit{b}\mathrm{\right]}$ to be Riemann integrable.

b) Using your definition from part (a), prove that any monotonically increasing func- tion on [0,1] is Riemann integrable.

62. Given the vector field $\mathrm{V}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{z}\mathrm{\right)}\mathrm{=}\mathrm{\left(}\mathrm{4}\mathit{y}\mathrm{,}\mathit{x}\mathrm{,}\mathrm{}\mathrm{2}\mathit{z}\mathrm{\right)}$ in 3-space, find the value of the integral

$\int {\int }_{\mathit{H}}$curl V$\mathrm{\cdot }\mathrm{n}\mathit{d}\mathit{A}$

where $\mathit{H}$ is the hemisphere ${\mathit{x}}^{\mathrm{2}}\mathrm{+}{\mathit{y}}^{\mathrm{2}}\mathrm{+}{\mathit{z}}^{\mathrm{2}}\mathrm{=}{\mathit{a}}^{\mathrm{2}}\mathrm{,}$ $\mathit{z}\mathrm{\ge }\mathrm{0}\mathrm{,}$ $\mathrm{n}$ is the unit outward normal and $\mathit{d}\mathit{A}$ is the element of area.

63. a) Let $\mathit{c}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ be a given smooth function and $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\not\equiv }\mathrm{O}$ satisfy the differential equation $\mathrm{-}\mathit{u}\mathrm{\prime }\mathrm{\prime }\mathrm{+}\mathit{c}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathit{u}\mathrm{=}{\mathrm{\right\}}}_{\mathrm{\setminus }\mathit{u}}$ on the bounded interval $\mathrm{\Omega }\mathrm{=}\mathrm{\left\{}\mathit{a}\mathrm{<}\mathit{x}\mathrm{<}\mathit{b}\mathrm{\right\}}$ with $\mathit{u}\mathrm{=}\mathrm{0}$ on the boundary of $\mathrm{\Omega }$. Show that

${\mathrm{\right\}}}_{\mathrm{\setminus }}\mathrm{=}\frac{{\int }_{\mathrm{\Omega }}\mathrm{\left(}{\mathit{u}}^{\mathrm{\prime }\mathrm{2}}\mathrm{+}\mathit{c}{\mathit{u}}^{\mathrm{2}}\mathrm{\right)}\mathit{d}\mathit{x}}{{\int }_{\mathrm{\Omega }}{\mathit{u}}^{\mathrm{2}}\mathit{d}\mathit{x}}$

b) Let $\mathit{c}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ be a given smooth function and $\mathit{u}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{\not\equiv }\mathrm{O}$ satisfy the differential equa- tion $\mathrm{-}\mathrm{\left(}{\mathit{u}}_{\mathit{x}\mathit{x}}\mathrm{+}{\mathit{u}}_{\mathit{y}\mathit{y}}\mathrm{\right)}\mathrm{+}\mathit{c}\mathit{u}\mathrm{=}{\mathrm{\right\}}}_{\mathrm{\setminus }\mathit{u}}$ on a bounded set $\mathrm{\Omega }\mathrm{\subset }{\mathbb{R}}^{\mathrm{2}}$ with $\mathit{u}\mathrm{=}\mathrm{0}$ on the boundary of $\mathrm{\Omega }$. Show that

${\mathrm{\right\}}}_{\mathrm{\setminus }}\mathrm{=}\frac{\int {\int }_{\mathrm{\Omega }}\mathrm{\left(}\mathrm{|}\mathrm{\nabla }\mathit{u}{\mathrm{|}}^{\mathrm{2}}\mathrm{+}\mathit{c}{\mathit{u}}^{\mathrm{2}}\mathrm{\right)}\mathit{d}\mathit{x}\mathit{d}\mathit{y}}{\int {\int }_{\mathrm{\Omega }}{\mathit{u}}^{\mathrm{2}}\mathit{d}\mathit{x}\mathit{d}\mathit{y}}$

64. Investigate the continuity and differentiability of

$\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\left\{\begin{array}{cc}\mathrm{|}\mathit{x}{\mathrm{|}}^{\mathit{p}}\mathrm{cos}\frac{\mathrm{1}}{\mathit{x}}& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{}\mathit{x}\mathrm{\ne }\mathrm{0}\mathrm{,}\\ \mathrm{0}& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{}\mathit{x}\mathrm{=}\mathrm{0}\mathrm{,}\mathrm{}\mathrm{\text{'}}\end{array}$

where $\mathit{p}$ is a real number.

