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These are the notes of Past Paper of Multivariable Calculus. Key important points are: Vector Algebra, Constant Force, Work Done by Force, Area of Parallelogram, Vector Projection, Orthogonal Projection, Parametric Curve, Parametric Equation, Point of Intersection
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This exam contains 6 problems, worth a total of 100 points. Only work submitted on this booklet will be considered. Please write the answers in the corresponding boxes. SHOW YOUR WORK.
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Problem 2. (5+5= 10 points) Throughout this problem ~v = ใ 1 , โ 2 , 3 ใ and P = (2, โ 1 , 4).
(a) If Q = (5, โ 2 , 1), find the vector projection of P Q~ onto ~v.
Answer: (b)proj~v P Q~ =
(b) Let ` denote the line through P parallel to ~v, and consider the point Q = (5, โ 2 , 1).
Find the point R on such that RQ~ is perpendicular to ~v. (R is the orthogonal projection of Q onto.) Hint: Draw a figure; you may want to use part (b).
Answer: (c) R =
Problem 3. (7+8+5=20 points) Consider the parametric curve
~r(t) = ใt , t^2 , t^3 ใ
(a) Find the parametric equation ~q(s) of the tangent line to the curve at a time t = T.
Answer: (a) ~q(s) =
(b) If T = 2, find the point of intersection of the tangent line of part (a) with the xy plane.
Answer: (b)
(c) Set up an explicit definite integral equal to the length of the arc of the curve from the origin to the point (2, 4 , 8). You do not have to evaluate the integral. Circle your answer.
Problem 5. (5+5+10=20 points) Throughout this problem
f (x, y, z) = cos(xy) + z^2
(a) Compute โf at an arbitrary point (x, y, z).
Answer: (a)
(b) Find the directional derivative of f at (1, ฯ/ 2 , 1) in the direction of the vector 4~ฤฑ โ 3 ~k.
Answer: (b)
(c) Find the equation of the plane tangent to the surface cos(xy) + z^2 = 1 at the point (1, ฯ/ 2 , 1).
Answer: (c)
Problem 6. (5x4=20 points) On the following page you will find five plots. Assume that the scales of the x and y axes are the same. You are also given the following statements about plots: (1) This plot represents the graph of the function f 1 (x, y) =
x^2 + y^2. (2) This is a contour plot of the function f 1 (x, y) =
x^2 + y^2. (3) These are curves with equations sin(x^2 + y^2 ) = c, for several evenly-spaced values of the constant c. (4) This is a portion of the graph of the function f 2 (x, y) = sin(x^2 + y^2 ). (5) This is part of the image of the parametric surface ~r(s, t) = (s, t, sin(s^2 + t^2 )). (6) These are some level curves of the function f 3 (x, y) = x^2 + 2y^2. (7) This is a portion of the graph of the function f 4 (x, y) = (x^2 โ y^2 )/(1 + x^2 ) (8) This represents a level surface of the function g(x, y, z) = z โ (x^2 โ y^2 )/(1 + x^2 ). (9) These are some level curves of the function f 4 (x, y) = (x^2 โ y^2 )/(1 + x^2 ) (10) This is a portion of the graph of the function f 5 (x, y) = x + y(1 โ y). (11) These are some level curves of the function f 5 (x, y) = x + y(1 โ y). (12) This is a portion of the graph of the function f 6 (x, y) = 2 cos(2x)^2 + y^2. (13) These are some level curves of the function f 6 (x, y) = 2 cos(2x)^2 + y^2.
Fill in the following table with the numbers of the above statements that apply to each plot.
Note: Not all statements apply to some plot, and some plots may have more than one statement that applies to them.
Plot No. Statements that apply 1 2 3 4 5
