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Calculus Problems

Math 504-505 Jerry L. Kazdan

1. Sketch the points (x,y) in the plane R2 that satisfy |y-x|2.

2. A certain function f(x) has the property that 0xf(t)dt=excosx+C. Find both f and the constant C.

3. Compute limx0(cosxcos2x)1/x2

4. Sketch the curve that is defined implicitly by x3+y3-3xy=0. Calculate y(O) .

5. Calculate n=114n2-1.

6. Determine the indefinite integral log(1+x2)dx.

7. Let f(x) be a smooth function for 0x1. If f(x)=0 for all 0x1, what can you conclude? Prove all your assertions.

8. Solve the initial value problem (1+ex)yy=ex with y(1)=1.

9. Let the continuous function f(Θ), 0Θ2π 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 g(x):=x3(1-x) . Without computation, show that g(c)=0 for some 0<

c<1.

b) Let h(x) :=x3(1-x)3. Show that h(x) has exactly three distinct roots in the interval 0<x<1.

c) Let p(x):=(ddx)4(1-x2)4. Show that p is a polynomial of degree 4 and that it has 4 real distinct zeroes, all lying in the interval -1<x<1.

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11. In R2, let Q1=(x1,y1), Q2=(x2,y2) , and Q3=(x3,y3) , where x1<x2<x3.

a) Show there is a unique quadratic polynomial p(x) that passes through these points:

p(xj)=yj, j=1,2,3.

b) If y1>y2 and y3>y2 and f(x) is any smooth function that passes through these three points, show there is some point c(x1,x3) where f(c)>0. Even better, for some c, f(c)p, so p is the optimal constant. [Remark: It is enough to consider the special case where x2=0 and y2=0. Then write x1=-a<0,

x3=b>0].

12. a) If f(x)>0 is continuous for all x0 and the improper integral 0f(x)dx exists, then limxf(x)=0. Proof or counterexample.

b) If f(x)>0 is continuous for all x0 and the improper integral 0f(x)dx exists, then f(x) is bounded. Proof or counterexample.

13. Find explicit rational functions f(x) and g(x) with the following Taylor series: f(x)= 1nxn, g(x)=1n2xn.

14. a) Let x=(x1,x2) be a point in R2 and consider R21(1+||x2)pdx. For which p does this improper integral converge?

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

c) Repeat part a) where xR3 and the integral is over R3 instead of R2.

15. Compute R21[1+(2x+y+1)2+(x-y+3)2]2dxdy.

16. Let v(x,t) :=x-2tx+2tg(s)ds, where g is a continuous function. Compute v/t and v/x.

17. Let H(t) :=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):=x3+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 d2<-4c3/27].

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b) Generalize to the real polynomial p(x) :=ax3+bx2+cx+d(a0) by a change of variable reducing to the above special case.

19. If b0, show that for every real c the equation x5+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 tR. 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(O)=2g(1)=0g(4)=6. Show that at some point 0<c<4 one has g(c)>0. Better yet, find a number m>0 so that g(c)m>0.

Is it true that g must be positive at at least one point 0<c<1 ? Proof or counterex- ample.

22. Let f(x) be a differentiable function for all real x with the property that f(x)<1 for all x. Show has at most one fixed point, that is, at most one point c where f(c)=c.

23. Let g be a differentiable function with the properties g(a)=0, g(b)=0, and g(x) 0 for all x[a, b]. What can you deduce about g(x) for x[a, b] ? Justify your conclusions.

24. Let v(x) be a smooth real-valued function for 0x1. If v(O)=v(1)=0 and v(x)>0 for all 0x1, show that v(x)0 for all 0x1.

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)f(tb+(1-t)a) for all 0t1.

26. a) Find an integer N so that 1+12+13+...+1N>100. b) Find an integer N so that 1+12+13+...+1N>100.

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27. Let c be any complex number. Show that limncnn!=0.

28. a) Show that sinx is not a polynomial.

b) Show that sinx 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 ex is not a rational function.

29. Let f(x) be a differentiable function of x:=(x1,x2,x3) for all xR3. If f is ho- mogeneous of degree k in the sense that f(cx)=ckf(x) for all c>0, show that xf(x)=kf(x) (Euler).

30. The Gamma function is defined by Γ(x) :=0e-ttx-1dt.

a) For which real x does this improper integral converge?

b) Show that Γ(x+1)=xΓ(x) and deduce that Γ(n+1)=n! for any integer n0.

31. Say γ(t) : RR2 defines a smooth curve in the plane.

a) If γ(0)=0 and sqrt(t)c, show that for any T0, γ(T)cT. Moreover, show that equality can occur if and only if one has γ(t)=cvt where v is a unit vector that does not depend on t.

b) If γ(0)=0, sqrt(0)=0 and sqrt(t)12, give an upper bound estimate for γ(2). When can this upper bound be achieved?

