Mathematics
Calculus Flashcards
36 calculus terms every Calc 1-2 exam expects you to define
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All 36 Mathematics Calculus flashcards
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- Antiderivative
- A function F whose derivative is the given function f, so that F'(x) = f(x). Any two antiderivatives of the same function differ only by an added constant.For example: x^3/3 is an antiderivative of x^2, and so is x^3/3 + 7.
- Chain Rule
- A rule for differentiating a composite function: the derivative of f(g(x)) is f'(g(x)) times g'(x). Multiply the outer derivative, evaluated at the inner function, by the inner derivative.For example: d/dx[sin(x^2)] = cos(x^2) times 2x.
- Concavity
- The direction a curve bends. A graph is concave up where f''(x) > 0, lying above its tangent lines, and concave down where f''(x) < 0, lying below its tangent lines.For example: f(x) = x^2 is concave up everywhere because f''(x) = 2 > 0.
- Continuous Function
- A function is continuous at a point when the limit there exists, the function is defined there, and the two are equal, so the graph has no break, jump, or hole at that point.For example: Polynomials are continuous everywhere; f(x) = 1/x is continuous on its domain but not at x = 0.
- Critical Point
- A point in the domain of f where the derivative is zero or undefined. Critical points are the only candidates for local maxima and minima in the interior of the domain.For example: f(x) = x^3 - 3x has f'(x) = 3x^2 - 3 = 0 at x = 1 and x = -1, its two critical points.
- Definite Integral
- The signed area between a function's graph and the x-axis over an interval [a, b], defined as the limit of Riemann sums. It evaluates to a number, not a function.For example: The integral of x from 0 to 1 equals 1/2, the area of the triangle under y = x.
- Derivative
- The instantaneous rate of change of a function, defined as the limit of the difference quotient as the interval shrinks to zero. Geometrically it is the slope of the tangent line.For example: The derivative of position with respect to time is velocity.
- Difference Quotient
- The expression [f(x + h) - f(x)] / h, giving the average rate of change of f over an interval of width h. Its limit as h approaches 0 defines the derivative.For example: For f(x) = x^2, the difference quotient simplifies to (2xh + h^2)/h = 2x + h, which tends to 2x.
- Differentiability
- A function is differentiable at a point when its derivative exists there, which requires a smooth graph with no corner, cusp, or vertical tangent. Differentiability implies continuity, but not the reverse.For example: f(x) = |x| is continuous at x = 0 but not differentiable there because of the corner.
- Extreme Value Theorem
- If a function is continuous on a closed, bounded interval [a, b], then it attains both an absolute maximum and an absolute minimum value somewhere on that interval.For example: On [0, 2pi], sin x reaches a maximum of 1 and a minimum of -1; the theorem guarantees such values exist.
- Fundamental Theorem of Calculus, Part 1
- It states that if g(x) is the integral of a continuous f from a to x, then g'(x) = f(x). Differentiation undoes integration, so an integral with a variable upper limit is an antiderivative.For example: d/dx of the integral of cos t from 0 to x equals cos x.
- Fundamental Theorem of Calculus, Part 2
- It states that the integral of f from a to b equals F(b) - F(a), where F is any antiderivative of f. This turns computing an area into evaluating an antiderivative at two points.For example: The integral of sin x from 0 to pi is [-cos x] from 0 to pi = 1 + 1 = 2.
- Implicit Differentiation
- A technique for finding dy/dx when y is not isolated: differentiate both sides of the equation with respect to x, applying the chain rule to every y term, then solve algebraically for dy/dx.For example: For x^2 + y^2 = 25, differentiating gives 2x + 2y(dy/dx) = 0, so dy/dx = -x/y.
- Improper Integral
- A definite integral with an infinite limit of integration or an unbounded integrand, evaluated as the limit of proper integrals. It converges if that limit is finite and diverges otherwise.For example: The integral of 1/x^2 from 1 to infinity equals the limit of (1 - 1/b) as b grows, which is 1, so it converges.
- Inflection Point
- A point where a curve changes concavity, from concave up to concave down or the reverse. The second derivative is zero or undefined there and changes sign across it.For example: f(x) = x^3 has an inflection point at x = 0, where f'' changes from negative to positive.
- Integration by Parts
- An integration technique derived from the product rule: the integral of u dv equals uv minus the integral of v du. Choose u to be the factor that simplifies when differentiated.For example: The integral of x times e^x dx = x e^x - integral of e^x dx = x e^x - e^x + C.
- Intermediate Value Theorem
- If f is continuous on [a, b] and N lies between f(a) and f(b), then there is at least one c in (a, b) with f(c) = N. It guarantees that roots exist.For example: Since f(x) = x^3 + x - 1 is -1 at x = 0 and 1 at x = 1, it has a root somewhere between 0 and 1.
- Limit
- The value a function f(x) approaches as x gets arbitrarily close to a fixed number, whether or not f is defined there. Written lim x->a f(x) = L.For example: lim x->2 of (x^2 - 4)/(x - 2) = 4, even though the expression is undefined at x = 2.
