Any review sheet is a compilation of personal opinions about the relative importance of various parts of the material. These are jason's. It would be crazy to rely on any single resource to study for your Exam, including this little study guide!

Material covered on Exam 1:

Chapter 1, sections 1.1-1.8

Chapter 2, sections 2.1-2.6
with a ``lighter'' emphasis on sections 1.7-1.8 which discuss continuity and limits.

Section 1.1-Functions, linearity, proportionality

Some phrases which should give you that comfortable and confident familiar feeling: ``...is a function of...'', ``is not a function of,'' ``is inversely proportional to the square root of'', ``has as its domain [4,6]'', ``is not in the range of'', ``constant rate of change'', ``by a table, a graph, a formula, and a verbal description'', ``the difference quotient '', ``is decreasing for negative x'', ``find the vertical intercept'', etc.

What's the domain of ? What's its range? How can you tell from a table whether a function is linear? If it is linear, how can you find its formula? How is slope related to increasingness/decreasingness of a linear function? Can you read the slope from a graph? Can you read the y-intercept? What's a constant of proportionality, and why might you have to use one? And if you use one, how do you usually calculate its value? How can you read domain and range from a graph? What does the Greek capital Delta mean?

Section 1.2-Exponentials

Exponential functions have constant what? Here's one people seem to get wrong more often than right: What's a continuous growth rate? What's a growth factor? If the growth rate is 0.5, what's the growth factor? Which one appears on the general formula and which one appears in ? Speaking of these two, why do we have two formulas for an exponential function? If I give you one, can you convert it to the other? I like to think about this by analogy with the two forms of linear functions: point-slope and slope-intersept form. What does concave-down mean? How can you see it on a graph? on a table? What's that ? Yes, yes, I know it's the initial value, but what does it represent in the context of any particular problem? How can you tell at a glance whether a formula represents exponential growth or decay? Write an exponential function that starts really big and decays really slowly.

Can you tell from a table whether a function might be exponential? How? If it is exponential, how can you find its formula? (Careful, this is really easy in some examples, but can be hard in others!) From two points on a graph can you find the formula? It's fun to ask (ala 1.4.50) ``When will the population reach 5 million,'' or ``When will the amount of C-14 reach 99.5 percent,'' which require you to use logs to solve for the independent variable. How do you recognize when you'll have to solve such an equation? How do you set up the equation? How do you solve it? Which kinds of real-world situations tend to be modeled by exponentials? by sinusoidals? by linear functions? What are half-life and doubling time? From an exponential formula, can you calculate them? If you don't know (as in a couple group homework problems), is it safe to just assume it's 1?

Section 1.3-Shifts/stretches, Compositions, and Inverses.

How does the graph of f(x) relate to the graphs of f(x+4), f(x-2), f(4x) and 4f(x)? What does this relationship tell you when f(x) is not given graphically, but is described via a real-world relationship between two variables? (e.g. problem 1.5.34.d) Inverses and composition have to be understood in terms of real-world situations. If I give you a story problem involving functions f and g, representing realistic relationships between quantities, what does f(g(x)) tell us? What about ? How can we find a formula for , given one for f? Problem 1.3.36 is a good summary of the tabular skills necessary here. How does the graph of the inverse of f relate to that of f? (Oddly, many students confuse the transformation ``flip across the line x=y'' with the transformation ``rotate 180 degrees''. Do you understand these two different things?)

Section 1.4-Logarithms

There's more than one way to do it. In this case, there's a perfectly intuitive way (log base ten) and a more theoretically useful way (natural logs). This section is not just about the rules of logarithms. Know those rules of logarithms! Can you solve equations using logs? Which equations demand the use of logs, and which ones don't? What's the relationshp between this section and inverse functions?

Section 1.5-Trigonometric functions

What's the difference between trigonometric functions and sinusoidal functions? In the general formula , what does each letter mean, in the context of any given problem? Is B the period? How can you graph such a function, based only on the values of A,B, and D? What kinds of systems are modeled by such functions? Why does tan(x) have vertical asymptotes? What do the graphs of sin(x) and cos(x) look like? In particular, what are the horizontal intercepts of each? What's a radian?

