University Level Calculus formulas and notes of the most important concepts Flashcards
(26 cards)
What is the derivative of f(x) = x^n?
f’(x) = n*x^(n-1)
True or False: The integral of a function gives the area under its curve.
True
Fill in the blank: The limit of a function as x approaches a value is denoted as _____.
lim x→a f(x)
What is the power rule for integration?
∫x^n dx = (x^(n+1))/(n+1) + C, for n ≠ -1
What is the Fundamental Theorem of Calculus?
It states that differentiation and integration are inverse processes.
What is the second derivative test used for?
To determine the concavity of a function and classify critical points.
What does the Mean Value Theorem state?
If a function 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).
What is the formula for the area under a curve from a to b?
A = ∫[a, b] f(x) dx
True or False: The derivative of sin(x) is cos(x).
True
What is the limit definition of the derivative?
f’(x) = lim h→0 (f(x+h) - f(x)) / h
What is the chain rule?
If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).
What is the formula for the derivative of e^x?
The derivative of e^x is e^x.
What is a critical point?
A point where f’(x) = 0 or f’(x) is undefined.
Fill in the blank: The Taylor series of a function f(x) around a point a is _____.
f(x) = Σ (f^n(a)/n!)(x-a)^n
What is the formula for the product rule?
If u and v are functions, then (uv)’ = u’v + uv’.
What is L’Hôpital’s Rule used for?
To evaluate limits of indeterminate forms like 0/0 or ∞/∞.
What is the derivative of ln(x)?
The derivative of ln(x) is 1/x.
True or False: The integral of cos(x) is sin(x) + C.
True
What is the formula for the quotient rule?
If u and v are functions, then (u/v)’ = (u’v - uv’)/v^2.
What is the relationship between the first and second derivatives in terms of concavity?
If f’‘(x) > 0, the function is concave up; if f’‘(x) < 0, it is concave down.
What is the formula for finding the length of a curve y = f(x) from x = a to x = b?
L = ∫[a, b] √(1 + (dy/dx)^2) dx
Fill in the blank: The Riemann sum is an approximation of the _____ of a function.
integral
What does it mean for a function to be differentiable at a point?
The function has a defined derivative at that point.
What is the formula for the derivative of tan(x)?
The derivative of tan(x) is sec^2(x).
