Study with the several resources on Docsity
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Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Past Exam of Multivariable which includes Vertices, Parallelogram, Vector, Coordinate Equation, Vector Field, Equation, Value, Function etc. Key important points are: Point, Vector Perpendicular, Equation, Normal Vector, Parametric Equation, Plane, Magnitude, Midpoint, Origin, Sphere Centered
Typology: Exams
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Exam 1- In-Class Portion
Show all your work to receive full credit for a problem.
(a) Find a vector perpendicular to both a and b.
(b) Let x 0 = (1, 2 , 1). Find the equation for the plane going through the point x 0 and with normal vector the one you found in part (a).
(c) Find a parametric equation for the plane in part (b).
(a) What is the magnitude of a?
(b) What is the midpoint between a and the origin?
(c) Write the equation for the sphere centered at the origin and containing a.
(d) What would the equation you found in part (c) be if you were to write it in spherical coordinates?
1.' .. I.' 12 yU u 0.' 0.' 0.2-
-u -~. -!..~ ~^02 0.'^ 0.'^ 0.' :t: '
/ -0.2- -0.' /(^ -0.'
-0.' y -I -1.2- (^) J. -IA /
1IL
U
u
0.'
" (^) 0.' y 1 0.2- -0.2- -o.s (^) -06-0' -1 (^) -u -"
(a) Explain why p is a quadratic form.
(b) Find a symmetric matrix S such that p(x) = xT^ Sx.
(c) Use Sylvester’s Theorem to show that this quadratic form is positive definite. What does this tell you about the graph of p(x, y)?