32. Let r(t) define a smooth curve that does not pass through the origin.

a) If the point a=r(t0) 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 r(t0) is perpendicular to the tangent vector r(t0) .

b) What can you say about a point b=r(t1) that is furthest from the origin?

33. Consider two smooth plane curves γ1, γ2:(0,1)R2 that do not intersect. Suppose P1 and P2 are points on γ1 and γ2, respectively, such that the distance |P1P2| is mini- mal. Prove that the straight line P1P2 is normal to both curves.

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34. Let h(x,y,z)=0 define a smooth surface in R3 and let P:=(a,b, c) be a point 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 r(t) describe a smooth curve and let V be a fixed vector. If r(t) is perpendicular to V for all t and if r(O) is perpendicular to V, show that 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 ut=3ux. Show that u also satisfies the wave equation utt=9uxx.

37. Let u(x,y) be a smooth function.

a) If ux=0 with u(O,y)=sin(3y) , find u(x,y) .

b) If ux=2xy with u(O,y)=sin(3y) , find u(x,y) .

c) If ux+uy=0 with u(O,y)=sin(3y) , find u(x,y) . Is there more than one such function?

d) If ux+uy=3-2xy with u(O,y)=sin(3y) , find u(x,y) . Is

e) If ux-2uy=0 with u(O,y)=sin(3y) , find u(x,y) . Is there more than one such function?

38. Let r:=xi+yj and V(x,y) :=p(x,y)i+q(x,y)j be (smooth) vector fields and C a smooth curve in the plane. In this problem I is the line integral I=CV. dr. 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 divV=0, show that the line integral CVNds=DVNds= [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 conservation of mass, since divV=0 means there are no sources or sinks in the region Q.]

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40. (Integration by Parts for Multiple Integrals) Let F be a smooth vector field and u a smooth scalar-valued function.

a) Prove the identity (uF)=uF+u. 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 C with unit outer normal N. Also, let u be a scalar-valued function, and F a vector field. Prove the identity

q)uFdA=CuFNds-q)uFdA.

Notice that for a function of one variable with q) being the interval {a<x<b}, 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 F=v. Show the above formula for integration by parts becomes (say in two dimensions)

q)uvdA=CuvNds-q)uvdA.

This is Green's theorem. To what does this reduce for functions on one variable? e) As a short application using this, say u(x,y) is a harmonic function in a bounded region q), so Δu:=u=uxx+uyy=0. One can think of u(x,y) as being the equilibrium temperature of q). Let C is the boundary of q). If u=0 on C, it is plausible that one must have u(x,y)=0 throughout 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 u=0?

41. Let q) be a bounded region in the plane, and let B be its boundary.

a) Use the divergence theorem (or any related formula you know) to show that for any smooth function v(x,y)

bΔvdxdy=BvNds

where v/N:=vn is the outer normal directional derivative on B.

b) Let u(x,y,t) be a solution of the heat equation ut=Δu for (x,y) in q). Assume that the boundary, B, is insulated, so the outer normal derivative there is zero: UN=0 on B.

Show that Q(t) :=bu(x,y,t)dxdy is a constant.

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42. Continuing the notation of the previous problem, say that instead the temperature u(x,y,t)=0 for all points (x,y) on the boundary B.

a) Show that the function E(t) :=12q)u2(x,y,t)dxdy has the property that dE/dt 0.

b) Use this to show that with these zero boundary conditions, if the initial temperature is zero, u(x,y,O)=0, then u(x,y,t)=0 for all t0.

43. Let u(x,y,t) describe the motion of a vibrating drumhead q). A reasonable mathe- matical model shows that u satisfies the wave equation utt=Δu in q) with boundary condition u(x,y,t)=0 for all (x,y) on the boundary q).

Physical reasoning leads one to define the energy as

E(t) :=12q)(ut2+|u|2)dA.

a) Show that energy is conserved: E(t)=E(0) . [HINT: Show dE/dt=0.]

b) If in addition one knows that the initial position u(x,y,O)=0 and that the initial velocity ut(x,y,O)=0, show that E(t)=0 for all t and deduce that u(x,y,t) 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 h(t)ch(t) , where c is a constant, show that h(t)ecth(0) for all t0.

45. Say u(t) satisfies u+b(t)u+c(t)u=0, where b(t) and c(t) are bounded functions. Let E(t) :=12(u2+u2) .

a) Show that E(t)γE(t) , where Y is a constant.

b) Deduce that if u(O)=0 and u(O)=0, then u(t)=0 for all t.

46. Let ν:={u(x)C2(R)|u+u=0}. Prove that dimν=2. Prove all of your assertions in detail.

47. The solutions to the following matrix differential equation

X={3-111}X

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

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48. Consider the differential equation X(t)=AX(t) where

A=(00-10-20100)

Which of the following assertions are correct-and why?

a) There is a solution of the form X(t)=U, where U is a real constant (non-zero) vector.

b) There is a solution of the form X(t)=Ve2t, where V is a real constant (non-zero) vector.

c) There is a solution of the form X(t)=Ve-2t, where V is a real constant (non-zero) vector.

d) There is a complex solution of the form X(t)=Weit, where W is a constant (non- zero) vector.

e) All of the solutions of this equation remain bounded as t.

