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Calculus Problems
Math 504-505 Jerry L. Kazdan
1. Sketch the points $(x,y)$ in the plane $\mathbb{R}^{2}$ that satisfy $|y-x|\leq 2.$
2. A certain function $f(x)$ has the property that $\displaystyle \int_{0}^{x}f(t)dt=e^{x}\cos x+C$. Find both $f$ and the constant $C.$
3. Compute $\displaystyle \lim_{x\rightarrow 0}(\frac{\cos x}{\cos 2x})^{1/x^{2}}$
4. Sketch the curve that is defined implicitly by $x^{3}+y^{3}-3xy=0$. Calculate $y'(\mathrm{O})$ .
5. Calculate $\displaystyle \sum_{n=1}^{\infty}\frac{1}{4n^{2}-1}.$
6. Determine the indefinite integral $\displaystyle \int\log(1+x^{2})dx.$
7. Let $f(x)$ be a smooth function for $0\leq x\leq 1$. If $f'(x)=0$ for all $0\leq x\leq 1$, what can you conclude? Prove all your assertions.
8. Solve the initial value problem $(1+e^{x})yy'=e^{x}$ with $y(1)=1.$
9. Let the continuous function $f(\Theta), 0\leq\Theta\leq 2\pi$ represent the temperature along the equator at a certain moment, say measured from the longitude at Greenwich.. Show there are antipodal points with the {\it same} temperature.
10. a) Let $g(x):=x^{3}(1-x)$ . Without computation, show that $g'''(c)=0$ for some $0<$
$$
c<1.
$$
b) Let $h(x) :=x^{3}(1-x)^{3}$. Show that $h'''(x)$ has exactly three distinct roots in the interval $0y_{2}$ and $y_{3}>y_{2}$ and $f(x)$ is any smooth function that passes through these three points, show there is some point $c\in(x_{1},x_{3})$ where $f''(c)>0$. Even better, for some $c, f''(c)\geq p''$, so $p''$ is the optimal constant. [Remark: It is enough to consider the special case where $x_{2}=0$ and $y_{2}=0$. Then write $x_{1}=-a<0,$
$$
x_{3}=b>0].
$$
12. a) If $f(x)>0$ is continuous for all $x\geq 0$ and the improper integral $\displaystyle \int_{0}^{\infty}f(x)dx$ exists, then $\displaystyle \lim_{x\rightarrow\infty}f(x)=0$. Proof or counterexample.
b) If $f(x)>0$ is continuous for all $x\geq 0$ and the improper integral $\displaystyle \int_{0}^{\infty}f(x)dx$ exists, then $f(x)$ is bounded. Proof or counterexample.
13. Find {\it explicit} rational functions $f(x)$ and $g(x)$ with the following Taylor series: $f(x)= \displaystyle \sum_{1}^{\infty}nx^{n}, g(x)=\displaystyle \sum_{1}^{\infty}n^{2}x^{n}.$
14. a) Let $x=(x_{1},x_{2})$ be a point in $\mathbb{R}^{2}$ and consider $\displaystyle \int_{\mathbb{R}^{2}}\frac{1}{(1+||x\Vert^{2})^{p}}dx$. For which $p$ does this improper integral converge?
b) This integral can be computed explicitly. Do so.
c) Repeat part a) where $x\in \mathbb{R}^{3}$ and the integral is over $\mathbb{R}^{3}$ instead of $\mathbb{R}^{2}.$
15. Compute $\displaystyle \iint_{\mathbb{R}^{2}}\frac{1}{[1+(2x+y+1)^{2}+(x-y+3)^{2}]^{2}}dxdy.$
16. Let $v(x,t) :=\displaystyle \int_{x-2t}^{x+2t}g(s)ds$, where $g$ is a continuous function. Compute $\partial v/\partial t$ and $\partial v/\partial x.$
17. Let $H(t) :=\displaystyle \int_{a(t)}^{b(t)}f(x,t)dx$, where $a(t), b(t)$ , and $f(x,t)$ are smooth functions of their variables. Compute $dH/dt.$
18. a) Let $p(x):=x^{3}+cx+d$, where $c$, and $d$ are real. Under what conditions on $c$ and $d$ does this has three distinct real roots? $[$ANSWER: $c<0$ and $d^{2}<-4c^{3}/27].$
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b) Generalize to the real polynomial $p(x) :=ax^{3}+bx^{2}+cx+d(a\neq 0)$ by a change of variable reducing to the above special case.
