Math 2300| Calc 3 | Practice Exam 2 PDF

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MATH 2300

Practice Problems for Exam 2

#1. Find and sketch the domain of the function: f (x, y) = ln xy #2. Let f (x, y) = ex cos(xy). (a) Show that f is differentiable at the origin. (b) Find its linearization L(x, y) at the origin. (c) Use the linearization you found to approximate f (.01, .02) #3. Find an equation for the tangent plane to the surface x2 y cos z + xy3 z = 1 at the point (1, 1, 0). p #4. Find an equation for the tangent plane to the surface z = 4 − x2 − 2y2 at the point (1, −1, 1).

#5. Use the Chain Rule to find

∂z ∂z ∂z , , ∂r ∂s ∂t

at the point (1, 2, 0) where z = xy , x = rest and y = rset .

#6. Let f (x, y, z) = x2 + 3xy − z 2 + 2y + z + 4 and P0 (1, 1, 1). (a) Compute the directional derivative of f at P0 in the direction of ~v = ~i + ~j + ~k. (b) Find the direction in which f increases most rapidly at P0 and compute the directional derivative in that direction. (c) Find the direction in which f increases most slowly at P0 and compute the directional derivative in that direction. x3 3

#7. Find and classify the critical points of the function f (x, y) = y3 −

− 3y2 +

x2 2

+ 5.

#8. Find the minimum and maximum values of f (x, y, z) = x − 2y + 3z on the sphere x2 + y2 + z 2 = 1. #9. Find the extreme values of f (x, y, z) = #10. Evaluate the iterated integral:

Z

0

1 x

1 Z x2

#11. Evaluate the double integrals: (a)

−

1 y

+

1 z

subject to the constraint

1 x2

+ y12 +

1 z2

= 3.

(x + 2y) dy dx

0

Z

0

8Z 2

cos(x4 ) dx dy

(b)

√ 3y

Z 1Z 0

y/x3

#12. Find the volume of the solid that lies below the surface z = 4e xy-plane that is bounded by the curves y = x7 , x = 1 and y = 0.

1

x3 sin(y3 ) dy dx x2

and above the region in the

#13. Find the volume of the solid that lies below the paraboloid z = x2 + y2 and above the region in the xy-plane that is bounded by the curves y = x2 , x = y2 ....