MA 227 Calculus III - Midterm Test II, Exams of Advanced Calculus

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MA 227, Calculus - III. Midterm test - II November 5, 2003.

Student’s Name

. (Please, print)

GIVE REASONS FOR YOUR ANSWERS!

TEST 1:

HW:

The Final Grade for TEST 1:

I. (10%) Suppose z = f (x, y), x = g(s, t), y = h(s, t), g(1, 2) = 3, gs(1, 2) = − 1 , gt(1, 2) = 4, h(1, 2) = 6, hs(1, 2) = − 5 , ht(1, 2) = 10, fx(3, 6) = 7, fy(3, 6) = 8. Find ∂z/∂s and ∂z/∂t when s = 1, t = 2.

IV. (15%) Find the directions in which the function f (x, y) = x^2 + sin y increases and decreases most rapidly at the point (1, 0). Then find the derivative of the function in these directions.

V. (15 %) Find all the local maxima, local minima and saddle points of the function f (x, y) = x^3 + y^3 + 3x^2 − 3 y^2 − 8.

VIII (15 %) Calculate the iterated integral: ∫ (^1) 0

∫ (^1) 0 e^3 x−ydydx.