AP CALCULUS AB REVIEW OF THE DERIVATIVE, RELATED RATES, & PVA

The Derivative The derivative has three major interpretations: the instantaneous rate of change, the slope of the graph, and the slope of the tangent line. The derivative is a limit. There are two forms : In order for the derivative to exist at a point, the left and right hand derivatives must match. The derivative does not exist for values of x where the function has a sharp turn, discontinuity, vertical tangent, or vertical asymptote. [Ex] See Board For Examples.

Derivative Rules, Implicit Differentiation, Tangent Lines, and Inverses Make sure you know all the basic rules: power, product, quotient, chain. Review the last page of the notes from Section 5.6. There is a summary of all the rules that we have covered. For implicit differentiation add a dy/dx term to all y terms that you take the derivative of. Then isolate and solve for dy/dx. For tangent line problems: (i) find the derivative, (ii) plug in the point given (you may need to find the y-coordinate), (iii) use point-slope form to find the equation of the line. If g(x) is the inverse of f(x) then the derivative of g(x) at x = a is given by: [Exs] See Board.

Related Rates Guidelines For RR Problems: (1)Identify all given quantities and quantities you are trying to find. Draw a diagram. (2)Write an equation involving the variables whose rates of change you are either given or are trying to solve for. (3)Using chain rule implicitly differentiate both sides of the equation with respect to time. (4)Substitute in the given values to solve for the rate of change you are finding. [Ex] See Board

PVA Position: s(t) Velocity: v(t) = s′(t) Acceleration: a(t) = v′(t) = s′′(t) Things You Want to Find: (1) Initial position: evaluate s(0) (assuming t≥0) (2) When the object is at the starting point: set s(t) = s(0) and solve for t. (3) When the object is at rest: set v(t) = 0 (4) When the object is moving right: v(t) > 0 (5) When the object is moving left: v(t) < 0 (6) When the object is changing direction: find where v(t) changes sign. This will have to be at a time where v(t) = 0. We denote these times of change tc. (7) Average velocity on the interval [t1, t2]: ∆s/∆t = s(t2) –s(t1) t2 – t1 (8) Displacement on the interval [t1, t2]: ∆s = s(t2) –s(t1) (9) When the acceleration is 0: set a(t) = 0 (10) Total distance traveled: │s(t1) - s(tc)│+│s(tc) - s(t2)│** (11) Sketch the motion: (i) Find s(0) and any times where the particle changes direction, tc. (ii) Evaluate s(tc) and plot the points on a number line. (iii) Use arrows to show the direction of travel. (iv) Make sure to label the time next to each point you plot. (12) Extreme points: Use the sketch to determine where the object reaches its most extreme values on the given interval (that is what are the maximum and minimum values the object reaches. [ Ex] See Board