Math E-21a Practice Exam #1 Solutions Fall 2009
1) True/False questions (circle one) you do not need to show your reasoning: a) The vector v 1, 2, 4 is perpendicular to the plane with equation 4 x 2 y 2 z 100 . False: In fact, the vector v 1, 2, 4 is para
Math E-21a Fall 2009 Practice Exam #2 solutions
1) The level sets of a function f ( x, y ) are shown. Determine the signs of the following derivatives at the point P. Indicate either +, , or 0 in the spaces provided.
fx =
+ fy = f xx = f yy = f xy = +
2)
Math E-21a Fall 2009 HW #1 problems Due in class on Thurs, Sept 10: Section 9.1: 8. Find the distance from (3, 7,5) to each of the following. (a) The xy-plane (b) The yz-plane (c) The xz-plane (d) The x-axis (e) The y-axis (f) The z-axis 12. Find an equat
Math E-21a Things to know for the 2nd midterm exam Higher order derivatives, interpretation of 2nd derivatives, equality of mixed partials (Clairauts Theorem), quadratic approximation. Extrema of functions and the 2nd Derivative test; local maxima, local
Math E-21a Fall 2009 Exam #2 solutions
1) Find all critical points of the function f ( x, y ) x 3 y 2 6 xy 6 x 3 y 2 and, for each point, determine whether it is a local maximum, a local minimum, or a saddle point. Solution: The critical points (stationar
HW #8 Solutions Math E-21a Fall 2009 Problem 1:
Problem 2:
Problem 3:
Problem 4:
1
Problem 5:
Problem 6: [Note: The figure shown is a rather poor one. Yours is probably a lot better.]
Heres an alternative: Use the parameterization from part (a) which give
Math E-21a Exam #1 solutions Fall 2009 1) Given the vectors u = 2, 2,1 , v = 1,1, 4 , and w = 2,1, 2 , find the following quantities and complete the box. Show all work below.
a) The angle between the vectors u and v Solution: cos
uv 224 4 0.314269 , so
Math E-21a Fall 2009 HW #5 problems Problems to turn in on Thurs, Oct 8: Section 11.2: In problems 10 and 11, find the limit, if it exists, or show that the limit does not exist. 6 x3 y xy 10. lim 11. lim 4 4 ( x , y ) (0,0) 2 x y ( x , y ) (0,0) x2 y 2 S
Math E-21a Fall 2009 HW #8 problems Read sections 11.8 and 12.1 (and maybe 12.2-12.3) and do the following problems: Problems to turn in on Thurs, Oct 29: 1. (Prob. 11.8/16) Use the Method of Lagrange Multipliers to find the maximum and minimum values of
Math E-21a Fall 2009 HW #13 problems To be turned in Thurs, Dec 10: Section 12.6: 2. Find the area of the part of the plane 2 x 5 y z 10 that lies inside the cylinder x 2 y 2 9 .
4. Find the area of the part of the plane with vector equation r (u , v) 1 v
Math E-21a Fall 2009 HW #7 problems Problems to turn in on Thurs, Oct 22: Section 11.7: In problems 6-14, find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function wi
Math E-21a Some useful facts f dx f dy + = f v for a path in R2; Basic Chain Rule: d [ f ( x(t ), y (t ) ] = dt x dt y dt d f ( x(t ), y (t ), z (t ) = f dx + f dy + f dz = f v for a path in R3. [ ] dt x dt y dt z dt Directional Derivative of a function f
Math E-21a Practice Final Exam solutions Fall 2009
1) True or False (circle one you need not show your work): a) The dot product of two nonzero vectors gives the cosine of the angle between these two vectors. FALSE - u v u v cos , so in general the dot pr
Math E-21a Practice Final Exam Fall 2009
The following problems appeared on previous Final Exams. Neither the number of problems nor the type of problems are necessarily indicative of problems that will appear on this semesters exam, but they should provi
Math E-21a Midterm Exam 1 Topics Fall 2009
1. Given vectors in R2 or R3, do addition, scalar multiplication, dot and cross products, scalar and vector projection. 2. Manipulate vector expressions symbolically (distributive law, triple product, A (B C), et
Math E-21a Mega-List of Things You May Want to Know Fall 2009 1. Write equations for surfaces in R3 that are characterized geometrically by distances (spheres, cylinders, etc.), and determine their intersection with specified planes. 2. Sketch or identify
Math E-21a Fall 2009 HW #12 problems
Problems due Thurs, Dec 3:
Section 13.4: 4. Evaluate the line integral by two methods: (a) directly and (b) using Greens Theorem. x dx y dy , C consists of the line segments from (0,1) to (0,0) and from (0,0) to (1,0)
Math E-21a Fall 2009 HW #2 Solutions Proof of the Pythagorean Theorem Perhaps the easiest way to prove this is with areas: The area of the larger square is the sum of the areas of the smaller square and the four right triangles. This gives us:
b a c c b c
Math E-21a Fall 2009 HW #2 problems Problems to turn in on Thurs, Sept 17: Basic Concept Problems: (a) Prove the Pythagorean Theorem. [Hint: This can be done, for example, by considering the areas of plane figures like squares and triangles.] (b) Prove th
Math E-21a Fall 2009 HW #3 problems Problems to turn in on Thurs, Sept 24: Section 9.6: 11. Sketch the graph of the function f ( x, y ) 6 3x 2 y . 12. Sketch the graph of the function f ( x, y ) cos x . 34. Find an equation for the surface consisting of a
Math E-21a Fall 2009 HW #4 problems Problems to turn in on Thurs, Oct 1: Note: Bold indicates vector quantities. Section 10.2: 38. Prove formula 3 of Theorem 3: d f (t )u(t ) f (t )u(t ) f (t )u(t ) where f (t ) is s scalar-valued function dt and u(t ) is
Math E-21a Fall 2009 HW #6 problems Problems to turn in on Thurs, Oct 15: Section 11.3: z z and : yz ln( x z ) 42. Use implicit differentiation to find x y 48. Find all the second partial derivatives: f ( x, y ) ln(3 x 5 y ) 62. Level curves are shown for
Math E-21a Fall 2009 HW #7 Solutions Section 11.7:
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50. Since the plane passing through (1, 2, 3) must cut out a tetrahedron in the 1st octant, its intercepts must all be positive call them a, b, and c. The volume of this tetrahedron will be V 1 ab