49. Consider the second order differential equation X=AX where A is a symmetric 2×2 matrix.

a) Find the general solution if A=(500-3).

b) Find the general solution if A=(1441). [Suggestion: First diagonalize A, so D:=R-1AR is diagonal. Then make the change of variables X=RY to obtain a simpler differential equation for Y(t).]

c) Find the general solution if A=(-211-2).

50. For which complex numbers z does the series 1ne-nz converge?

51. a) Let u(x1, ...,xn) be a smooth function that depends only on the distance r= x12++xn2. Show that

2uxj2=xj2r2d2udr2+(r2-xj2)r3dudr , and hence Δu=urr+n-1rur.

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b) Find all harmonic functions (these are the solutions of Δu=0) that depend only on r.

52. a) Find the equation of the tangent plane to the surface x2+xy+y3-z2=2 at the point (1, 1, 1) .

b) Say the function T=x2+xy+y3-z2 gives the temperature at the point (x,y,z) . At the point (1, 1, 1) , in which direction should one move so that the temperature increases fastest?

53. Let ψ(t) be a scalar-valued function with a continuous derivative for 0<t< and let X=(x,y,z)R3. Define the vector field F(X):=ψ(X)X for all X0. Show that this vector field is conservative by finding a scalar-valued function φ(r) with the property that F(X) :=φ(X) . In particular, this shows that every centralforce field is conservative.

54. Let q) be a bounded region in the plane with smooth boundary B. Show that

Area (q) ) =12Bxdy-ydx.

Use this to find the area inside the ellipse (x,y)=(acos6, b sin6) for 0Θ2π.

55. If {bj}>0, prove the arithmetic-geometric mean inequality

(b1b2...bn)1/nb1+b2+...+bnn.

When is there equality?

56. Let 0<c<1. Show that sct1-c<cs+(1-c)t for all s, t>0, st (if s=t, then this becomes an equality).

57. Let p, q1 with 1p+1q=1. Show that xyxpp+yqq for all x, y>0.

58. Let P1,...Pn be n3 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 : R2R that has exactly three critical points, all non- degenerate, one being a local max, one a local 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 limλ01|sinλx|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 V(x,y,z)=(4y,x, 2z) in 3-space, find the value of the integral

Hcurl VndA

where H is the hemisphere x2+y2+z2=a2, z0, 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)O satisfy the differential equation -u+c(x)u=}u on the bounded interval Ω={a<x<b} with u=0 on the boundary of Ω. Show that

}=Ω(u2+cu2)dxΩu2dx

b) Let c(x,y) be a given smooth function and u(x,y)O satisfy the differential equa- tion -(uxx+uyy)+cu=}u on a bounded set ΩR2 with u=0 on the boundary of Ω. Show that

}=Ω(|u|2+cu2)dxdyΩu2dxdy

64. Investigate the continuity and differentiability of

f(x)={|x|pcos1xfor x0,0for x=0, '

where p is a real number.

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65. Determine the radius of convergence of the series

n=0xn2n! .

66. Calculate limx{c1/3[(x+1)2/3-(x-1)2/3]}.

67. Prove that the function f(x)=(1+1X)x is monotone increasing for x>0.

68. Compute dxsinx+cosx.

69. Find the critical points of each of the following functions defined on the plane R2. Also, where possible, classify these critical points as local maxima, minima, or sad- dles.

a) f(x,y)=x4+y4-4xy+1

b) g(x,y)=x2y2

c) cosx1+y2

70. Find an example of a smooth function f(x,y) defined on the whole plane R2 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 ex=1+x+x22!+x33!+... to show that e is irrational.

a) Show that 2<e<3, so e is not an integer.

b) Assume e=p/q is rational with p and q integers with q2. Use Taylor series with q terms and remainder Rq to show that eq!=N+ecq+1, where N is an integer and 0<c<1.

c) Deduce that ecq+1 is an integer. Show this contradicts ec<e1<3 and q+13.

72. Let h(x)0 be a continuous monotonically decreasing function for 0x with the property that limxh(x)=0. Show that the improper integral 0h(x)sinxdx exists.

73. Let fC2(a,b) and say x0(a,b) . If f(x0)>0, give an analytic proof that near x0 the graph of y=f(x) lies above its tangent line at (x0, f(x0)) .

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74. Let Ik=-x2ke-x2dx, k0.

a) Show that Ik=2k-12Ik-2.

b) Compute I2, I3, I4, I5, I6, and I7. [You may use that -e-x2dx=π].

[Last revised: January 25, 2013]

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