This homework is due October 5th. You may work in groups or individually. Please, bring a physical copy of the set to class. Do not email it to me. The use of writing softwares (like sharelatex, greenleaf, or authorea) is strongly recommended, especially for those of you with bad handwriting.
Algebra
1. List the properties of powers
2. The surface area of a sphere with radius r is \(4\pi r^2\). By what factor will the surface area increase if the radius is quadrupled? By what factor will it increase if the radius increases by a factor of \(x?\)
3. What is the general formula for the calculation of interest? What should the interest be if I wanted to obtain $1 million in 40 years, starting with $3,000 today?
Logic, proofs, and set theory
1. Make a table with two columns. On the first column write the following symbols we encountered in class: \(\forall,\ \in,\ \notin,\ \subset,\Sigma,\ \Longrightarrow,\ \Longleftrightarrow,\ :\) .
2. Prove that the difference between \(\frac{1}{n}\sum_{i=1}^nx_i\) and the arithmetic mean of x, \(\mu_x\), is equal to zero. First, explain the steps in plain English, and then write it down using standard mathematical formulas.
3. What is Newton's binomial formula? Use the formula to derive the following: \((a+b)^7\)
4. Provide an example of necessary condition, one of sufficient condition, and one of necessary and sufficient condition. Rewrite your examples using the symbols from above.
Equations
1. A firm is characterized by flat average cost and marginal cost curves, where AC=MC=17. in order to operate at all for one period, the firm must incur in the fixed cost FC=500. Draw the situation faced by this firm. How many units would the firm supply at the market price P=32 if it had to meet the profit target \(\Pi=15320\) (do not worry if you find that your result contains decimals)? What if the price were P=20?
2. Derive the reduced form of the simple macroeconomic problem. Based on this form, what restrictions must we impose the value of the marginal propensity to consume. Provide a mathematical justification and an economic justification for your answer.
3. Solve the following equations:
a. \(3x^2-5x+15=0\)
b. \(4x^2=1+3x\)
c. \(2Q-14Q^2=0\)
4. Solve the following systems of equations
a. \(P=a-bQ\) \(P=\alpha +\beta Q\)
b. \(3x+2y=3\), \(6y+6x=-1\)
Univariate functions
1. Use the point-slope formula to derive (and draw) the function for the following combinations:
a. Slope= -4, point= (3, -12)
b. +.7, (0, 1)
2. A firm has a cost function of the form: \(C=3 Q+ 1.5 Q^2\) and faces a demand function of the form: \(P=1500-4Q\) . Write the formula for the profits of this firm. Calculate the profit maximizing quantity supplied by this firm.
3. Write the rules for the natural log and then solve the following:
a. \(4e^2x=24\)
b. \(2e^-3x-e^4x=7\)