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Calculus Flashcards

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Calculus

46 flashcards

The limit of a function is the value that the function approaches as the input value gets closer and closer to some value, even if the function is not defined at that point.
The limit from the right refers to the limit as the input value approaches from larger values, while the limit from the left refers to the limit as the input value approaches from smaller values. For a limit to exist, these two limits must be equal.
The derivative of a function is the rate of change of the function with respect to one of its variables. It represents the slope of the tangent line to the function's graph at a given point.
The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1), where n is any real number except 0.
The derivative of a constant function is always zero, since the rate of change of a constant is zero.
The chain rule is used to differentiate a composite function. If y = f(g(x)), then y' = f'(g(x))g'(x).
An integral is the inverse operation of differentiation. It is a way to find the accumulated value of a function over an interval, representing the area under the curve of the function.
The fundamental theorem of calculus states that differentiation and integration are inverse operations. It relates the derivative of an integral to the original function and vice versa.
A definite integral has specified limits of integration and gives the exact value of the area under the curve. An indefinite integral does not have limits and gives the antiderivative or a family of functions.
A series is an infinite sum of terms that follow a specific pattern. Series are used to represent functions, approximate values, and solve differential equations.
A convergent series has a finite sum, meaning that the terms approach a specific value as more terms are added. A divergent series has an infinite sum and does not approach a specific value.
A Taylor series is an infinite series representation of a function around a given point, using the derivatives of the function at that point. It is useful for approximating functions and solving differential equations.
The Mean Value Theorem for integrals states that if a function is continuous on a closed interval [a, b], then there exists at least one point c in the interval such that the integral from a to b is equal to the function evaluated at c multiplied by (b - a).
Position is the antiderivative of velocity, and velocity is the antiderivative of acceleration. In other words, velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time.
An exact differential equation can be solved by finding an exact solution that satisfies the equation. An inexact differential equation cannot be solved exactly and requires approximation methods like series solutions or numerical methods.
A partial derivative is the derivative of a multivariable function with respect to one of its variables, treating all other variables as constants.
A local maximum or minimum is the highest or lowest point in a specific region of the function's domain, while a global maximum or minimum is the highest or lowest point on the entire domain.
A critical point is a point where the derivative is equal to zero or undefined. An inflection point is a point where the concavity of the function changes from positive to negative or vice versa.
Integration and differentiation are inverse operations, as stated by the fundamental theorem of calculus. If F(x) is the antiderivative of f(x), then F'(x) = f(x).
A separable differential equation can be separated into two parts, one containing only the variable and the other containing only the derivative, which can be solved by integration. A non-separable differential equation cannot be separated in this way and requires other solution methods.
An ordinary differential equation involves one independent variable and its derivatives, while a partial differential equation involves multiple independent variables and their partial derivatives.
A first-order differential equation involves only the first derivative of the unknown function, while a second-order differential equation involves the second derivative of the unknown function.
A linear differential equation is one where the unknown function and its derivatives appear linearly (not multiplied or raised to powers). A non-linear differential equation involves the unknown function and its derivatives in a non-linear way.
A homogeneous differential equation has only the unknown function and its derivatives on one side of the equation, with the other side being zero. A non-homogeneous differential equation has additional terms that are not just functions of the unknown variable and its derivatives.
A Riemann sum is an approximation of the definite integral by dividing the area under the curve into a finite number of rectangular regions and summing their areas. A definite integral is the exact value of the area under the curve.
A continuous function is a function whose graph can be drawn without lifting the pencil from the paper. A discontinuous function has points where the graph must be lifted or moved, creating a break or jump in the graph.
A piecewise function is defined by multiple sub-functions, each applying to a different range of the input variable. A composite function is a function formed by combining two or more functions, where the output of one function becomes the input of the next.
An absolute maximum is the highest point on the entire domain of a function, while an absolute minimum is the lowest point on the entire domain of a function.
A secant line intersects a curve at two points, while a tangent line touches the curve at only one point and is perpendicular to the radius of curvature at that point.
A bounded function is a function whose values are constrained within a specific range, while an unbounded function can take on arbitrarily large or small values.
A periodic function is a function that repeats its values at regular intervals, while a non-periodic function does not exhibit this repeating behavior.
An even function is a function that satisfies f(-x) = f(x) for all x in its domain, while an odd function satisfies f(-x) = -f(x) for all x in its domain.
A one-sided limit considers the behavior of a function as the input approaches a particular value from only one direction (either from the left or from the right), while a two-sided limit considers the behavior from both directions.
A convergent sequence or series has a finite limit as the number of terms increases, while a divergent sequence or series does not have a finite limit and continues to grow or oscillate without bound.
A power series is an infinite series representation of a function using powers of the independent variable, while a Fourier series is an infinite series representation of a periodic function using sine and cosine terms.
A Maclaurin series is a specific type of Taylor series where the center point is zero, meaning that the infinite series representation is centered around x = 0.
A geometric series is a series where each term is obtained by multiplying the previous term by a constant factor, while an arithmetic series is a series where each term is obtained by adding a constant value to the previous term.
A parametric equation represents a curve or surface using two or more equations involving a parameter, while an explicit equation directly relates the dependent and independent variables without using a parameter.
A polar equation represents a curve or surface using polar coordinates (radius and angle), while a Cartesian equation represents a curve or surface using rectangular coordinates (x and y).
A singular point of a differential equation is a point where the equation or its solution becomes undefined or behaves in an unusual way, while a regular point is a point where the equation and its solution are well-behaved.
A first-order differential equation involves only the first derivative of the unknown function, while a higher-order differential equation involves second or higher derivatives of the unknown function.
A linear function is a function whose graph forms a straight line, while a non-linear function is a function whose graph is not a straight line.
A polynomial function is a function that can be expressed as a sum of powers of the independent variable, while a rational function is a function that can be expressed as a ratio of two polynomials.
A trigonometric function is a function that involves sine, cosine, tangent, or their reciprocals, while an exponential function is a function that involves a variable exponent.
A logarithmic function is the inverse of an exponential function, while a hyperbolic function is a function that involves hyperbolic sine, cosine, or tangent.
A transcendental function is a function that cannot be expressed using a finite combination of algebraic operations (addition, subtraction, multiplication, division, and root extraction), while an algebraic function can be expressed in this way.