Calculus Concepts
Fundamental ideas in differential and integral calculus.
What is a derivative?
The instantaneous rate of change of a function with respect to a variable. Geometrically, the slope of the tangent line at a point. Notation: f'(x) or dy/dx.
State the Power Rule
d/dx [x^n] = n·x^(n-1). E.g. d/dx [x³] = 3x².
What is an integral?
The reverse of differentiation. A definite integral ∫ₐᵇ f(x)dx gives the net area under a curve between x=a and x=b. An indefinite integral gives a family of antiderivatives.
State the Fundamental Theorem of Calculus
Part 1: If F is an antiderivative of f on [a,b], then ∫ₐᵇ f(x)dx = F(b) − F(a). Part 2: d/dx ∫ₐˣ f(t)dt = f(x).
What is a limit?
The value a function approaches as the input approaches some value. Written lim(x→c) f(x) = L. Foundation of both derivatives and integrals.
What is the Chain Rule?
d/dx [f(g(x))] = f'(g(x)) · g'(x). Used when differentiating composite functions.
What is the Product Rule?
d/dx [u·v] = u'v + uv'. Used when differentiating the product of two functions.
What is L'Hôpital's Rule?
If lim f/g yields 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the latter limit exists.
What is a Taylor Series?
An infinite sum representing a function as a power series: f(x) = Σ f⁽ⁿ⁾(a)/n! · (x-a)ⁿ. Approximates smooth functions around a point a.
What is the difference between definite and indefinite integrals?
Indefinite integral: ∫f(x)dx = F(x) + C (antiderivative, no bounds). Definite integral: ∫ₐᵇ f(x)dx = a specific number (area/accumulation between bounds a and b).