MAT251 Multivariable Calculus III MIDTERM Name ______________________
Prof. PORTER FALL 2006
1. What is Multivariable Calculus? Give a good reason why you might want you to know this material?
Given Points A(1,2,3) B(-1,2,0) C(0,1,4) answer the following:
2. Use vectors to find the angle between AB and AC.
Find the equation of a plane containing the three points given above.
3. Given D( 0,3,-2). Does AB cross CD? If so, where?
4. Draw on x-y plane and label the level curves for 0, 1, and 2
Use r(t) = 3cos2ti+3sin2tj+1.75tk to solve the following:
5. Find the distance (to 2 decimal places) from r(0) to r(3π)?
6. Find the speed, direction, and acceleration at t = 3π
7. Find the limit as (x,y) goes to (0,0) along the paths x = 0 and y = x for the function:
a) f(x,y) =
x=0 path Answer___________ y=x path Answer___________
b) If f(x,y) is defined as below, then is the function continuous at (0,0)?
8. Given g(x,y) = siny ex and x(u,v) = u + 3v and y(u,v) = 2u - v
Find ,,,, and in terms of u and v.
in terms of u and v Answer___________
9. Given PV = nRT the Ideal Gas Law. Give the total differential for P if n and R are constant.
Use the total differential to estimate the maximum error for P when V= 2atm ±.05atm and T= 275K ± 1.5K
10. Use the Implicit Differentiation to determine the partial derivative of y with respect to x when
siny2+ 2xz + yez = 9 and y(x,z)
11. Find the directional derivative of the function f(x,y) = x2 +9y2 at the point (0,1) in the direction of the point (2,1) ?
What is the largest directional derivative at the point (0,0) in an y direction ?
What does the result mean?
12. Find the equation of the tangent plane to the surface z2 = x3 + y3 +xyz at the point (0,0,0)
13. For the equation z = 3x – x3 - 2y2 + y4, find all the critical values.
Determine if the critical values are a max., min., or saddle point.
14. If f(x,y,z) = 2x – 3y + 4z, Find the maximum for f if x2 + y2 + z2 = 1 using Lagrange multipliers .