380 Problem Set 1

380 Problem Set 1 - e = d y = 4 x.5 e 2 3 3 log 4 2 x x y f...

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Problem Set 1- Mathematics Review-bivariate calculus R. Pope Economics 380 1. Suppose y=f(x)=10+3x 2 . a) What value of x corresponds to a stationary point (where the derivative is zero)? b) Does the value of x in a) correspond to a minimum or a maximum? (roughly graph and use calculus for your arguments.) c) How does the value 10 and 3 affect the value of x that yields the minimum or maximum of f(x)? (Do they affect the solution?) 2. a) Using the rules of logarithms (base 10 or e), write y=5 x .5 in logarithms where x is a variable which is positive. b) Using the properties of exponents, .5 1 10 y x - = in a way that doesn’t use a negative exponent on x. 3. Calculate the derivative (dy/dx) of the following: a: y = 4 ln(x) b. .4 x y e = c. 2 3 x y
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Unformatted text preview: e = d. y = 4 x .5 e. 2 3 3 log( ) 4 ( ) 2 x x y f x x + + = = (Also, what must the domain of f(x) be in order that f(x) be defined?) 4. Taxes are assigned by the formula: T = -50+.001I 2 , where I is income and T are taxes. a. Calculate taxes paid at I=$0, $100, $1000. b. At these same income levels, calculate the marginal tax rate (dT/dI) and the average tax rate (T/I) and distinguish between the two. c. Graph the tax (T) and after-tax income (I-T) against income I (on the horizontal axis). d. What income, I*, maximizes after-tax income (I-T)? Interpret the first order conditions (equation giving the optimal solution) in terms of marginal benefit and marginal cost to an earner of another dollar of income, I?...
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