CLEP Calculus flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Limit definition

The limit of f(x) as x approaches a equals L when f(x) can be made arbitrarily close to L by taking x sufficiently close to a, but not equal to a.

One-sided limits

The left-hand limit uses x approaching a from below and the right-hand limit uses x approaching from above. A two-sided limit exists only when both are equal.

Indeterminate form 0/0

A quotient limit where substitution yields zero over zero. It signals possible cancellation or use of L'Hôpital's Rule; the value is not automatically undefined.

L'Hôpital's Rule

For a 0/0 or ∞/∞ indeterminate limit, lim f/g equals lim f'/g' when that derivative limit exists. Differentiate numerator and denominator separately, not as a quotient.

Squeeze theorem

If g(x) ≤ f(x) ≤ h(x) near a and both g and h have limit L at a, then f also has limit L at a. Useful for oscillating functions like sin(1/x).

Limit of sin x over x

As x approaches 0, (sin x)/x approaches 1. This fundamental trig limit underlies the derivative formulas for sine and cosine.

Limits at infinity

For rational functions, compare polynomial degrees of numerator and denominator to determine the horizontal asymptote: lower degree gives 0, equal gives leading-coefficient ratio, higher diverges.

Continuity at a point

f is continuous at a when f(a) is defined, the limit as x approaches a exists, and that limit equals f(a). All three conditions must hold.

Derivative as a limit

f'(x) equals the limit as h approaches 0 of [f(x+h) − f(x)]/h. It gives the instantaneous rate of change and tangent-line slope.

Power rule

The derivative of x raised to a constant power n is n times x raised to n minus one. Valid for all real exponents where the expression is defined.

Product rule

The derivative of a product f·g equals f' times g plus f times g'. Both factors must be differentiated, one at a time.

Quotient rule

The derivative of f/g equals (f'g − fg') divided by g squared. Order matters in the numerator; reversing it flips the sign.

Chain rule

The derivative of f(g(x)) equals f'(g(x)) times g'(x). Differentiate the outer function, keep the inner unchanged, then multiply by the inner derivative.

Derivative of sine

The derivative of sin x is cos x, and the derivative of cos x is negative sin x. These follow from the special limit of (sin h)/h.

Derivative of e to the x

The derivative of e^x is e^x itself, and the derivative of ln x is 1/x for x > 0. These are the key exponential and logarithmic derivatives.

Critical point

A point in the domain where f'(x) equals zero or f' is undefined. Critical points are candidates for local extrema but need not be extrema.

First derivative test

If f' changes from positive to negative at a critical point, f has a local maximum there; a change from negative to positive indicates a local minimum.

Second derivative test

At a point where f' equals zero, f'' positive means local minimum, f'' negative means local maximum, and f'' zero is inconclusive.

Concavity

A function is concave up where f'' is positive and concave down where f'' is negative. Inflection points occur where concavity changes sign.

Mean Value Theorem

If f is continuous on [a,b] and differentiable inside, some interior point c has f'(c) equal to the average slope [f(b) − f(a)]/(b − a).

Implicit differentiation

Differentiate both sides of an equation with respect to x, treating y as a function of x and applying the chain rule, then solve algebraically for dy/dx.

Related rates procedure

Write an equation relating the quantities, differentiate both sides with respect to time t, then substitute the given instantaneous values to solve for the unknown rate.

Linear approximation

Near x = a, f(x) is approximated by f(a) + f'(a)(x − a). This is the tangent-line approximation, accurate when x is close to a.

Antiderivative power rule

The integral of x^n dx is x^(n+1)/(n+1) plus C for n not equal to negative one. When n equals negative one, the result is ln|x| plus C.

Fundamental Theorem of Calculus

If F is an antiderivative of continuous f, the definite integral of f from a to b equals F(b) − F(a), linking area accumulation to antiderivatives.

Accumulation function derivative

The derivative with respect to x of the integral from a to x of f(t) dt equals f(x). With an upper limit g(x), multiply by g'(x).

Integration by substitution

Reverse the chain rule: let u equal the inner function, replace dx with du divided by du/dx, integrate in u, then back-substitute. For definite integrals, also convert the limits.

Definite integral as signed area

The definite integral counts area above the x-axis as positive and below as negative. Total geometric area requires integrating absolute value or splitting at zeros.

Area between curves

For curves where top(x) is above bottom(x) on [a,b], the enclosed area equals the integral from a to b of top(x) minus bottom(x) dx.

Disk method volume

Revolving a region about the x-axis gives volume equal to pi times the integral from a to b of the square of the radius function f(x) dx.

Riemann sum

A sum of f evaluated at sample points times subinterval width approximates a definite integral. Its limit as widths shrink to zero defines the integral exactly.

Integral of e to the x

The integral of e^x dx is e^x plus C, and the integral of 1/x dx is ln|x| plus C. The absolute value is needed because 1/x is defined for negative x.