Midterm 1, Topics by Worksheet
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 apply correctly 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 quadratic 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 1-4)
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 (say, ΔA) how do you compute the overall amount A?
Understand Delta (Δ) notation. Understand when a quantity is an increment of another.
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 and Speeds
How do you compute the incremental rate of change of y with respect to x? How do you do you compute the overall rate of change of same?
New concept: Average Revenue. Similarly: Average Cost
What kind of rates are these? Of what with respect to what?
Do you understand the GPA example and how it relates to D vs t or to TR vs q?
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 5-9)
WS 5: Lagging Car (intro to functional
Understand functional notation. Why is it useful?
How do you write in functional notation that the red car is always behind the green car by 5 miles? By 10 minutes? That the red car traveled at an average trip speed of 0.75 miles per minute during the first 8 minutes? Etc.
WS 6: Three Languages
Understand how to translate back and forth between English, Graph Language, and Functional Notation the main concepts so far:
a) Change (increments) in an overall amount (ΔA)
b) Rates of change (both overall and incremental) of A
Understand and be able to recognize the Patterns given in lecture. Be able to do translations like those in the handout table or in the text
WS 7: Increments and Reference Lines
Given a graph of, say, f(x), take an English question about f(x), or Δf, or a rate of change of f (x) (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 8: Analysis of Cost I
New concepts: Fixed Cost, Breakeven Price, Variable Cost , Average Variable Cost & Shutdown Price
For each of these: know definitions, formulas (eg: VC(q)=TC(q)-FC, AC(q)=TC(q)/q, etc) and be able to compute them from a table or to read them off graphs. Why is BEP the smallest value of AC ?
Why is SDP the smallest value of AVC ?
WS 9: Analysis of Cost II
Be able to apply the 3 graph methods to determine maximum profit:
1) max vertical distance between TR and TC
2) finding parallel tangents to graphs of TR and TC (because we want: MR=MC, going from MR>MC to MR<MC)
3) Intersection of graphs of MR and MC (because we want: MR=MC, going from MR>MC to MR<MC)
Also: Find the BEP as the crossing
point of MC and AC & find SDP as the crossing point of MC and AVC. .