AP Calculus Theorems Flashcards
This set goes over all those pesky theorems, rules, and properties that are useful to know when it comes to the AP test.
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| 9860907636 | Definition of Continuity | 1. lim x→c f(x) exists. 2. f(c) exists. 3. lim x→c f(x) = f(c) | 0 | |
| 9860907637 | When does the limit not exist? | 1. f(x) approaches a different number from the right as it does from the left as x→c 2. f(x) increases or decreases without bound as x→c 3. f(x) oscillates between two fixed values as x→c | 1 | |
| 9860907638 | Intermediate Value Theorem | If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b) then there is at least one number c in [a, b] such that f(c) = k | 2 | |
| 9860907639 | Definition of a Derivative | lim h→0 (f(x+h) - f(x)) / h | 3 | |
| 9860907640 | Product Rule | d/dx (f(x) g(x)) = f(x)g'(x) + g(x) f'(x) | 4 | |
| 9860907641 | Quotient Rule | d/dx (g(x)/ h(x)) = (h(x) g'(x) - g(x) h'(x))/ h(x)^2 | 5 | |
| 9860907642 | Chain Rule | d/dx f(g(x)) = f'(g(x)) g'(x) | 6 | |
| 9860907643 | Extrema Value Theorem | If f is continuous on the closed interval [a, b], then f has both a maximum and a minimum on the interval. | 7 | |
| 9860907644 | The first derivative gives what? | 1. critical points 2. relative extrema 3. increasing and decreasing intervals | 8 | |
| 9860907645 | The second derivative gives what? | 1. points of inflection 2. concavity | 9 | |
| 9860907646 | Rolle's Theorem | Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f'(c)= 0 | 10 | |
| 9860907647 | Mean Value Theorem | f'(c) = (f(b) - f(a))/ (b - a) | 11 | |
| 9860907648 | Fundamental Theorem of Calculus | The integral on (a, b) of f(x) dx = F(b) - F(a) | 12 | |
| 9860907649 | Mean Value Theorem (Integrals) | The integral on (a, b) of f(x) dx = f(c) (b - a) | 13 | |
| 9860907650 | Average Value Theorem | 1/ (b-a) times the integral on (a, b) of f(x) dx | 14 | |
| 9860907651 | Second Fundamental Theorem of Calculus | If f is continuous on an open interval containing a, then for every x in the interval the derivative of the the integral of f(x) dx on said interval is equal to f(x) | 15 | |
| 9860907652 | Derivative of an Inverse Function | g'(x) = 1/ f'(g(x)) where g(x) is the inverse of f(x) | 16 |