19. If $b\geq 0$, show that for every real $c$ the equation $x^{5}+bx+c=0$ has exactly one real root. What if $b<0$? Say as much as you can.
20. Let $f(t)$ and $g(t)$ be smooth increasing functions of $t\in \mathbb{R}$. Proof or counterexam- ple:
a) $f(t)+g(t)$ is an increasing functions of $t.$
b) $f(t)g(t)$ is an increasing functions of $t.$
c) If $f(t)>0$ and $g(t)>1$ then $f(t)^{g(t)}$ is an increasing functions of $t.$
21. Let a smooth function $g(x)$ have the three properties: $g(\mathrm{O})=2g(1)=0g(4)=6.$ Show that at some point $00$. Better yet, find a number $m>0$ so that $g''(c)\geq m>0.$
Is it true that $g''$ must be positive at at least one point $00$ for all $0\leq x\leq 1$, show that $v(x)\leq 0$ for all $0\leq x\leq 1.$
25. If a smooth curve $y=f(x)$ has the property that $f''(x)>0$, show that the chordjoining two points of the curve lies above the curve:
$tf(b)+(1-t)f(a)\geq f(tb+(1-t)a)$ for all $0\leq t\leq 1.$
26. a) Find an integer $N$ so that $1+\displaystyle \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{N}}>100.$ b) Find an integer $N$ so that $1+\displaystyle \frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{N}>100.$
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27. Let $c$ be any complex number. Show that $\displaystyle \lim_{n\rightarrow\infty}\frac{c^{n}}{n!}=0.$
28. a) Show that $\sin x$ is not a polynomial.
b) Show that $\sin x$ is not a rational function, that is, it cannot be the quotient of two polynomials.
c) Let $f(t)$ be periodic with period 1, so $f(t+1)=f(t)$ for all real $t$. If $f$ is not a constant, show that it cannot be a rational function. that is, $f$ cannot be the quotient of two polynomials.
d) Show that $e^{x}$ is not a rational function.
29. Let $f(x)$ be a differentiable function of $x:=(x_{1},x_{2},x_{3})$ for all $x\in \mathbb{R}^{3}$. If $f$ is {\it ho}- {\it mogeneous of degree} $k$ in the sense that $f(cx)=c^{k}f(x)$ for all $c>0$, show that $x\cdot\nabla f(x)=kf(x)$ (Euler).
30. The {\it Gamma function} is defined by $\Gamma(x) :=\displaystyle \int_{0}^{\infty}e^{-t}t^{x-1}dt.$
a) For which real $x$ does this improper integral converge?
b) Show that $\Gamma(x+1)=x\Gamma(x)$ and deduce that $\Gamma(n+1)=n!$ for any integer $n\geq 0.$
31. Say $\gamma(t)$ : $\mathbb{R}\rightarrow \mathbb{R}^{2}$ defines a smooth curve in the plane.
a) If $\gamma(0)=0$ and $\Vert\sqrt{}(t)\Vert\leq c$, show that for any $T\geq 0, \Vert\gamma(T)\Vert\leq cT$. Moreover, show that equality can occur if and only if one has $\gamma(t)=cvt$ where $v$ is a unit vector that does not depend on $t.$
b) If $\gamma(0)=0, \sqrt{}(0)=0$ and $\Vert\sqrt{}'(t)\Vert\leq 12$, give an upper bound estimate for $\Vert\gamma(2)\Vert.$ When can this upper bound be achieved?
