Mathematics: Calculus Flashcards

Q: What is 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. The derivative of position with respect to time is velocity.

Q: Define 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. d/dx[x^3] = 3x^2, and d/dx[x^(1/2)] = (1/2)x^(-1/2).

Q: Define 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. d/dx[x^2 times sin x] = 2x sin x + x^2 cos x.

Q: Define 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. d/dx[sin(x^2)] = cos(x^2) times 2x.

Q: Define 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. f(x) = x^3 - 3x has f'(x) = 3x^2 - 3 = 0 at x = 1 and x = -1, its two critical points.

Q: Define 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. f(x) = x^3 has an inflection point at x = 0, where f'' changes from negative to positive.

Q: Define 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. 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.

Q: Define 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. Drive 120 miles in 2 hours and your speedometer must read exactly 60 mph at some instant.

Question 1

What is Limit?

Accepted Answer

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.

Question 2

Define One-Sided Limit.

Accepted Answer

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.

Question 3

Define Continuous Function.

Accepted Answer

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.

Question 4

Define Squeeze Theorem.

Accepted Answer

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.

Question 5

What is Derivative?

Accepted Answer

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.

Question 6

Define Difference Quotient.

Accepted Answer

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.

Question 7

What is Differentiability?

Accepted Answer

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.

Question 8

Define Power Rule.

Accepted Answer

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.

Question 9

Define Product Rule.

Accepted Answer

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.

Question 10

Define Quotient Rule.

Accepted Answer

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.

Question 11

Define Chain Rule.

Accepted Answer

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.

Question 12

Define Implicit Differentiation.

Accepted Answer

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.

Question 13

Define Critical Point.

Accepted Answer

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.

Question 14

Define Inflection Point.

Accepted Answer

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.

Question 15

What is Concavity?

Accepted Answer

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.

Question 16

Define Extreme Value Theorem.

Accepted Answer

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.

Question 17

Define Intermediate Value Theorem.

Accepted Answer

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.

Question 18

Define Rolle's Theorem.

Accepted Answer

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.

Question 19

Define Mean Value Theorem.

Accepted Answer

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.

Question 20

Define Related Rates.

Accepted Answer

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.

Question 21

What is Optimization?

Accepted Answer

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.

Question 22

What is Antiderivative?

Accepted Answer

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.

Question 23

Define Definite Integral.

Accepted Answer

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.

Question 24

Define Riemann Sum.

Accepted Answer

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.

Question 25

Define Fundamental Theorem of Calculus, Part 1.

Accepted Answer

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.

Question 26

Define Fundamental Theorem of Calculus, Part 2.

Accepted Answer

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.

Question 27

What is u-Substitution?

Accepted Answer

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.

Question 28

Define Integration by Parts.

Accepted Answer

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.

Question 29

Define Partial Fraction Decomposition.

Accepted Answer

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.

Question 30

Define Improper Integral.

Accepted Answer

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.

Question 31

What is Sequence?

Accepted Answer

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.

Question 32

What is Series?

Accepted Answer

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.

Question 33

Define Ratio Test.

Accepted Answer

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.

Question 34

Define Taylor Series.

Accepted Answer

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.

Question 35

Define Maclaurin Series.

Accepted Answer

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.

Question 36

Define Radius of Convergence.

Accepted Answer

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.