- Maclaurin Series
- A Taylor series centered specifically at a = 0, with terms f^(n)(0)/n! times x^n. It is the most common form because expanding about zero keeps the coefficients simple.For example: The Maclaurin series for sin x is x - x^3/3! + x^5/5! - ...
- Mean Value Theorem
- If f is continuous on [a, b] and differentiable on (a, b), then some c in (a, b) satisfies f'(c) = [f(b) - f(a)] / (b - a): the instantaneous rate equals the average rate somewhere.For example: Drive 120 miles in 2 hours and your speedometer must read exactly 60 mph at some instant.
- One-Sided Limit
- The value a function approaches as x nears a point from a single direction: the left-hand limit uses x < a, the right-hand limit uses x > a. Both must agree for the two-sided limit to exist.For example: For f(x) = |x|/x, the left-hand limit at 0 is -1 and the right-hand limit is 1, so lim x->0 does not exist.
- Optimization
- The process of finding the largest or smallest value of a quantity subject to constraints, done by expressing it as one function, locating its critical points, and testing them against the endpoints.For example: Maximizing the area of a rectangle with a fixed perimeter of 40 gives a 10-by-10 square.
- Partial Fraction Decomposition
- A method of rewriting a rational function as a sum of simpler fractions with linear or irreducible quadratic denominators, so that each piece can be integrated separately.For example: 1/[x(x + 1)] = 1/x - 1/(x + 1), which integrates to ln|x| - ln|x + 1| + C.
- Power Rule
- A differentiation rule stating that the derivative of x^n is n times x^(n-1) for any real exponent n. It is the most frequently used rule for polynomials and roots.For example: d/dx[x^3] = 3x^2, and d/dx[x^(1/2)] = (1/2)x^(-1/2).
- Product Rule
- A rule for differentiating a product of two functions: the derivative of f(x)g(x) equals f'(x)g(x) + f(x)g'(x). It is not simply the product of the two derivatives.For example: d/dx[x^2 times sin x] = 2x sin x + x^2 cos x.
- Quotient Rule
- A rule for differentiating a ratio: the derivative of f(x)/g(x) is [f'(x)g(x) - f(x)g'(x)] divided by [g(x)]^2, valid wherever g(x) is not zero.For example: d/dx[x/(x + 1)] = [1(x + 1) - x(1)]/(x + 1)^2 = 1/(x + 1)^2.
- Radius of Convergence
- The number R such that a power series with terms c_n(x - a)^n converges for every x with |x - a| < R and diverges when |x - a| > R. It is often found with the ratio test.For example: The series of x^n has radius of convergence 1, converging on the interval (-1, 1).
- Ratio Test
- A convergence test for a series: compute L, the limit of |a_(n+1) / a_n|. The series converges absolutely if L < 1, diverges if L > 1, and the test is inconclusive if L = 1.For example: For the series of 1/n!, the ratio is 1/(n + 1), which tends to 0 < 1, so the series converges.
- A class of problems where two or more quantities change with time and are linked by an equation. Differentiating that equation with respect to time relates their rates of change.For example: For a ladder sliding down a wall, differentiating x^2 + y^2 = L^2 relates how fast the base moves to how fast the top drops.
- Riemann Sum
- An approximation of a definite integral formed by dividing an interval into subintervals, multiplying each width by a sample function value, and adding the products. Its limit as the widths shrink is the integral.For example: A right Riemann sum for the integral of x^2 on [0, 1] with four rectangles uses heights at x = 0.25, 0.5, 0.75, and 1.
- Rolle's Theorem
- If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there is at least one c in (a, b) where f'(c) = 0. It is the special case of the Mean Value Theorem.For example: f(x) = x^2 - 2x on [0, 2] has f(0) = f(2) = 0, and f'(1) = 0.
- Sequence
- An ordered infinite list of numbers a1, a2, a3, ... defined by a formula or rule. It converges if its terms approach a single finite limit as n grows without bound.For example: The sequence a_n = 1/n converges to 0; the sequence a_n = (-1)^n diverges.
- Series
- The sum of the terms of a sequence, written as sigma of a_n. It converges when its sequence of partial sums approaches a finite limit, and its value is that limit.For example: The geometric series with terms (1/2)^n starting at n = 1 converges to 1.
- Squeeze Theorem
- If g(x) <= f(x) <= h(x) near a point and g and h both approach the same limit L there, then f is forced to approach L as well. Also called the sandwich theorem.For example: Because -x^2 <= x^2 sin(1/x) <= x^2, the Squeeze Theorem gives lim x->0 of x^2 sin(1/x) = 0.
- Taylor Series
- A representation of a function as an infinite power series with terms f^(n)(a)/n! times (x - a)^n, built from the function's derivatives at a single point a. It matches the function near a.For example: The Taylor series of e^x about a = 0 is 1 + x + x^2/2! + x^3/3! + ...
- u-Substitution
- An integration technique that reverses the chain rule: substitute u for an inner function, replace dx using du = u'(x) dx, integrate in terms of u, then convert back to x.For example: The integral of 2x(x^2 + 1)^5 dx becomes the integral of u^5 du with u = x^2 + 1, giving (x^2 + 1)^6 / 6 + C.