Mathematics 241, section 1 – Multivariable Calculus Information on Final Exam December 6, 2010

General Information

The final exam for this course will be given at 3:00 p.m. on Tuesday, December 14. The exam will be comprehensive, meaning that it will cover all the material we have studied since the beginning of the semester (see list of topics to be covered below). I will write the exam to be doable in about twice the time for one of the midterm exams (i.e. about 100 minutes) in-class midterms. However, you will have the full two and one half-hour (i.e. 150 minute) period 3:00 - 5:30 p.m. to work on the exam if you need that much time.

Review Session

I will be happy to run an (optional) review session to help you prepare for the final exam, but due to the timing of our exam and several other commitments I have, I think it will have to be on Monday, December 13. I think 9:00a.m. that day will work well.

Topics to be Covered

a) Vectors, the dot product, lengths and angles, orthogonal projections

b) The cross product, equations of lines and planes

c) Parametric curves and motion, tangent vectors and tangent lines to parametric curves d) Vector fields, flow lines, critical points (including the “sink-source-saddle-center” clas-

sification), stability of critical points

e) Limits and continuity for f(x, y)

f) Directional and partial derivatives (know the definition of D_{u}f(x_{0}, y_{0}) and how to compute it using the definition). Differentiability for f(x, y) (know the definition and what it means)

g) The gradient vector field ∇f(x, y); the geometric meaning of the vector ∇f(x_{0}, y_{0}); the relation between contours of f(x, y) and flow lines of ∇f(x, y); the relation between ∇f(x_{0}, y_{0}) and directional derivatives.

h) Critical points of f(x, y) and their relation with critical points of ∇f(x, y) – the “First Derivative Test”

i) The tangent plane to a graph z = f(x, y).

j) The Second Derivative Test for critical points of f(x, y).

k) The Lagrange Multiplier method for constrained max-min problems

l) Riemann sums and double integrals over rectangular regions in R^{2}

m) Double Integrals over non-rectangular regions via iterated integrals with variable limits of integration

n) Polar coordinates in R^{2}, and polar double integrals

o) Triple Integrals,

p) Applications of double and triple integrals to mass, volume, centers of mass, etc.

q) Cylindrical and spherical coordinates in R^{3}. r) Line integrals and Green’s theorem.

Note – some of the exam questions may ask you to “put together” several of these topics.

Suggestion on How to Study

Begin by reviewing the class notes, the problem sets and computer labs, but don’t stop there. Look at the in-class midterm problems and try to work out solutions for those again, especially the ones you had difficulty with the first time around! As you do this, refer to your previous work, my comments on your exam sheet, and the posted solutions as needed. But be sure you can solve problems like those on the midterm exams on your own; the large majority of the final exam problems will be similar. Try a selection of the practice/review problems from the review sheets for Exams 1,2,3 again also.

Then (and only then) take a look at the following questions from previous MATH 241 finals and try to work out solutions:

Practice Final Questions

I. A) Find the equation of the plane containing the point (0, 2, −1) and perpendicular to

the line passing through the points (0, 1, −2) and (3, 2, 2). B) Give a parametrization of the line from part A. C) At what point does the line from part A intersect your plane?

II. All parts of this problem refer to the parametric curve

α(t) = (t^{2} + t − 2, t^{3} + t^{2} − 2t)

(defined for all t ∈ R). A) Thinking of α(t) as the position of a moving object as a function of time, at how many

different times is the object at the location (x, y) = (0, 0)? What times are they? B) Find all times at which the instantaneous velocity of the object is a horizontal vector. C) A thin wire has the shape of the portion of the curve α for t ∈ [0, 2] and density

d(t) = t^{2}. Set up an integral to compute its total mass. (You do not need to evaluate!)

III. The following plot shows the vector field

F (x, y) = (x^{2} − x, y^{2} − 1)

and some flow lines: A) Using the component functions of F, determine all the critical points of F. B) Using the plot, classify each as a either a sink, a source, a saddle point, or a center. C) Verify that α(t) = (0, \frac{e^{−t}−e^{t}}{e^{−t}+e^{t}}) is a flow line of the vector field F.

IV. Both parts of this question refer to f(x, y) = x^{4} − 3xy.

A) At the point (1, 2), what is the direction of maximum rate of increase of f? Express

your result as a unit vector. B) Find the equation of the plane in R^{3} through the point (1, 2, f(1, 2)), containing the

vectors (1, 0, \frac{∂f}{∂x}(1, 2)) and (0, 1, \frac{∂f}{∂y} (1, 2)).

V. The vector field plotted above in question III is the gradient vector field of the function

f(x, y) = y^{3}/3 + x^{3}/3 − y − x^{2}/2

A) Using the plot of ∇f(x, y), and your answers to III B, classify each of the critical

points of f as a maximum, minimum, or neither. B) Apply the Second Derivative Test to each of the critical points and check your re-

sponses to part A.

VI. In the vicinity of Snowshoe Pass, which is located at coordinates (0, 0), the elevation of the terrain is given by the function f(x, y) = xy + 0.9 (x, y, f in miles). A group of hikers walks along the path given by the equation 9x^{2} +(y −1)^{2} = 1, making one complete circuit of the path. What are the maximum and minimum elevations that the hikers reach?

VII. A region R in the plane is the set of all (x, y) satisfying x^{2} + y^{2} ≤ 16, y ≥ 0, and y ≥ \sqrt{3}x. A) Set up iterated integral(s) to compute the total accumulation of a function f(x, y)

over R using rectangular coordinates. (Use any convenient ordering of the variables: dx dy or dy dx).

B) Repeat part A, but using polar coordinates. C) A thin plate has the shape of R and density d(x, y) = y^{2} at the point (x, y). Compute

its center of mass.

VIII. A hole is drilled through a solid sphere of radius 3 centered at the origin in R^{3}. A) If all the material that was at points satisfying x^{2} + y^{2} ≤ 1 is removed, what is the

volume of the remaining solid? (Hint: use cylindrical coordinates.) B) If all the material that was at points with z^{2} > x^{2} +y^{2} is removed, what is the volume

of the remaining solid? (Hint: use spherical coordinates.)

IX. Let α be the simple, closed, positively oriented curve in the plane given by the semicircle of radius 4 with center at (0, 0) and y ≥ 0 (traversed counterclockwise), followed by the portion of the x-axis from (−4, 0) to (4, 0). Let F (x, y) = (x^{2} − y^{2}, xy). A) Compute the line integral ∫_{α} F · T ds directly from the parametrizations. B) Verify that the conclusion of Green’s theorem holds for this F and α by computing

an appropriate double integral.