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AP Calculus AB
47 flashcards
The limit of a function is the value that the function approaches as the input value gets closer and closer to some particular value.
There are one-sided limits (left-hand and right-hand limits) and two-sided limits.
The derivative of a function at a point is defined as the limit of the difference quotient as the step size approaches zero.
The main rules are the power rule, product rule, quotient rule, and chain rule.
It relates differentiation and integration: if a function is continuous, then its indefinite integral is the antiderivative, and the definite integral is the area under the curve.
A definite integral has limits and represents the area under a curve. An indefinite integral does not have limits and represents the antiderivative.
Substitution, integration by parts, partial fractions, and using tables of integrals.
It states that for a continuous function on a closed interval, there exists at least one point where the function equals the average value of the function over that interval.
Average rate of change measures change over an interval, while instantaneous rate of change measures the rate at a single point using the derivative.
Finding velocity, acceleration, related rates, optimization problems, and curve sketching with derivatives.
Finding the area under and between curves, volumes of solids of revolution, average value of a function, and solving separable differential equations.
It uses linear approximations to iteratively compute approximate values at successive points from an initial value.
An increasing function has a positive derivative, while a decreasing function has a negative derivative.
A function is continuous if the limit of the function exists at that point and equals the function value.
Checking for removable discontinuities, vertical and horizontal asymptotes, infinite limits and jump discontinuities.
A function is differentiable if its derivative exists at that point.
Checking if the limit definition of the derivative exists by examining the difference quotient limit.
Intervals of increase/decrease, local max/min, points of inflection, asymptotes, periodicity.
An absolute extremum is the overall max or min on an entire domain, while a relative extremum is a local max or min.
Take the derivative, set it equal to 0, solve for critical points, then use the Second Derivative Test.
It determines concavity and the nature of relative extrema using the sign of the second derivative.
Periodicity, symmetry, asymptotes, critical points, and derivatives of sine, cosine, tangent, etc.
Substitute u for an expression in the integral, then integrate with respect to u, and finally substitute back.
One part (dv) is easy to integrate, while the other part (u) has an antiderivative.
Find the top curve's antiderivative and the bottom curve's antiderivative, then evaluate them at the limits.
Problems like finding rate of change of area/volume with changing measurements, rate of shadow length, etc.
An improper integral has infinite limits or is over an interval containing a discontinuity. It calculates area under certain curves.
A differential equation where one variable appears in only one term, allowing separation of variables.
Use the chain rule, reciprocal derivatives, and relationships between inverse trig functions.
Use tests like the nth term test, integral test, p-series test, alternating series test, ratio test, etc.
Even functions are symmetric about the y-axis, while odd functions are symmetric about the origin.
A function is one-to-one if each output value corresponds to at most one input value in the entire domain.
L'Hopital's Rule is used to evaluate limits of indeterminate forms by replacing them with the limits of their derivatives.
Absolute convergence means the series converges absolutely. Conditional means it only converges if terms are arranged in the given order.
A sequence is a list of terms, while a series is the sum of the terms of a sequence.
Partial fraction decomposition breaks a fraction into a sum of simpler fractions. The method solves for coefficients in this decomposition.
Take the derivative of the function and evaluate it at the given point.
Use methods like the Trapezoidal Rule or Simpson's Rule that approximate the region as a sum of trapezoids or parabolic segments.
An absolutely convergent series will converge regardless of how its terms are rearranged. Its terms must satisfy the absolute convergence test.
The derivative at a point is the slope of the tangent line, which can be constructed geometrically.
Use the relations x=rcos(theta), y=rsin(theta) to convert between polar and Cartesian, and parametric gives x(t), y(t).
Position is given directly, velocity is the derivative of position, acceleration is the derivative of velocity.
It states that a continuous function must have a derivative equal to the slope between any two points in its domain.
Local extrema are relatively max/min in an open interval, while global extrema are overall max/min of the entire function.
Properties include base conversions, laws of logs/exponents, derivatives, limits, and graphical behavior.
Calculate the antiderivative of sqrt(1 + (dy/dx)^2), then evaluate between limits to get arc length.
An implicit function relates variables through an equation not solved for one variable explicitly.