AP CALC AB

Estimate of the rate of change

The formula to estimate the rate of change is (b(32) - b(31.999)) / (32 - 31.999).

Squeeze theorem

If g(x) ≤ f(x) ≤ h(x) and lim x→a g(x) = L and lim x→a h(x) = L, then lim x→a f(x) = L.

Types of discontinuities - Hole

A removable discontinuity found when factors in both the numerator and denominator cancel out.

Types of discontinuities - VA (Vertical Asymptote)

A non-removable discontinuity found when the denominator equals 0 and cannot be canceled out.

Types of discontinuities - Jump

A non-removable discontinuity that occurs when a function jumps from one value to another.

Types of discontinuities - Oscillate

A non-removable discontinuity where the function oscillates between two values.

Continuous at x = c

For f(x) to be continuous at x = c, f(c) must be defined, the limit as x approaches c must exist, and f(c) must equal lim x→c f(x).

Limit of infinity theorem

Both lim x→∞ (1/x^r) and lim x→-∞ (1/x^r) equal 0, where r is a rational number.

Average rate of change (secant slope)

Calculated as (f(a+h) - f(a)) / ((a+h) - a).

Instantaneous rate of change (tangent slope)

Defined as lim h→0 (f(a+h) - f(a)) / h.

Equation of tangent line

Given by y - y1 = m(x - x1).

Power rule

The formula for differentiation where d/dx of x^n = nx^(n-1).

Chain rule

For composition of functions, dy/dx = dy/dg * dg/dx.

Implicit differentiation

The process where d/dx of y^2 = 2y(dy/dx).

First derivative test

Determines relative minima or maxima based on sign changes of the first derivative.

Candidates for absolute extrema

End points of the domain and critical points where d/dx does not exist or d/dx = 0.

Second derivative test

If d/dx(c) = 0 and f''(c) > 0, then there is a relative minimum at x = c; if f''(c) < 0, then there is a relative maximum.

Point of inflection

Occurs when the second derivative changes sign, indicating a change in concavity.

Application of L'Hôpital's rule

Lim x→a of f(x)/g(x) equals 0/0, take derivatives of both numerator and denominator.

Motion of acceleration

If a(t) > 0, the object accelerates right or up; if a(t) < 0, it accelerates left or down; if a(t) = 0, it's constant velocity.

Relative extrema

Points where the function has local maximums or minimums which occur at critical points.

Log properties

ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), ln(a^n) = n*ln(a).