ECON $100$: Intermediate Microeconomics

Question #$1$: Exponents

$($a$)$ Use the rules of exponents to compute the value of the following:

$($i$)(5^{\frac{1}{3}})({200}^{\frac{1}{3}})$

$($ii$)\frac{3^{\frac{5}{2}}}{3^{\frac{1}{2}}}$

$($iii$)(\frac{8}{27}{)}^{\frac{2}{3}}$

$($b$)$ Use the rules of exponents to simplify the following algebraic expressions:

$($i$)\frac{x^4{y}^5}{x{y}^2}$

$($ii$)(5x^3{y}^4{)}^0$

$($iii$)x^5(\frac{y^2}{x}{)}^3$

$($iv$)\frac{-x^{3}y}{-xy}$

$($v$)\frac{x^{\frac{1}{3}}}{x^3}$

Question #$2$: Derivatives

Take the derivative of the following functions:

$($a$)x^{-\frac{1}{4}}$

$($b$)(x+1)(x^3+5x+2)$

$($c$)(x^2+3)(x^2-3{)}^{10}$

$($d$)\frac{(x-1{)}^4}{(x+2{)}^2}$

$($e$)(\frac{4-x}{x^2}{)}^3$

$($f$)\frac{1}{x}ln(x+1)$

Question #$3$: Optimization Problem

Suppose that the profit for a firm is given by the following equation: $(Q)=\mathrm{.}0\mathrm{.}01Q^2+50Q-10,000$

Where $Q=$ number of units produced. Find the value of Q that will maximize profits for the firm.

Question #$4$: Calculus Review

Find $($dy$/$dx$)$ for the following functions:

$($a$)y=4x^3-3x^2+2x-$

$($b$)y=(x^2+5{)}^{15}$

$[$Hint: Use Chain rule$]$

$($c$)y=(2x^3-5x)(3x+1)$

$[$Hint: Use Product Rule$]$

$($d$)y=\frac{x_2}{x^2-2}$

$[$Hint: Use Quotient Rule$]$

$($e$)y=ln(x^3+5x^2+8)$

$[$Hint: Use the chain rule and the fact that $\frac{d}{dx}(lnx)=\frac{1}{x}$