The Mean Value Theorem (6.5)

1. Explain, in plain English, the result of the Mean Value Theorem, and given an outline of the proof or a picture, explain the proof.
2. Define what it means for a function to be increasing or decreasing.
3. Explain, using algebra, how the MVT connects the sign of the derivative with a function's increase/decrease.
4. Apply the MVT to velocity problems, including problems requiring a difference of functions.

Maxima and Minima (5.1, 5.2, 5.3)

1. Define critical point, local minimum and local maximum.
2. Identify critical points and local extrema on continuous functions, including piecewise functions.
3. Explain the First and Second Derivative Tests and when to use them.
4. Use the First and Second Derivative Test to find local and absolute extrema.
5. Prove that certain simple functions have no local extrema.
6. Given a differentiable function, find its intervals of increase/decrease.
7. Solve simple optimization problems by finding an absolute maximum or minimum on an interval.

Concavity (5.4)

1. Define concave up, concave down and inflection point.
2. Explain how concavity is deduced from the sign of the second derivative.
3. Give examples of functions demonstrating the distinction between the qualities of concave up/down and increasing/decreasing.
4. Given a function, identify potential inflection points, actual inflection points, and intervals of concavity.

Asymptotes and Limits (revisted) (5.5)

1. Define vertical asymptote and horizontal asymptote.
2. Give examples of simple functions with vertical or horizontal asymptotes.
3. Identify the asymptotes of a function.
4. Explain and understand what it means for the value of a limit to be infinite, or for the limit to be taken at infinity.
5. State and apply L'Hopital's Rule.

Function Sketching + Antidifferentiation

1. Given the algebraic expression for a function, sketch its graph, with accurate asymptotes, intervals of increase/decrease, extrema, concavity, and inflection points.
2. Sketch a function given intervals of continuity and differentiability, as well as extrema.
3. Find an algebraic expression for a function given critera similar to above.
4. Given the graph of a derivative, draw a sketch of the original function. (Or, given a graph of a function, draw a sketch of its anti-derivative.
5. Given a simple function, find its general anti-derivative; if given an initial value, find its specific anti-derivative.

Optimization (6.1)

1. Solve optimization problems that can be reduced to analyzing a function of one variable.
2. Understand how functions model real-world phenomena, and evaluate the suitability of a function model

Related Rates (6.2)

1. Solve standard related rates problems, using a list of steps.
2. Give examples of "real life" related rates problems.
3. Evaluate the plausibility of a model involving derivatives.

Linear Approximation (6.4)

1. Find the linear approximation to a function f at a, including identifying good candidates for a.
2. Sketch the graph of a linear approximation to a function at a point, determine whether it is an over or under estimate.
3. Use linear approximations to estimate predictions of mathematical models.
4. Understand different degrees of approximation.