241ex1e - Calculus I(Math 241 Test 1 Problem 1[5 Points Suppose that x2 y 2 Find dy/dx in terms of x and y Solution Differentiating both sides of

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Calculus I (Math 241) – Test 1 Problem 1. [5 Points] Suppose that ( x 2 + y 2 ) 2 = 5 x 2 y. Find dy/dx in terms of x and y . Solution : Di±erentiating both sides of the equation we ²nd: 2 ( x 2 + y 2 ) [2 x + 2 yy ] = 10 xy + 5 x 2 y . Bringing all terms with a factor y to the left side of the equation, and all terms without such a factor to the right, we obtain: b 4 y ( x 2 + y 2 ) - 5 x 2 B y = 10 xy - 4 x ( x 2 + y 2 ) . and y = 10 xy - 4 x ( x 2 + y 2 ) 4 y ( x 2 + y 2 ) - 5 x 2 . Problem 2. [5 Points] Find lim x 0 tan x 3 x . Solution : We calculate: lim x 0 tan x 3 x = lim x 0 sin x x · 1 3 cos x = 1 · 1 3 = 1 3 . Problem 3. [10 Points] Show that the equation x = 2 sin x has a solution with x [ π/ 2 , π ]. Name the theorem that you apply, and tell why its assumptions hold. Hint: Equivalently you may show that the function f ( x ) = x - 2 sin x has a zero in the interval [ π/ 2 , π ].
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This note was uploaded on 07/26/2009 for the course MATH 215 taught by Professor Heiner during the Spring '09 term at Hawaii.

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241ex1e - Calculus I(Math 241 Test 1 Problem 1[5 Points Suppose that x2 y 2 Find dy/dx in terms of x and y Solution Differentiating both sides of

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