Calculus BC Must Knows Flashcards
d/dx (sinx)
cosx
d/dx(cosx)
-sinx
What is the Fundamental Theorem of Calculus?
It states that if F is an antiderivative of f on [a, b], then ∫ from a to b of f(x)dx = F(b) - F(a).
What is the formula for the derivative of a power function?
If f(x) = x^n, then f’(x) = n*x^(n-1).
True or False: The limit of a function as x approaches a can be found using the value of the function at a.
False.
What is the formula for the area under a curve?
A = ∫ from a to b of f(x)dx.
What is L’Hôpital’s Rule used for?
It is used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞.
Fill in the blank: The Taylor series for e^x is _____ .
∑ from n=0 to ∞ of (x^n/n!).
What is the formula for the derivative of the natural logarithm?
If f(x) = ln(x), then f’(x) = 1/x.
What is the equation of the Mean Value Theorem?
If f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) such that f’(c) = (f(b) - f(a)) / (b - a).
True or False: The second derivative test can determine if a critical point is a local maximum or minimum.
True.
What is the formula for the arc length of a curve y = f(x) from x = a to x = b?
L = ∫ from a to b of √(1 + (f’(x))^2) dx.
What is the formula for integration by parts?
∫ u dv = uv - ∫ v du.
Fill in the blank: The derivative of sin(x) is _____ .
cos(x).
What is the power series representation of sin(x)?
∑ from n=0 to ∞ of ((-1)^n * x^(2n+1)) / (2n+1)!
What is the formula for the volume of a solid of revolution using the disk method?
V = π ∫ from a to b of [f(x)]^2 dx.
True or False: The convergence of a series can be tested using the Ratio Test.
True.
What is the formula for the derivative of cos(x)?
-sin(x).
Fill in the blank: The limit definition of the derivative is _____ .
f’(x) = lim (h -> 0) of (f(x+h) - f(x)) / h.
What is the formula for the geometric series?
S = a / (1 - r) for |r| < 1.
What is the formula for the nth term of a Taylor series?
f^(n)(a) / n! * (x - a)^n.
What is the difference between a definite and an indefinite integral?
A definite integral has limits and yields a number, while an indefinite integral has no limits and yields a function.
Fill in the blank: The integral of cos(x) is _____ .
sin(x) + C.
What is the formula for the surface area of a solid of revolution?
SA = 2π ∫ from a to b of f(x)√(1 + (f’(x))^2) dx.
