Problem Sets

Suggestions to the Student

The problems we choose from the book are a bit different from the usual calculus textbook problems.  They are not intended to be harder although some may well be.  They are intended, instead, to help you better understand the concepts of calculus and how to apply them.  None of these problems asks simply for a computation, and some ask for no computation at all.  Instead, they may ask you to do one of the following:  Apply a concept or technique you have just learned in a mildly novel context; combine concepts or techniques that you have seen only in isolation before; give a graphical interpretation of the behaviour of a function; make an inference, from a graph or a table of data, about a function or a physical relationship.

When you begin working on these problems,  you may feel that you do not know how to get started on a problem or where you should end up.  That's only natural.  In fact, some of the problems can be approached in a variety of ways and have no single answer.  Since the purpose of all the problems in this volume is to help you develop a better understanding of calculus, a good way to get started is to see if you understand the question.  Talk it over with a classmate and see if the two of you have the same interpretation.  If you don't check in the textbook to see if you have the right meanings for the crucial words in the problem.  Draw a picture, if possible, to illustrate the problem.  If you encounter a function that is hard to graph, use a computer or a graphing calculator to draw the graph.  In fact, all uses of computers and calculators are legitimate in working on these problems. If you are still stuck, talk it over some more with a classmate or ask for a discussion in class, but be prepared to offer the thoughts you have developed about the problem.

The keys to getting the most out of these problems are thinking, discussing and writing.  When you recognize a concept or technique that is likely to be involved in a problem, ask yourself what you know about it and how it might be applied, and be prepared to reread your textbook or lecture notes to refresh your understanding  Then test your ideas by discussing them with a classmate or in class. Finally, write up your conclusions in complete English sentences that convey your understanding as clearly as you know how.  With practice, you will discover that discussing and writing promote clear thinking and thus help you develop a better understanding of the material that you are studying.  

Assignments

Assignment 1
9.1 #1:  Suppose that the level curves of a function g(x,y) are horizontal lines.  What does that imply about g?  Suppose that the level curves of the function f are parallel straight lines.  Must f be a plane?  Justify your answer.
 #2:  Use the distance formula in 3-dimensions to justify:  (a) If P ≠ Q, then the distance between P and Q is positive, (b) the distance between P and Q = the distance between Q and P, (c) if M is the midpoint of P and Q, then the distance between P and M equals the distance between M and Q, which equals half the distance between P and Q.  Note: mostly these are verifications only, but with general points, not particular ones.

9.2 #1:  Prove properties 2 and 6 for 2-dimensional vectors (on p. 25).  You argument should look like those above the list of properties.  Be explicit about where you use properties for real numbers along the way.  #15 from textbook. 
 
9.3 #1:  Let v = <1, 2, 3> and w = <2, 3, -4>.  (a) Find the cosine of the angle between v and w.  Is it obtuse or acute?  How do you know?  (b)  Find the vector projection of v in the direction of w.  (c) Find a nonzero vector x that is perpendicular to both v and w.  There are infinitely many answers.  Perhaps use what must be true about dot products.  #12 from textbook.

9.4  #1 (a) Explain how the cross product can be used to determine whether three points in space are collinear.  (b) Describe a method for determining whether four points lie in the same plane.

#2 Suppose v and w are two different nonzero vectors.  If u x v = u x w, what must be true about u?

Now that your work is done, here are solutions to the problems for assignment one.


Assignment 2
9.5 #1 Let P0 and P1 be points in space, n a unit vector, and ∏ the plane n.(X-P0)= 0 (that is dot, I don't know how to write it centered).  Show that the (perpendicular) distance between the point P1 and the ∏ is d = |n.P0 - n.P1|.
#2 Let l1 be the line x = <1, 1, 2> + t<3, -1, 4> and l2 be the line (x-1)/6 = y/(-2) = (z-3)/8. 
(a) Are the lines l1 and l2 parallel?  Explain.
(b) The point P = (1, 1, 2) is on l1, and the point Q = (1, 0, 3) is on l2.  Find |Q-P|
(c) Find the scalar projection of Q-P in the direction of v = <3, -1,4>
(d) Find the distance between l1 and l2.

9.6
#1 (ok, some of these questions are mixing and matching sections, but fortunately we will have done all of them) An astronaut is flying in a spacecraft along the path described by r(t) = (t^2 - t, 2+t,-3/t), where t is given in hours.  The engines are shut off when the spacecraft reaches the point (6,5,-1).  Where is the astronaut 2 hours later? 
#12 from textbook

9.7 #1 Suppose that P(0) = u and P'(t) = tv, where u and v are constant vectors.  Describe the curve traced by P(t).
#2 Consider the vector-valued function p(t) = (cos t, sin t, t)
 (a) Plot the curve defined by p(t) for 0 ≤ t ≤ 4π
 (b) Find a vector equation for the tangent line l at t = π
 (c) On one set of axes, plot both the curve and the tangent line from part (a).  (Try to do as well as you can with maple … at least something)
 (d) Show that p(t) = (cos t, sin t, t) has constant speed.  Find the arclength from t = 0 to t = 4π. 

