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The september 29, 2006 exam for the multivariable calculus course taught by prof. P. Wong. The exam covers topics such as finding the area of parallelograms, transformations, level curves, and parametrized curves. Students are required to explain their work and provide reasons for their answers.
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EXAM I - SEPTEMBER 29, 2006
Instruction: Read each question carefully. Explain ALL your work and give reasons to support your answers. Advice: DON’T spend too much time on a single problem.
Problems Maximum Score Your Score
1
2 EXAM I - SEPTEMBER 29, 2006
(5 pts) (i) Find the area of P.
(3 pts) (ii) Suppose T (x 1 , x 2 ) = (5x 1 + 4x 2 , 5 x 1 + 3x 2 ). Find the associated matrix A such that T (x) = Ax where x = (x 1 , x 2 ).
(4 pts) (iii) What are the vertices of the image T (P )?
(4 pts) (iv) What is the area of T (P )?
(4 pts) (v) What is the angle of the parallelogram P at the vertex (6, 3)? (You may express it in terms of inverse trig function.)
4 EXAM I - SEPTEMBER 29, 2006
x
y
(5 pts) Describe or sketch the set of points in R^3 that satisfy the equation f (x, y) = x^2 + 2y^2 − 1 (or the graph of z = f (x, y))
(5 pts) (iii)What are the cylindrical coordinates of the point (1, 2 , 8)?
(5 pts) (iv) Write the equation z = x^2 + 2y^2 − 1 in spherical coordinates.
MATH206A MULTIVARIABLE CALCULUS - PROF. P. WONG 5
v(t) = i + (1 + t)j + cos tk.
(10 pts) (i) Find the position r(t) of the particle with the initial point r(0) = i + k.
(10 pts) (ii) Give an equation (in vector form) of the line tangent to the path of the particle at the point r(π).
