AP Calculus BC Flashcards
(25 cards)
What is the formula for the derivative of a function?
The derivative of a function f(x) is defined as f’(x) = lim(h -> 0) [f(x+h) - f(x)] / h.
True or False: The Fundamental Theorem of Calculus connects differentiation and integration.
True.
What is the formula for the integral of a power function?
The integral of x^n is (x^(n+1))/(n+1) + C, for n ≠ -1.
Fill in the blank: The limit definition of e is lim(n -> ∞) (1 + 1/n)^n = _____ .
e.
What is the formula for the Taylor series expansion of a function f(x) around a point a?
f(x) = f(a) + f’(a)(x-a) + f’‘(a)(x-a)^2/2! + f’’‘(a)(x-a)^3/3! + …
What is the result of the integral from 0 to 1 of x^2 dx?
1/3.
True or False: The Mean Value Theorem states that if f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that f’(c) = (f(b) - f(a))/(b - a).
True.
What is the formula for the area under a curve from a to b?
A = ∫[a to b] f(x) dx.
What is the derivative of sin(x)?
The derivative of sin(x) is cos(x).
What is the formula for the second derivative test for local extrema?
If f’‘(c) > 0, then f has a local minimum at c; if f’‘(c) < 0, then f has a local maximum at c.
What is the formula for the volume of a solid of revolution using the disk method?
V = π ∫[a to b] [f(x)]^2 dx.
What is the derivative of e^x?
The derivative of e^x is e^x.
Fill in the blank: The formula for the sum of a geometric series is _____ .
S = a/(1 - r) for |r| < 1.
What is the formula for the integral of e^(kx)?
The integral of e^(kx) is (1/k)e^(kx) + C.
True or False: The product rule states that the derivative of a product of two functions is the product of their derivatives.
False.
What is the formula for the arc length of a curve y = f(x) from x = a to x = b?
L = ∫[a to b] √(1 + (f’(x))^2) dx.
What is the formula for the limit definition of a derivative at a point x = a?
f’(a) = lim(h -> 0) [f(a+h) - f(a)] / h.
What is the formula for the chain rule?
If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).
Fill in the blank: The formula for the nth term of a Taylor series centered at a is _____ .
f^(n)(a)/n! * (x-a)^n.
What is the formula for integration by parts?
∫u dv = uv - ∫v du.
What is the derivative of ln(x)?
The derivative of ln(x) is 1/x.
What is the formula for the limit of sin(x)/x as x approaches 0?
lim(x -> 0) sin(x)/x = 1.
True or False: The squeeze theorem can be used to evaluate limits.
True.
