I've recently joined a group of AP Calc teachers who share ideas, ask questions, clarify concepts, discuss scoring protocals, etc. I've learned quite a bit from these folks and hope to share their insights with my students and blog followers. Here are some of the last minute ideas to insure success on the AP Calculus AB test.
* When calculating “total distance” use absolute value of the integral. Even if walking backwards, the distance traveled is still positive.
* Average velocity -- is an integral of the endpoint velocities, ∫[v(b) - v(a)], divided by the difference b-a. (Sum of velocities over the interval divided by the length of time of the interval.)
Average acceleration -- use [v(b) - v(a)] divided by b-a. Think: v(b) IS the integral of v’(b), the integral of acceleration. ∫[v’(b) - v’(a)] / b-a (Sum of accelerations over the interval divided by the length of time of the interval.)......This interpretation is closely related to....
Average rate of change -- is an integral of [f’(b) - f’(a)] divided by the difference b-a.
∫[f’(b) - f’(a)]/b-a (Sum of the rates over the interval divided by the length of time of the
* A particle is speeding UP if the velocity and acceleration are both either positive or negative.
A particle is slowing DOWN if the velocity and acceleration are opposite signs.
* A function must be continuous if it is differentiable, but continuity does NOT guarantee differentiability: think of sharp corners and piecewise functions.
* Concavity Theorem: The graph of f is concave UP at (c,f(c)) if f”(c)>0 and concave DOWN if f"(c)<0.
* Intermediate Value Theorem: IVT. (using the abbreviation is acceptable)
Use it to explain that there is an X value, c between a and b, that gives a Y value between the Y values of a and b.
* Mean Value Theorem (think Difference Quotient): MVT (abbreviation is acceptable)
There is at least one point on the graph at which the derivative (slope) of the curve is equal (parallel) to the "average" derivative of the arc (also known as the slope of the secant).
- If f’ is positive on a closed interval, then f is increasing. (positive slope)
- If f’ is negative, f is decreasing. (negative slope)
* Volume of rotations around vertical lines (x= or the y axis), use x= equations where the variable is y. Indicate domain of integrals as the highest and lowest y values.
* Volume of rotations around horizontal lines (y= or the x axis), use y= equations where the variable is x. Indicate domain of integrals as the highest and lowest x values.
* When finding absolute min/max values, be sure to check the endpoints of a closed interval.
* Explanations need to rely on mathematical reasoning; using a visual interpretation of “the graph” is not sufficient.
Use this collection of last minute reminders from hundreds of AP Calc teachers to earn your best score on the College Board test.