Review: Study Guide 1

Midterm 1 will take place in class on Tuesday, September 24. You will only need a sharpened pencil for the exam. The exam consists of two parts:

I created this study guide to support students who may find the math skills challenging, even if they understand the economics concepts well. I know that the math can sometimes feel like a big barrier to success in this class, so my goal is to provide clear explanations and plenty of practice to help you feel more confident with these skills.

Classwork 1: Quarto Formatting

There were no math skills that were required to solve this classwork.

Classwork 2: R as a Graphing Calculator

There were no math skills that were required to solve this classwork.

Classwork 3: Budget Constraints

Part A: Slope-Intercept Form

First, here’s a chapter on Khan Academy on the concept of the slope-intercept form for an equation. Then, try these practice questions.

Practice Question: Rearrange the equation \(2 x_1 + 4 x_2 = 6\) into slope-intercept form with \(x_2\) as the dependent variable (on the y-axis).





Practice Question: Given the equation \(y = \frac{1}{2} x + 3.2\), what are the slope and y-intercept?





Practice Question: The budget line in slope-intercept form is \(x_2 = -\frac{p_1}{p_2} x_1 + \frac{m}{p_2}\). The slope of the budget line is:





Now you’re ready to re-attempt Part A of Classwork 3. Go ahead and give it a shot!

Part B: Intercepts and Sketching Budget Lines

Try these math practice questions to review how intercepts work:

Practice Question: If you have $6 to spend and candy bars cost $2 per bar, how many candy bars can you buy at maximum?





Practice Question: If you have $6 to spend and bottles of soda cost $1 per bottle, how many bottles of soda can you buy at maximum?





Practice Question: If you have $6 to spend, candy bars cost \(p_1 = 2\), and bottles of soda cost \(p_2 = 1\), what is the x-intercept and y-intercept for your budget constraint?





Now you’re ready to re-attempt Part B of Classwork 3. Go ahead and sketch that budget line!

Parts C and D: Analyzing Changes in Income and Price

Before diving into these parts, try these math practice questions:

Practice Question: If you have a fraction \(\frac{a}{b}\) and the numerator \(a\) increases, does the value of the fraction increase or decrease?



Practice Question: If the denominator \(b\) increases in \(\frac{a}{b}\), does the value of the fraction increase or decrease?



Practice Question: What’s steeper: a line with a slope of 1 or a line with a slope of 2?



Practice Question: What’s steeper: a line with a slope of -1 or a line with a slope of -2?



Now you’re ready to re-attempt Parts C and D of Classwork 3.

Classwork 4: Preferences

The only math skills required for classwork 4 are knowing how to graph points on a coordinate plane and to draw a line connecting them. Here’s a chapter from khan academy covering the idea.

Classwork 5: Utility

Parts A and B: Level Curves

Practice Question: An indifference curve is a level curve of a utility function like \(u(x_1, x_2) = x_1 x_2\). What’s another way to think about an indifference curve?





Practice Question: For the utility function \(u(x_1, x_2) = x_1 x_2\), what does the level curve for \(u = 6\) represent?





Practice Question: What happens to the level curves of \(u(x_1, x_2) = x_1 x_2\) as the utility level increases (e.g., from \(u = 1\) to \(u = 10\))?





Practice Question: For \(u(x_1, x_2) = x_1 x_2\), how would you solve for \(x_2\) in terms of \(x_1\) for a given utility level \(u\)?





Now you’re ready to try parts A and B of Classwork 5.

Part C: Rearranging Equations

Part C of Classwork 5 does not require that you understand anything about partial differentials, only that you can do this: take the equation \(0 = AB + CD\), and solve for \(D/B\). For some practice:

Practice Question: Given the equation \(0 = AB + CD\), which of the following is the correct first step to solve for \(D\)?





Practice Question: After subtracting \(AB\) from both sides, what is the equation?





Practice Question: If you have \(CD = -AB\), what is the next step to isolate \(D\)?





Practice Question: What is the value of \(D\) in terms of \(A\), \(B\), and \(C\)?





Practice Question: Now that you know \(D = \frac{-AB}{C}\), what is \(\frac{D}{B}\)?





Part D: Derivatives and Partial Derivatives

Derivatives

Here’s a chapter on Khan Academy.

Practice Question: What does the derivative of a function represent?


v


Practice Question: What is the derivative of the constant function \(y = 3\)?





Practice Question: If \(y = x\), what is the slope (derivative) at any point?





Practice Question: For the function \(y = 2x\), what is the slope (derivative) at any point?





Practice Question: For the function \(y = 2x + 1\), what is the slope (derivative) at any point?





Practice Question: For the quadratic function \(y = 2x^2 + 1\), what is the derivative (slope) at any point?





Practice Question: At the point \(x = 2\), what is the slope of the function \(f(x) = x^3 - 4x + 2\)?





Practice Question: What is the derivative of the function \(f(x) = \ln(x)\)?





Partial Derivatives

Here’s a chapter on Khan Academy.

Practice Question: What does a partial derivative represent?





Practice Question: For the function \(f(x, y) = 3x^2 + 4y\), what is the partial derivative of \(f\) with respect to \(x\)?





Practice Question: Given the function \(f(x, y) = x^2y + y^3\), what is the partial derivative of \(f\) with respect to \(y\)?





Practice Question: For the function \(f(x, y) = x^2 + y^2\), what are the partial derivatives of \(f\) with respect to \(x\) and \(y\)?





Practice Question: What is the partial derivative with respect to \(x\) of the function \(f(x, y) = .25 \ln(x) + .75 \ln(y)\)?





Classwork 6: Choice

Parts A and B: Solving 2 equations for 2 unknowns

Practice Question: Given the system of equations, solve for \(x\) and \(y\).

\[\begin{align} 2x + 3y &= 12 \\ x - y &= 3 \end{align}\]





Parts A and B: Partial Derivatives

Do the review questions on partial derivatives in the previous section (Classwork 5, Pat D: Derivatives and Partial Derivatives).

Classwork 7: Demand

Parts A and B: Slope-Intercept Form

Do the review questions on slope-intercept form in the previous section: Classwork 3: Part A.

Classwork 8: Revealed Preference

Coming soon!