Section 1.6-Power, Polynomial, and Rational functions

It's all about understanding the relationship between the graphs and the formulas. ``But I don't need to understand all that, because when I have the formula I can use my calculator to graph it,'' you say? Think again! Your calculator won't help you understand long-term behavior. It won't tell you which functions eventually grow faster than others as . It won't help you find a formula for a polynomial to fit a graph. etc. How many roots does a quadratic have? a cubic? a degree-four polynomial? How many times does each ``turn around''? Can you graph a degree-four polynomial, given its equation? Can you find an equation from its graph? an you predict its long-term behavior based on the leading term? Where does a rational function have vertical asymptotes? Horizontal asymptotes? How can you tell from the formula?

Section 1.7-Continuity

The official word on this section is that it will be tested "lightly if at all," meaning the coordinators reserve the right, but it's not going to be the basis of a 30-point question #10. Understand what continuity is, and what a continuous function looks like.

Section 1.8-Limits

Again, "lightly if at all." Know the difference between left- and right-hand limits, and know how to detect when a graph has a limit and when it does not. Complex limit calculations with formulas will not be required. Be able to understand and use limit notation and to investigate limits numerically and graphically. The "formal definition" of the limit (with epsilons and deltas) will not be tested.

Section 2.1-Velocity as a derivative

How do we measure velocity? Is the instantaneous velocity (Distance/Time)? How can we measure instantaneous velocity? How does the similarity between instantaneous velocity and average velocity over a short interval allow us to make various estimations? What are the various ways we can calculate or estimate instantaneous velocity from a formula, a graph, or a table?

Section 2.2-Derivatives in general (at a point)

What's the relationship between slope and the derivative? How can we understand average rate of change and instantaneous rate of change, graphically?
Understand the limit definition/calculation of the derivative in terms of slopes of secant lines approaching a tangent line. Be confident expressing information in the language of slopes of lines or rates of change, since the two are equivalent.

Section 2.3-The derivative as a function

If f is increasing, what does that tell you about f'? What if f is decreasing, or concave-up? If f' has a root, what can you conclude about f? And if f' is increasing?

Suppose you are faced with a story problem, describing motion. Can you graph velocity as a function of time? If the graph is given, can you identify what features correspond to which actions? If I graph dV/dt, for some funny variable V, can you make sense of the various features of the graph?

The same quantity which can be visualized as a slope on one graph, appears as a height on the graph of its derivative.

Section 2.4-Interpreting the derivative

This section may contain the deepest ideas of all: What is a derivative? If P is position and t is time, then dP/dt is a simple enough gadget. Even if v is velocity dv/dt is an intuitively clear thing. But what if P is pressure, and t is temperature? If P is population per square miles and t is altitude? How can we take a crazy function, never seen before, and understand the meaning of its derivative? The easiest problem to ask in this section takes one of two forms: "Here's a derivative. Explain what this means, in scientific terms," or "Here's some information. Tell me about derivatives." There are some recipes for solving these problems (e.g., Make good use of units, because everything's easier with units!), but there's no substitute for a real understanding of the meaning of the "'" in "f'". I like to say that 2.5 is the heart of Calculus.

Section 2.5-Second derivatives

Okay, so maybe it's not the heart of Calculus, but it still has some nifty stuff. Four central notions appear here: First, since the derivative function is a function, it has its own derivative. Voila! Second derivatives. Second, the sign of the second derivative tells us useful things about the graph of the original function. Third, if P(t) is position and t is time, then P''(t) is acceleration, which we kind of already understand. Finally, and most subtle, the second derivative is a first derivative of the first derivative. It's amazing how easy it is to miss simple manifestations of this fact. For example, If f' is increasing, then f'' is positive. That seems graphically complicated if you're thinking about the graph of f. But consider the graph of f' itself. Going up? Then its derivative is positive. Simple as pie. Many (All?) properties of the second derivative are really about it being a first derivative in its own right.

Section 2.6-Differentiability

Although there were few problems assigned on this section, you should know that there are some situations when even a continuous function will not be differentiable at a point or points--most specifically when the graph has a sharp corner at a point, or the tangent to a graph at a point is vertical. Of course, if a function is not continuous at a point, it is not differentiable there!

Happy studies!
-jason