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65. Determine the radius of convergence of the series

$\sum _{\mathit{n}\mathrm{=}\mathrm{0}}^{\mathrm{\infty }}\frac{{\mathit{x}}^{{\mathit{n}}^{\mathrm{2}}}}{\mathit{n}\mathrm{!}}$ .

66. Calculate $\underset{\mathit{x}\mathrm{\to }\mathrm{\infty }}{\mathrm{lim}}\mathrm{\left\{}{\mathrm{c}}^{\mathrm{1}\mathrm{/}\mathrm{3}}\mathrm{\left[}\mathrm{\left(}\mathit{x}\mathrm{+}\mathrm{1}{\mathrm{\right)}}^{\mathrm{2}\mathrm{/}\mathrm{3}}\mathrm{-}\mathrm{\left(}\mathit{x}\mathrm{-}\mathrm{1}{\mathrm{\right)}}^{\mathrm{2}\mathrm{/}\mathrm{3}}\mathrm{\right]}\mathrm{\right\}}\mathrm{.}$

67. Prove that the function $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{\left(}\mathrm{1}\mathrm{+}\frac{\mathrm{1}}{\mathit{X}}{\mathrm{\right)}}^{\mathit{x}}$ is monotone increasing for $\mathit{x}\mathrm{>}\mathrm{0}\mathrm{.}$

68. Compute $\int \frac{\mathit{d}\mathit{x}}{\mathrm{sin}\mathit{x}\mathrm{+}\mathrm{cos}\mathit{x}}\mathrm{.}$

69. Find the critical points of each of the following functions defined on the plane ${\mathbb{R}}^{\mathrm{2}}\mathrm{.}$ Also, where possible, classify these critical points as local maxima, minima, or sad- dles.

a) $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}{\mathit{x}}^{\mathrm{4}}\mathrm{+}{\mathit{y}}^{\mathrm{4}}\mathrm{-}\mathrm{4}\mathit{x}\mathit{y}\mathrm{+}\mathrm{1}$

b) $\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}\mathrm{=}{\mathit{x}}^{\mathrm{2}}{\mathit{y}}^{\mathrm{2}}$

c) $\frac{\mathrm{cos}\mathit{x}}{\mathrm{1}\mathrm{+}{\mathit{y}}^{\mathrm{2}}}$

70. Find an example of a smooth function $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{\right)}$ defined on the whole plane ${\mathbb{R}}^{\mathrm{2}}$ that has exactly three critical points, all non-degenerate, with one alocal maximum, one alocal minimum, and the third a saddle point.

71. Here we use the series ${\mathit{e}}^{\mathit{x}}\mathrm{=}\mathrm{1}\mathrm{+}\mathit{x}\mathrm{+}\frac{{\mathit{x}}^{\mathrm{2}}}{\mathrm{2}\mathrm{!}}\mathrm{+}\frac{{\mathit{x}}^{\mathrm{3}}}{\mathrm{3}\mathrm{!}}\mathrm{+}\mathrm{\text{...}}$ to show that $\mathit{e}$ is irrational.

a) Show that $\mathrm{2}\mathrm{<}\mathit{e}\mathrm{<}\mathrm{3}$, so $\mathit{e}$ is not an integer.