32. Let $\mathrm{r}(t)$ define a smooth curve that does not pass through the origin.
a) If the point $\mathrm{a}=\mathrm{r}(t_{0})$ is a point on the curve that is closest to the origin (and {\it not} an end point of the curve), show that the position vector $\mathrm{r}(t_{0})$ is perpendicular to the tangent vector $\mathrm{r}'(t_{0})$ .
b) What can you say about a point $\mathrm{b}=\mathrm{r}(t_{1})$ that is {\it furthest} from the origin?
33. Consider two smooth plane curves $\gamma_{1}, \gamma_{2}:(0,1)\rightarrow \mathbb{R}^{2}$ that do not intersect. Suppose $P_{1}$ and $P_{2}$ are points on $\gamma_{1}$ and $\gamma_{2}$, respectively, such that the distance $|P_{1}P_{2}|$ is mini- mal. Prove that the straight line $P_{1}P_{2}$ is normal to {\it both} curves.
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34. Let $h(x,y,z)=0$ define a smooth surface in $\mathbb{R}^{3}$ and let $P:=(a,b,\ c)$ be a point {\it not} on the surface. If $Q:(x,y,z)$ is a point on the surface that is closest to $P$, show that the line $PQ$ is perpendicular to the tangent plane to the surface at $Q.$
35. Let $\mathrm{r}(t)$ describe a smooth curve and let V be a fixed vector. If $\mathrm{r}'(t)$ is perpendicular to V for all $t$ and if $\mathrm{r}(\mathrm{O})$ is perpendicular to V, show that $\mathrm{r}(t)$ is perpendicular to V for all $t.$
36. Let $f(s)$ be any differentiable function of the real variable $s$. Show that $u(x,t) := f(x+3t)$ has the property that $u_{t}=3u_{x}$. Show that $u$ also satisfies the wave equation $u_{tt}=9u_{xx}.$
37. Let $u(x,y)$ be a smooth function.
a) If $u_{x}=0$ with $u(\mathrm{O},y)=\sin(3y)$ , find $u(x,y)$ .
b) If $u_{x}=2xy$ with $u(\mathrm{O},y)=\sin(3y)$ , find $u(x,y)$ .
c) If $u_{x}+u_{y}=0$ with $u(\mathrm{O},y)=\sin(3y)$ , find $u(x,y)$ . Is there more than one such function?
d) If $u_{x}+u_{y}=3-2xy$ with $u(\mathrm{O},y)=\sin(3y)$ , find $u(x,y)$ . Is
e) If $u_{x}-2u_{y}=0$ with $u(\mathrm{O},y)=\sin(3y)$ , find $u(x,y)$ . Is there more than one such function?
38. Let $\mathrm{r}:=x\mathrm{i}+y\mathrm{j}$ and $\mathrm{V}(x,y) :=p(x,y)\mathrm{i}+q(x,y)\mathrm{j}$ be (smooth) vector fields and $C$ a smooth curve in the plane. In this problem $I$ is the line integral $I=\displaystyle \int_{C}\mathrm{V}$. {\it d}r. For each of the following, either give a proof or give a counterexample.
a) If $C$ is a vertical line segment and $q(x,y)=0$, then $I=0.$
b) If $C$ is a circle and $q(x,y)=0$, then $I=0.$
c) If $C$ is a circle centered at the origin and $p(x,y)=-q(x,y)$ , then $I=0.$
d) If $p(x,y)>0$ and $q(x,y)>0$, then $I>0.$
39. Let $C$ denote the unit circle centered at the origin of the plane, and $D$ denote the circle of radius 5 centered at $(2,\ 1)$ , both oriented counterclockwise. Let $Q$ denote the ring region between these curves. If a vector field V satisfies $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{V}=0$, show that the line integral $\displaystyle \int_{C}\mathrm{V}\cdot \mathrm{N}ds=\int_{D}\mathrm{V}\cdot \mathrm{N}ds=$ [This extends immediately to the situation where $C$ and $D$ are more general curves and $Q$ is the region between them. For fluid flow it is an expression of {\it conservation of mass}, since $\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{V}=0$ means there are no sources or sinks in the region $Q.$]
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40. (Integration by Parts for Multiple Integrals) Let $\mathrm{F}$ be a smooth vector field and $u$ a smooth scalar-valued function.
