380 Problem Set 2

380 Problem Set 2 - Multivariate Calculus and Optimization...

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Multivariate Calculus and Optimization Problem Set 2 R. Pope Econ 380 1. Let y= 2 12 1 2 (, ) 6 l n () fxx x x  where ln is the natural logarithm, ,0 xx . Calculate: 1 _________________ f x 2 __________________ f x . b) How do you interpret these partial derivatives? 2. Calculate the differential of y (sometimes called the total derivative) in problem 1. a) ______________________ dy b) Explain what this differential means? c) i) If 20 3, x and x  how much would we expect y to increase by for a small change (in the infinitesimal calculus sense ) in x 1 and x 2, , dx and dx . ii) Use small numbers 1 x for 1 dx (e.g., .001) and 22 x for dx (e.g., .001) and use the differential formula in i) to get the approximate change in y. iii) What is the exact change in y using the numbers in ii). Calculate using f with 1 2 20, 3 20 and 3 x x and x x  ---that is calculate the exact change in y. iv) Why don’t you get the same answer in ii) as you did in iii)? 3. a) Give the equation for the function in problem 1. of all values of x 1 and x 2 that yield a value of y equal to 100 (leave it in implicit form if you wish)? This is called a level curve.
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380 Problem Set 2 - Multivariate Calculus and Optimization...

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