b) Assume $\mathit{e}\mathrm{=}\mathit{p}\mathrm{/}\mathit{q}$ is rational with $\mathit{p}$ and $\mathit{q}$ integers with $\mathit{q}\mathrm{\ge }\mathrm{2}$. Use Taylor series with $\mathit{q}$ terms and remainder ${\mathit{R}}_{\mathit{q}}$ to show that $\mathit{e}\mathrm{\cdot }\mathit{q}\mathrm{!}\mathrm{=}\mathit{N}\mathrm{+}\frac{{\mathit{e}}^{\mathit{c}}}{\mathit{q}\mathrm{+}\mathrm{1}}$, where $\mathit{N}$ is an integer and $\mathrm{0}\mathrm{<}\mathit{c}\mathrm{<}\mathrm{1}\mathrm{.}$

c) Deduce that $\frac{{\mathit{e}}^{\mathit{c}}}{\mathit{q}\mathrm{+}\mathrm{1}}$ is an integer. Show this contradicts ${\mathit{e}}^{\mathit{c}}\mathrm{<}{\mathit{e}}^{\mathrm{1}}\mathrm{<}\mathrm{3}$ and $\mathit{q}\mathrm{+}\mathrm{1}\mathrm{\ge }\mathrm{3}\mathrm{.}$

72. Let $\mathit{h}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\ge }\mathrm{0}$ be a continuous monotonically decreasing function for $\mathrm{0}\mathrm{\le }\mathit{x}\mathrm{\le }\mathrm{\infty }$ with the property that ${\mathrm{lim}}_{\mathit{x}\mathrm{\to }\mathrm{\infty }}\mathit{h}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{0}$. Show that the improper integral ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathit{h}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{sin}\mathit{x}\mathit{d}\mathit{x}$ exists.

73. Let $\mathit{f}\mathrm{\in }{\mathit{C}}^{\mathrm{2}}\mathrm{\left(}\mathit{a}\mathrm{,}\mathit{b}\mathrm{\right)}$ and say ${\mathit{x}}_{\mathrm{0}}\mathrm{\in }\mathrm{\left(}\mathit{a}\mathrm{,}\mathit{b}\mathrm{\right)}$ . If $\mathit{f}\mathrm{\prime }\mathrm{\prime }\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}\mathrm{\right)}\mathrm{>}\mathrm{0}$, give an analytic proof that near ${\mathit{x}}_{\mathrm{0}}$ the graph of $\mathit{y}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ lies above its tangent line at $\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}\mathrm{,}\mathrm{}\mathit{f}\mathrm{\left(}{\mathit{x}}_{\mathrm{0}}\mathrm{\right)}\mathrm{\right)}$ .

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74. Let ${\mathit{I}}_{\mathit{k}}\mathrm{=}{\int }_{\mathrm{-}\mathrm{\infty }}^{\mathrm{\infty }}{\mathit{x}}^{\mathrm{2}\mathit{k}}{\mathit{e}}^{\mathrm{-}{\mathit{x}}^{\mathrm{2}}}\mathit{d}\mathit{x}\mathrm{,}$ $\mathit{k}\mathrm{\ge }\mathrm{0}\mathrm{.}$

a) Show that ${\mathit{I}}_{\mathit{k}}\mathrm{=}\frac{\mathrm{2}\mathit{k}\mathrm{-}\mathrm{1}}{\mathrm{2}}{\mathit{I}}_{\mathit{k}\mathrm{-}\mathrm{2}}\mathrm{.}$

b) Compute ${\mathit{I}}_{\mathrm{2}}\mathrm{,}$ ${\mathit{I}}_{\mathrm{3}}\mathrm{,}$ ${\mathit{I}}_{\mathrm{4}}\mathrm{,}$ ${\mathit{I}}_{\mathrm{5}}\mathrm{,}$ ${\mathit{I}}_{\mathrm{6}}$, and ${\mathit{I}}_{\mathrm{7}}$. [You may use that ${\int }_{\mathrm{-}\mathrm{\infty }}^{\mathrm{\infty }}{\mathit{e}}^{\mathrm{-}{\mathit{x}}^{\mathrm{2}}}\mathit{d}\mathit{x}\mathrm{=}\sqrt{\mathit{\pi }}$].

[Last revised: January 25, 2013]

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