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ps1a - Problem Set#1 Answer Key This problem set is...

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Problem Set #1Answer Key This problem set is designed to give you a little bit of practice in comput- ing derivatives and solving some simple optimization problems. The focus of much of the problem set is on the power rule of differentiation, use of the (natural) logarithm, and taking derivatives of the logarithm of a given function. 1 Derivatives Find the derivatives of the following functions. 1. f ( x ) = x 8 . Answer: 8 x 7 . 2. f ( x ) = x 2 (1 + 2 x ) Answer: 2 x (1 + 3 x ) 3. f ( x ) = 4 x 1 2 Answer: 2 x x 4. f ( x ) = x 1 3 Answer: - 1 3 x 2 3 5. f ( x ) = 5 ln x Answer: 5 x 6. f ( x ) = 4 x x 2 . Answer: - 8 ( x 2) 2 7. f ( x ) = 15 x 2 + x 3 Answer: 30 x + 3 x 2 8. f ( x ) = e 2 x , where e is the base of the natural logs. Answer: 2 e ( 2 x ) 9. f ( x ) = ( x + ln x )(3 x - x 2 ) Answer: 5 x - 3 x 2 + 3 ln x - 2 ln xx 10. f ( x ) = ( ax - b ) cx 2 , for constants a, b, and c . Answer: cx (3 ax - 2 b ) 11. f ( x ) = ax b , for constants a and b . Answer: bax b 1 . 12. Consider the function ax b again. What is the derivative of the natural logarithm of this function? Answer: b x . 13. Once again, consider the function ax b . What if you take the derivative of this function, and then compute the quantity f ( x ) f ( x ) ? (That is, the derivative of the function divided by the function itself). Answer: we saw above that f ( x ) = bax b 1 . Therefore, f ( x ) f ( x ) = bax b 1 ax b = b x . In fact, d ln f ( x ) dx = f ( x ) f ( x ) by definition. 1
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2 Partial Derivatives For the following functions, find the partial derivatives with respect to both x and y . 1. f ( x, y ) = xy + y 2 Answer: f x = y, f y = x + 2 y . 2. f ( x, y ) = x 1 3 y 2 3 Answer: f x = 1 3 x 2 3 y 2 3 , f y = 2 3 x 1 3 y 1 3 3. f ( x, y ) = x a y b for constants a and b . Answer: f x = ax a 1 y b , f y = bx a y b 1 . 4. For the function f ( x, y ) = x a y b , develop an expression for the ratio of the two partial derivatives, f x f y , and simplify the result as much as possible. Answer: f x f y = ay bx 5. For the function f ( x, y ) = x a y b , take the natural log of the function and form the partial derivatives. Now, what happens when you form an expression for the ratio of the partial derivatives f x f y ? Answer: This will give you the exact same answer as the previous question. 6. Consider the function f ( x, y ) = ( βx α + δy α ) 1 α , where α, β, and δ are all given constants. Find the partial derivatives of this expression with respect to x and y . Form an expression for the ratio of the two partial derivatives and simplify as much as possible.
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