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.

 

 

 

 

 

 

 

Answer____________

 

Find the equation of a plane containing the three points given above.

 

 

 

 

 

 

 

Answer____________

 

 

3.      Given D( 0,3,-2). Does AB cross CD? If so, where?

 

 

 

 

 

 

 

Answer____________

 

 

 

 

 

 

4.      Draw on x-y plane and label the level curves for 0, 1, and 2

 

y

x

 

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)?

 

 

 

 

 

 

 

 

 

Answer___________

 

6.      Find the speed, direction, and acceleration at t = 3π

 

 

 

Speed Answer___________

 

 

 

Direction Answer___________

 

 

 

Acceleration Answer___________

 

 

 

 

 

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)?

 

 

 

Answer___________

 

 

 

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.

Answer___________

 

Answer___________

 

Answer___________

 

Answer___________

 

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.

 

 

 

 

Answer___________

 

Use the total differential to estimate the maximum error for P when V= 2atm .05atm and T= 275K 1.5K

 

 

 

 

 

Answer___________

 

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)

 

 

 

 

 

 

 

 

 

 

 

Answer___________

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) ?

 

 

 

Answer___________

 

What is the largest directional derivative at the point (0,0) in an y direction ?

 

 

 

Answer___________

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.

 

 

 

 

 

 

 

Answer___________

Determine if the critical values are a max., min., or saddle point.

 

 

 

 

 

Answer___________

 

14.    If f(x,y,z) = 2x 3y + 4z, Find the maximum for f if x2 + y2 + z2 = 1 using Lagrange multipliers .

 

 

 

 

 

 

 

 

 

 

 

 

Answer___________