Mathematics
Compute the derivatives of the following functions:
(a) = (! + 1)!
(b) =
!!!!
!!!!
(c) = 4! − 3(d) = ln(5 + 4)
(e) = 2! ** If you are not familiar with these questions, you may need to go back to your calculus textbooks to study the chain rules, or you may go to the following links to seek answers. **http://www.themathpage.com/acalc/exponential.htmhttp://www.ltcconline.net/greenl/courses/116/explog/l ogderivative.htmhttp://webspace.ship.edu/msrenault/GeoGebraCalculu s/derivative_intuitive_chain_rule.htmlQ2. Please answer the following questions:(a) In the equation = 13 + 8 where is a function of , please define: (i) What is the independent variable? (ii) (What is the dependent variable?
(iii) What is the intercept? (iv) What is the slope? (v) What is the value of when = 5?(b) Given an economic model of a market as:
Q = – P + W …(equation 1)P = Q +Z … (equation 2)Please define: (i) Which equation represents demand side?(ii) Which equation represents supply side?
(iii) What are endogenous variables in this
model?
(iv) What are exogenous variables in this
model?
(v) What is the reduced form of this model? In other words, please use the solve for the endogenous variables in terms of the
exogenous variables.
Q3. Please use your words to explain the following terms and make sure that you will give a descriptive example to support your explanation. All the terms can be found in chapter 1, but I am not asking you to copy the text from the textbook. I am creating this problem set to kindly see if you can apply these terms and give your own explanations and examples. Take (a) for example, after you explain opportunity cost in your own words, you can give the example like, “after you graduate with a BA/BS, the opportunity cost to go on to the graduate school is your work experience and salary.)(a) opportunity cost(b) the invisible hand(c) positive-normative distinction
Q4. Let QD = -‐‐5P + 54 and QS = P -‐‐ 6. Where QD : the quantity demanded of a good. P: the price of a good. Qs : the quantity supplied of a good.(a) Please graph the demand and supply curves and make sure that you put P on the vertical
axis and Q on the horizontal axis.
(b) What are the values of P and Q when QD = QS ?Q5. Suppose and economy has a production possibility frontier (PPF) characterized by the equation:
4X2 + Y2 = 16 (a) Please the intercepts of this equation. In other words, what is the value of X if Y =0?
What is the value of Y if X =0?
(b) Please sketch this equation and make sure that you label the vertical and horizontal axes and
the original point.
(c) Given X =1, Y = 4, what is the location of this point on the PPF graph? (In other words, is this point outside the PPF? On the PPF? Inside the PPF? Or the wrong quadrant to be on the
graph?)
(d) Given X = 3 and Y=2, what is the location of
this point on the PPF graph?
(e) What is the opportunity cost of going from 1 unit of X to 2 units of X (in terms of units of
Y)?
(f) What is the opportunity cost of going from 2 units of X to 3 units of X (in terms of units of
Y)?
(g) Is the opportunity cost of X in terms of Y
constant in this economy, or does it depend on the levels of output being produced? Please explain.
Q6. Consider the function: Y = X · Z. Let both X and Z be non-negative. (a) Please graph the Y = 4 contour line for this
function.
(b) What does the line X + 4Z =8 intersect the Y =4 contour line? (Hint: Solve the equation for X and substitute into the equation for the contour line. You should get only a single
point.)