9.8 #1 Which helix is longer, one of radius 5 centimeters and height 4 centimeters that makes three complete turns or one of radius 3 centimeters and height 4 centimeters that makes five complete turns?  Justify your answer. 
#2 Find the curvature of the curve y = x^3 at the point (1,1).  Use maple to draw both the curve and the osculating circle there. 

And for something new, here are solutions to assignment 2. 
 
Assignment 3

10.1 Textbook #12 and #14.

10.2  #1 The partial differential equation ∂u/∂t + ∂u/∂x = ku is used in population modeling.  Here u = u(x,t) is the number of individuals of age x at time t, and k is the mortality rate.  Show if a + b = k, then the function u(x,t) = e^(ax+bt) is a solution to this equation. 
Textbook #16

10.3  #1 Define f(x,y) = { xy(x^2 - y^2) / (x^2 + y^2) if (x,y) ≠ (0,0), but 0 if (x,y) = (0,0)
Calculate ∂f/∂x and ∂f/∂y (note:  you will need to use the limit definition).  Now calculate ∂f^2/∂x∂y(0,0), and ∂f^2/∂y∂x(0,0).  Show they are not equal.
Textbook #12

10.4  #1 Find a point on the surface x^2 + y^2 + 3z^2 = 8 where the tangent plane is parallel to the plane 2x + y + 3z = 0. 
Textbook #14

Look, and you will see … solutions to assignment 3.

Assignment 4

10.5  #1 Let z = ƒ(x,y) be a function of the Cartesian coordinates x and y.  Show that if the variable substitutions x = r cos ø and y = r sin ø are used to express ƒ in polar coordinates, then ∂^2ƒ/∂x^2 + ∂^2ƒ/∂y^2 = ∂^2ƒ/∂r^2 + 1/r ∂f/∂r + 1/r^2 ∂^2ƒ/∂ø^2.
Textbook #17

10.6 #1 Let g(x,y) = x^2 - 3xy + 6 and let C be the level curve of g that passes through the point (x0, y0).  (Feel free to let Maple give you a contour plot if you wish to see them.)
a.  Show that if g(x0,y0) ≠ 6, then (x0,y0) is a point on the curve described by the equation y = (x^2 + 6 - g(x0,y0))/(3x).
b. Suppose that g(x0,y0) ≠ 6.  Find a vector that is tangent to C at (x0,y0).  [Hint:  Use implicit differentiation.]
c. Suppose that g(x0,y0) = 6 and x0 ≠ 0.  Show that (x0,y0) is a point on the line y = x/3.
d. Suppose that g(x0,y0) =6 and x0 = 0.  Show that (x0,y0) is a point on the line x = 0.
e. Use parts b-d to show that in all cases the gradient of g(x0,y0) is perpendicular to C at (x0,y0).

#2 Suppose that f(x,y) = x^2y.  In what direction(s) from the point (1,2) is the rate of change 3?

10.7 #1 If a continuous function of one variable has at least two local maxima, then it must also have at least one local minimum (think about drawing this picture).  The situation is different for functions of two variables.  Show that f(x,y) = (x^2-1)^2+y^2 has exactly three critical points - two local minima and a saddle point. 
Choose one of #21, #22, #23 in the textbook.
 

10.8 #1 Create a contour plot for ƒ(x,y) = x^2 + xy + y^2.  We will consider minimize ƒ subject to the constraint g(x,y) = x + y - 2 = 0.  (There is no constrained maximum.)
a.  Carefully draw the constraint set g(x,y) = 0 into the picture.  Label some contour lines with their z-values. 
b.  Using the picture alone, estimate the points at which the constrained minimum occurs and the values of ƒ at this point. 
c.  Use the Lagrange multiplier condition to check your work in the previous part. 
Textbook #14

Want some more?  Here are solutions to assignment 4.
 
Assignment 5

11.1 #1
Let I = the double integral of (x^2 + y)dA over R, (I don't know how to type integral here, sorry) where R = [0,1] x [0,2]
a. Explain why I ≥ 0
b. Explain why I ≤ 6
c. Estimate I by calculating a double midpoint sum with four subdivisions (two in each direction).
Textbook #13

11.2 #1
a. Let R = [a,b] x [c,d] and f(x) and g(y) be functions such that the integral from a to b of f(x)dx = 29 and the integral from c to d of g(y)dy = 37.  Compute the double integral of f(x) + g(y)dA over R.
b. Give a geometric interpretation of the integral of h(x,y)dx from x=1 to x=4.
Textbook #12

11.3 #1 Aside from being difficult to type here (you'll see), this integral is difficult to evaluate in the order given.  Write it with the order of integration reversed and then evaluate the integral.  The original integral is:  the integral of 1/(ln y)dydx over 0 ≤ x ≤ 1 (outer bound) and e^x ≤ y ≤ e (inner bound).
Textbook #13

11.4 #1 Consider the triangle with vertices at (0,1), (0,-1), and (a,b), where (a,b) is any point in the xy-plane with a > 0.  (Every triangle is similar to a triangle like this.)
a.  Show by calculating an appropriate integral that the triangle has centre of mass at (a/3, b/3).  [Hint: write equations for the "top" and "bottom" edges of the triangle; use these lines in an iterated integral.]
b. The centroid of a triangle with vertices (a1,b1), (a2,b2), and (a3,b3) is defined to the point ((a1,b1) + (a2,b2) + (a3,b3))/3 = ((a1+a2+a3)/3, (b1+b2+b3)/3) that is the average of the three vertices.  Explain why the centre of mass of the triangle in part (a) is also the centroid. 
Textbook #13

To find what you can derive, please look at solutions to assignment 5.


