Math 111, Midterm 1, Main Topics for Review

Prologue

a) Straight Lines: The graph of a linear function f(x)=ax+b is a straight line.

Skills: Compute the slope, draw the graph, write the formula of a linear function (from two points or a point and the y-intercept), find intersections of two lines

b) Quadratics: The graph of a quadratic function f(x)=ax²+bx+c is a parabola.

Understand and be able to use the quadratic formula and the vertex formula. Know how to graph a quadratic function (read and understand Examples1-4 on pages 9-10. What if there are no roots? See Example 3!). Be able to solve quadratics equations and to find intersections of lines and parabolas.

c) Basic Algebra skills: know how to solve a linear equation, a quadratic equation, and a system of two equations.

Part I: information from Graphs and Tables (WS 1-5): be able to read and interpret a graph or a table of data.

WS 1: Introduction to Rates of Change

What is a rate of change? How many kinds are there? How do we compute them

a) from a table b) from a graph.

Understand the specific example of speed (as a rate of change of distance with respect to time). In particular understand Average Speed and Average Trip Speed

Come up with a different example (not involving speed!).

WS 2: Reservoir

Given a table or graph of increments, how do you compute the overall?

Understand Output O vs Input I, and how to use tables or graphs of these to compute the largest shortage, the largest surplus, and the amount we have to start with in order not to run out.

WS 3: Print Shop

New concepts:Total Revenue, Total Cost, Profit & Marginal Revenue, Marginal Cost.

For each of these: know definitions, formulas (eg: TR=pxq, Profit=TR-TC, etc) and understand how to compute them from a table or how to read them off a graph.

Also, what are the relationships between these concepts? In particular, how can you use MR and MC to determine the quantity q which maximizes your profit?

WS 4: Increments

Understand Delta (Δ) notation. Understand when a quantity is an increment of another. In particular, understand the analogy table on page 32.

WS 5: Increments and Rates of Change

How do you write incremental rate of change of y with respect to x in Delta notation? How do you do you write the overall rate of change of same?

New concept: Average Revenue.

Understand the extended analogy table on page 39.

Understand the GPA example.

Come up with another real-life example of an independent variable x and a dependent one y (y is a function of x). What are the analogous notions of overall and incremental quantities and rates of change for your example? How do you compute them from tables or graphs?

Part II: Functional Notation and Graphs (WS 6-10)

WS 6: Lagging Car (intro to functional notation)

Take a first look at functional notation. Why is it useful?

Understand horizontal and vertical shifts of graphs (eg. the graph for the distance from the starting line for the Purple car versus time is just the graph for the Red car shifted right by 5 horizontal units).

WS 7: Three Languages

Understand how to translate back and forth between English, Graphs, and Functions the main concepts so far:

a) Change (increments) of an overall quantity

b) Rates of change (overall and incremental)

Understand and be able to recognize the Patterns given in lecture. The same for the table we partially filled in. See if you can do the rest of it. (Answers on website)

WS 8: Increments and Reference Lines

Given a graph of, say, y versus x, take an English question about y, or a change in y, or a rate of change of y (overall or incremental) and translate it into graph language, then use your understanding of the graph to answer the question.

Be able to draw reference lines when needed, and use rolling ruler methods to answer your questions.

WS 9: Analysis of Cost I

New concepts: Fixed Cost, Breakeven Price, Variable Cost, Average Cost, Average Variable Cost & Shutdown Price

For each of these: know definitions, formulas (eg: VC=TC-FC, AC(q)=TC(q)/q, etc) and understand how to compute them from a table or how to read them off graphs.

WS 10: Analysis of Cost II (first part only)

Be able to apply the 3 graph methods to determine the maximum profit and the quantity where it is attained:

1) by finding max vertical distance between TR and TC

2) by finding parallel tangents to graphs of TR and TC

(because we want: MR=MC, going from MR>MC to MR<MC)

3) by finding the intersection of graphs of MR and MC

(because we want: MR=MC, going from MR>MC to MR<MC)

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