Explanation It is given that the function is f ( x , y ) = 2 sin 3 2 x 1 − 3 y . Here, the dependent variable is f and the independent variables are x and y . Differentiate f ( x , y ) = 2 sin 3 2 x 1 − 3 y partially with respect to x as follows. ∂ f ∂ x = ∂ ∂ x ( 2 sin 3 2 x 1 − 3 y ) = 2 1 − 3 y ∂ ∂ x ( sin 3 2 x ) = 6 ( sin 2 2 x ) 1 − 3 y ∂ ∂ x ( sin 2 x ) ( use chain rule ) = 6 ( sin 2 2 x ) cos 2 x 1 − 3 y ∂ ∂ x ( 2 x ) On further simplifications, ∂ f ∂ x = 6 ( sin 2 2 x ) cos 2 x 1 − 3 y ( 2 ) = 12 sin 2 2 x cos 2 x 1 − 3 y Thus, the partial derivative of f ( x , y ) = 2 sin 3 2 x 1 − 3 y with respect to x is 12 sin 2 2 x cos 2 x 1 − 3 y . Differentiate f ( x , y ) = 2 sin 3 2 x 1 − 3 y partially with respect to y as follows

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Chapter 29.3, Problem 21E

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The partial derivatives ∂f∂x and ∂f∂y of the function, f(x,y)=2sin32x1−3y.

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Chapter 29.3, Problem 20E

Chapter 29.3, Problem 22E

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== 4. [10] Let X be a RV. Suppose that E[X(X-1)] = 3 and E(X) = 2. (a) Find E[(4-2X)²]. (b) Find V(-3x+1).

Students were asked to prove the identity (sec x)(csc x) = cot x + tan x. Two students' work is given.Student AStep 1:1/Cos x * 1/sin x = cot x + tan xStep 2: 1/cos x sin x = cot x + tan xStep 3: (cos^2 x + sin^2 x)/cos x sin x = cot x + tan xStep 4: cos^2 x/cos x sin x + sin^2x/cos x sin x= cot x + tan xStep 5: cos x/sin x + sin x/cos x = cot x + tan xStep 6: cot x + tan x = cot x + tan xStudent BStep 1: sec x csc x = cos x/ sin xStep 2: sec x csc x = cos^2x/cos x sin x + sin^2x/cos x sin xStep 3: sec x csc x = cos^2x + sin^2x/cos x sin xStep 4: sec x csc x = 1/cos x sin xStep 5: sec x csc x = (1/cos x), (1/sin x)Step 6: sec x csc x = sec x csc xPart A: Did either student verify the identity properly? Explain why or why not. Part B: Name two identities that were used in Student A's verification and the steps they appear in.

2. [15] Let X and Y be two discrete RVs whose joint PMF is given by the following table: y Px,y(x, y) -1 1 3 0 0.1 0.04 0.02 I 2 0.08 0.2 0.06 4 0.06 0.14 0.30 (a) Find P(X ≥ 2, Y < 1). (b) Find P(X ≤Y - 1). (c) Find the marginal PMFs of X and Y. (d) Are X and Y independent? Explain (e) Find E(XY) and Cov(X, Y).

Chapter 29 Solutions

Pearson eText for Basic Technical Mathematics with Calculus -- Instant Access (Pearson+)

Ch. 29.1 - Practice Exercise If f(x, y) = 4xy2 – 3x2y, find...Ch. 29.1 - Prob. 1E Ch. 29.1 - Prob. 2E Ch. 29.1 - Prob. 3E Ch. 29.1 - Prob. 4E Ch. 29.1 - Prob. 5E Ch. 29.1 - Prob. 6E Ch. 29.1 - Prob. 7E Ch. 29.1 - Prob. 8E Ch. 29.1 - Prob. 9E

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The partial derivatives ∂ f ∂ x and ∂ f ∂ y of the function, f ( x , y ) = 2 sin 3 2 x 1 − 3 y .

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The partial derivatives ∂f∂x and ∂f∂y of the function, f(x,y)=2sin32x1−3y.

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Chapter 29 Solutions