Assignment 6

11.5 #1 Find the volume of the solid bounded by the paraboloids z = x^2 + y^2, and z = 2 - (x^2 + y^2).
Textbook #19 - but that seems too long - so - do a or c, and do b or d.  So, do half of the problem, but choose one of each kind.
 
11.6 Textbook #11

11.7 #1 Suppose that I = the triple integral of f(x,y,z)dV over the region R above the triangle in the xy-plane described by 0 ≤ x ≤ 2 and 0 ≤ y ≤ 2-x and below the cone z = 3√(x^2+y^2) i.e. z is 3 times the square root of x^2+y^2.  Sketch the region of integration R.  Let f(x,y,z) = xz.  Evaluate I.  
Textbook #15.

11.8 #1 Suppose that I = the triple integral of f(x,y,z)dV over the region R bounded by the cone z = √x^2+y^2 (again that is the square root of x^2 + y^2) and the paraboloid z = 2 - (x^2 + y^2).  Sketch the region of integration R.  Let f(x,y,z) = x + y + z.  Evaluate I.  [Hint:  what section is this in?]
Textbook #15 - find one of these three to evaluate using spherical coordinates.  Pick which you like.  The intent of the problem is one is easier in spherical coordinates.  Find that one, and do it. 

11.9 #1 Let I = the double integral of (x+y)dA over Rxy, the region bounded by 3x - 2y = 4, 3x - 2y = -2, x+y = -1, and x+y = 2.  Evaluate I in xy coordinates.  Next, let u = 3x - 2y, and v = x + y.  Draw the region Rxy and the corresponding region Ruv.  Finally, use the change of variables to evaluate I and notice how much it helps. 
Textbook #11.

Problems surely from the longer end of the mix can be read about in solutions to assignment 6.


Assignment 7

I guess the last chapter of our text isn't finished because it doesn't have problems yet.  So, here's more from this side:

12.1 #1 Consider ((ln y)^z,(xz/y)(ln y)^{z-1},xln(ln y)(ln y)^z).  Find a function u(x,y,z) so that it has gradient equal to the vector field.

#2 Show that F(x,y) = (y,2x) is not the gradient vector field of any continuously differentiable function.

12.2 #1 Let f(x,y) = (x,0). 
a. Draw the vector field f in the rectangle [-2,2] x [-2,2].
b. From the picture alone, what can you say of the sign of the integral along the curve X(t) = (cos t, sin t), 0 ≤ t ≤ 2π?  Do not compute.  Do explain. 
c. From the picture alone, what can you say of the sign of the integral along the curve X(t) = (1+cos t, sin t), 0 ≤ t ≤ 2π?  Do not compute.  Do explain. 

#2 Let f(x,y) =(x-y,x+y).  Plot this vector field by hand or with maple on the rectangle [-4,4] x [-4,4].
Consider the curve X(t) = (2sint,2cost), 0 ≤ t ≤ π/2 at the point (√2,√2).  Is the scalar component of the vector field in the direction tangent to the curve at the point positive, negative, or zero?  Explain and illustrate your reasoning. 

12.3 #1 Evaluate the line integral along the curve X(t) = (t^3, 3), 1≤t≤4 of f(x,y) = (3,e^{-x^2}).

#2 Let f(x,y) = (y,x).  Evaluate the line integral of this field along the following paths:
a. the line segment from (0,0) to (2,4) parametrised by X(t) = (t,2t), 0≤t≤2.
b. the line segment from (0,0) to (2,4) parametrised by X(t) = (t^2,2t^2), 0≤t≤√2.
c. the curve y=x^2 from x=0 to x=2 parametrised by X(t) = (t,t^2)
d. the curve y=x^2 from x=0 to x=2 parametrised by X(t) = (t^2,t^4)

12.4 #1 Suppose f is a conservative vector field.  Is the following statement always true, never true, or sometimes true?  Explain your answer. 
    Suppose that the line integral along the line segment from (-2,0) to (2,0) is 11.  If we integrate along the path (-t,4-t^2), -2 ≤ t ≤ 2 then we get -11. 

#2 Let f=(xy + y cos(xy), xy + x cos(xy)).  Explain why the line integral around any closed curve of f is the same as the line integral of (xy,xy). 


I hope you're glad that they don't go to eleven, here are your last solutions to assignment 7.