MAT251 Multivariable Calculus III MIDTERM Name ______________________

Prof. PORTER FALL 2006

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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 e^x 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 + ye^z = 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___________