a) Prove the identity $\nabla\cdot(u\mathrm{F})=\nabla u\cdot \mathrm{F}+u\nabla$. F. Compare this with the special case of a function of one variable.
b) Let {\it q}) be a bounded region in the plane whose boundary is the curve $C$ with unit outer normal N. Also, let $u$ be a scalar-valued function, and $\mathrm{F}$ a vector field. Prove the identity
$$
\int\int_{q)}u\nabla\cdot \mathrm{F}dA=\oint_{C}u\mathrm{F}\cdot \mathrm{N}ds-\int\int_{q)}\nabla u\cdot \mathrm{F}dA.
$$
Notice that for a function of one variable with {\it q}) being the interval $\{a0$, prove the arithmetic-geometric mean inequality
$$
(b_{1}b_{2}\cdots b_{n})^{1/n}\leq\frac{b_{1}+b_{2}+\cdots+b_{n}}{n}.
$$
When is there equality?
56. Let $00, s\neq t$ (if $s=t$, then this becomes an equality).
57. Let $p, q\geq 1$ with $\displaystyle \frac{1}{p}+\frac{1}{q}=1$. Show that $xy\displaystyle \leq\frac{x^{p}}{p}+\frac{y^{q}}{q}$ for all $x, y>0.$
58. Let $P_{1},\ldots P_{n}$ be $n\geq 3$ points on a circle and let $Q$ be the polygon obtained by connect- ing these successive points. How should the points be situated to maximize the area of $Q$?
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59. a) Find a smooth function $f$ : $\mathbb{R}^{2}\rightarrow \mathbb{R}$ that has exactly three critical points, all non- degenerate, one being a local $\displaystyle \max$, one a local $\displaystyle \min$, and the third a saddle.
b) Show there is no such $f(x,y)$ of the form $f(x,y)=g(x)+h(y)$ .
60. Compute $\displaystyle \lim_{\lambda\rightarrow\infty}\int_{0}^{1}|\sin\lambda x|dx$ (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 $[a,b]$ 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 $\mathrm{V}(x,y,z)=(4y,x,\ 2z)$ in 3-space, find the value of the integral
$\displaystyle \int\int_{H}$curl V$\cdot \mathrm{n}dA$
where $H$ is the hemisphere $x^{2}+y^{2}+z^{2}=a^{2}, z\geq 0, \mathrm{n}$ is the unit outward normal and $dA$ is the element of area.
63. a) Let $c(x)$ be a given smooth function and $u(x)\not\equiv \mathrm{O}$ satisfy the differential equation $-u''+c(x)u=\}_{\backslash u}$ on the bounded interval $\Omega=\{a0.$
68. Compute $\displaystyle \int\frac{dx}{\sin x+\cos x}.$
69. Find the critical points of each of the following functions defined on the plane $\mathbb{R}^{2}.$ Also, where possible, classify these critical points as local maxima, minima, or sad- dles.
a) $f(x,y)=x^{4}+y^{4}-4xy+1$
b) $g(x,y)=x^{2}y^{2}$
c) $\displaystyle \frac{\cos x}{1+y^{2}}$
70. Find an example of a smooth function $f(x,y)$ defined on the whole plane $\mathbb{R}^{2}$ 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 $ e^{x}=1+x+\displaystyle \frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots$ to show that $e$ is irrational.
a) Show that $20$, give an analytic proof that near $x_{0}$ the graph of $y=f(x)$ lies above its tangent line at $(x_{0},\ f(x_{0}))$ .
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74. Let $I_{k}=\displaystyle \int_{-\infty}^{\infty}x^{2k}e^{-x^{2}}dx, k\geq 0.$
a) Show that $I_{k}=\displaystyle \frac{2k-1}{2}I_{k-2}.$
b) Compute $I_{2}, I_{3}, I_{4}, I_{5}, I_{6}$, and $I_{7}$. [You may use that $\displaystyle \int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}$].
[Last revised: January 25, 2013]
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