AP Calculus BC

Squeeze Theorem

If h(x) ≤ ƒ(x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and lim x→c h(x) = lim x→c g(x) = L, then lim x→c ƒ(x) exists and equals L.

Continuity

A function ƒ is continuous at c if:

a. ƒ(c) is defined

b. lim x→c ƒ(x) exists

c. lim x→c ƒ(x) = ƒ(c)

Intermediate Value Theorem

If ƒ is continuous on an interval [a,b] and k is any number between ƒ(a) and ƒ(b), there is at least one number c in [a,b] such that ƒ(c) = k.

Definition of the derivative

ƒ'(x) = lim h→0 (ƒ(x+h) - ƒ(x))/h

Power rule

d/dx xⁿ = nx^(n-1)

Product rule

d/dx ƒ(x)g(x) = ƒ'(x)g(x) + ƒ(x)g'(x)

Quotient rule

d/dx ƒ(x)/g(x) = (g(x)ƒ'(x) - ƒ(x)g'(x))/g²(x)

Chain rule

d/dx ƒ(g(x)) = ƒ'(g(x))g'(x)

Implicit differentiation

d/dx ƒ(y) = ƒ'(y)(dy/dx)

Definition of a minimum

ƒ(c) is the minimum of ƒ on an interval if ƒ(c) ≤ f(x) for all x contained in the interval.

Definition of a maximum

ƒ(c) is the maximum of ƒ on an interval if ƒ(c) ≥ f(x) for all x contained in the interval.

Extreme Value Theorem

If ƒ is continuous on a closed interval, then ƒ has both a minimum and maximum on the interval.

Critical number

A number c is called critical if ƒ'(c) = 0 or ƒ is not differentiable at c.

Points tested for minima/maxima

Critical points and enpoints.

Local minimum/maximum

A point where ƒ' changes from positive to negative or vice versa.

Inflection point

A point where ƒ" changes from positive to negative or vice versa.

Antiderivative

A function F(x) that satisfies F'(x) = ƒ(x)

Definition of the indefinite integral

The collection of all antiderivatives of a function, e.g. ∫ƒ(x)dx = F(x) + C, since all functions of the form F(x) + C satisfy (F(x) + C)' = ƒ(x).

Differential equation

An equation involving the derivative(s) of a function.

Definition of the definite integral

∑ƒ(ci)∆xi, as ∆x→0

FUNDAMENTAL THEOREM OF CALCULUS TURN UP DRANK FADED MILLI BILLI BEAN

∫ƒ(x)dx from a to b = F(b) - F(a), where F is an antiderivative of ƒ

Mean Value Theorem / Average value of a function

∫ƒ(x)dx from a to b = ƒ(c)(b-a)

or

(∫ƒ(x)dx)/(b-a) = ƒ(c)

Second part of the Fundamental Theorem of Calculus

d/dx (∫ƒ(t)dt from a to x) = f(x)

U-substitution

∫F'(g(x))g'(x)dx = F(u) + C = F(g(x)) + C

Change of variables for definite integrals

∫ƒ(g(x))g'(x)dx from a to b = ∫ƒ(u)du from u(a) to u(b)

Disk method

V = π∫r(x)²dr

Washer method

V = π∫(R(x)² - r(x)²)dr

Integration by parts

∫udv = uv - ∫vdu

L'Hopital's rule

lim x→c ƒ(x)/g(x) = ƒ'(x)/g'(x), given that f(x)/g(x) is indeterminate at c

Improper Integrals

∫ƒ(x)dx from a to INF = lim c→INF of ∫ƒ(x)dx from a to c

Monotonic sequences

A sequence is monotonic if it's terms are either nondecreasing or nonincreasing.

Bounded Monotonic Sequences

If a sequence is bounded and monotonic, then it converges.

Convergent series

An infinite series is convergent if the sequence of partial sums is convergent.

Sum of a geometric series

∑arⁿ = a + ar + ar² + ... = a/(1-r)

Nth term test

If lim n→∞ aⁿ ≠ 0, then the infinite series ∑aⁿ diverges.

Integral test

If ƒ is positive, continuous, and decreasing for x ≥ 1 and aⁿ = ƒ(n), then the series ∑aⁿ and the improper integral of ∫ƒ either both converge or both diverge.

P-series

The series ∑n^(-p) converges if p > 1, and diverges if p ≤ 1.

Direct Comparison test

Let the sequences a and b be defined so 0 < a ≤ b.

1. If ∑bⁿ converges, then ∑aⁿ converges.

2. If ∑aⁿ diverges, then ∑bⁿ diverges.

Limit Comparison test

Let sequences aⁿ > 0 and bⁿ > 0. If lim n→∞ a/b = L, where L is finite and positive, then the two sequences either both converge or both diverge.

Alternating Series test

An alternating series ∑(-1)ⁿaⁿ converges if lim n→∞ aⁿ = 0 and aⁿ⁺¹ ≤ aⁿ for all n.

Alternating Series remainder

For a convergent alternating series, the absolute value of the remainder in approximating the sum with the first n partial sums is less than or equal to the value of the first neglected term.

Absolute convergence

If the series ∑|aⁿ| converges, then ∑aⁿ converges.

Conditional convergence

A series is conditionally convergent is ∑aⁿ converges but ∑|aⁿ| diverges.

Ratio test

∑aⁿ converges absolutely is lim n→∞ |aⁿ⁺¹/aⁿ| < 1, diverges if this limit is > 1, and is inconclusive if the limit = 1.

Root test

∑aⁿ converges absolutely if lim n→∞ nth root(|aⁿ|) < 1, diverges if the limit is > 1, and is inconclusive if the limit = 1.

Taylor series

ƒ(x) = ∑ƒⁿ(c)(x-c)ⁿ/n! + R(x), where ƒⁿ(c) is the nth derivative of ƒ at c.

Lagrange error bound

R(x) = ƒⁿ⁺¹(z)(x-c)ⁿ⁺¹/(n+1)!, where ƒⁿ⁺¹(z) is the maximum value of the (n+1)th derivative of ƒ.

Power series

Any series of the form ∑a(x-c)ⁿ. This series is said to be centered at c.

Finding the radius of convergence of a power series

Perform a ratio test on the terms of the series as the term number approaches infinity. Set the value that you end up with as < 1, and solve.

Finding the interval of convergence of a power series

Find the radius of convergence, then manipulate this, and test the endpoints.

Parametric form of the derivative

dy/dx = (dy/dt)/(dx/dt) = y'(t)/x'(t)

Parametric form of the second derivative

d/dt (dy/dx) / (dx/dt)

Parametric arc length

L = ∫sqrt((x'(t))² + ((y'(t))²)dt from t = a from t = b

Parametric speed

speed = sqrt((x'(t))² + ((y'(t))²)

Polar coordinate conversion

r = ƒ(θ)

x = r cos(θ)

y = r sin(θ)

tan(θ) = y/x

r² = x² + y²

Polar area

A = ½∫ƒ(θ)²dθ

Polar arc length

L = ∫sqrt((ƒ(θ)² + (ƒ'(θ))²)dθ from θ₁ to θ₂