AP Calc AB Flashcards

1

Intermediate Value Theorem

If f is continuous on [a, b], then f takes on every y value between f(a) and f(b).

Definition of Derivative

Alternate Definition of Derivative/Derivative at a Point

Power Rule

Product Rule

Quotient Rule

d/dx sinx

cosx

d/dx cosx

-sinx

d/dx tanx

sec²x

d/dx cotx

-csc²x

d/dx secx

secxtanx

d/dx cscx

-cscxcotx

Chain Rule

d/dx sin^-1x

d/dx cos^-1x

d/dx tan^-1x

d/dx cot^-1x

d/dx sec^-1x

d/dx csc^-1x

d/dx e^x

e^x

d/dx a^x

a^xln(a)

d/dx lnx

1/x

d/dx log_bx

1/xln(b)

Extreme Value Theorem

If f is continuous on [a, b], then f has both an absolute maximum and absolute minimum on that interval.

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

Average Value

If f is integrable on [a, b], its average value on [a, b] is 1/(b - a) ∫ from a to b of f(x)dx.

The Fundamental Theorem of Calculus, Part 1

If f is continuous on [a, b], then the function F defined by F(x) = ∫ from a to x of f(t)dt has a derivative F'(x) = f(x).

The Fundamental Theorem of Calculus, Part 2

If F is any antiderivative of f on [a, b], then ∫ from a to b of f(x)dx = F(b) - F(a).

Y changes at a rate proportional to the amount

present

Fundamental Theorem in Morgan Language

Given a velocity v(t) and a position at a given time... ex: when t = 8, the position is 4. Find the position at 2 seconds.

Displacement

The change in position of an object over time, calculated as the difference between initial and final positions.

Total Distance Traveled

The integral of the absolute value of velocity over a given time interval.

Derivative of Inverses

If f is the inverse of g, then d/dx[f(g(x))] = 1/g'(f(g(x))).