AP Calclus BC review

Derivative of sin(u)

d/dx(sin u) = cos(u) * u'

Derivative of cos(u)

d/dx(cos u) = -sin(u) * u'

Derivative of tan(u)

d/dx(tan u) = sec²(u) * u'

Derivative of sec(u)

d/dx(sec u) = sec(u)tan(u) * u'

Derivative of csc(u)

d/dx(csc u) = -csc(u)cot(u) * u'

Derivative of cot(u)

d/dx(cot u) = -csc²(u) * u'

Derivative of ln(u)

d/dx(ln u) = (1/u) * u'

Derivative of e^u

d/dx(e^u) = e^u * u'

Product Rule

d/dx(uv) = uv' + vu'

Quotient Rule

d/dx(u/v) = (vu' - uv') / v²

Mean Value Theorem (MVT) Requirements

f(x) is continuous on [a, b] and differentiable on (a, b).

Mean Value Theorem (MVT) Formula

There exists a 'c' in (a, b) such that f'(c) = [f(b) - f(a)] / (b - a).

Extreme Value Theorem (EVT)

If f(x) is continuous on a closed interval [a, b], then f(x) must have both an absolute maximum and an absolute minimum on that interval.

Critical Point

f'(x) = 0 or undefined.

Point of Inflection

f''(x) = 0 or undefined AND concavity changes.

Integration by Parts Formula

∫ u dv = uv - ∫ v du (Use LIATE to choose u).

Taylor Series Expansion

f(x) ≈ f(c) + f'(c)(x-c) + f''(c)/2! * (x-c)² + ... + fⁿ(c)/n! * (x-c)ⁿ

Ratio Test for Convergence

Series converges if lim |aₙ₊₁ / aₙ| < 1 as n → ∞. If = 1, test is inconclusive.

Maclaurin Series for e^x

e^x = 1 + x + x²/2! + x³/3! + ...

Maclaurin Series for sin(x)

sin(x) = x - x³/3! + x⁵/5! - ... (odd powers)

Maclaurin Series for cos(x)

cos(x) = 1 - x²/2! + x⁴/4! - ... (even powers)

Area inside a Polar Curve

A = ½ ∫ [r(θ)]² dθ from α to β.

Slope of a Parametric Equation (dy/dx)

dy/dx = (dy/dt) / (dx/dt)

What is Part 1 of the Fundamental Theorem of Calculus?

d/dx [∫ (from a to x) f(t) dt] = f(x)

What is Part 2 of the Fundamental Theorem of Calculus?

∫ (from a to b) f(x) dx = F(b) - F(a)

What is Euler's Method formula?

y_new = y_old + Δx * f'(x_old, y_old)

What is the logistic growth differential equation?

dP/dt = kP(1 - P/L)

What is the carrying capacity in logistic growth?

L (The limit as t → ∞ is L)

What is the arc length formula for a function?

∫ √(1 + [f'(x)]²) dx

What is the arc length formula for parametric equations?

∫ √([x'(t)]² + [y'(t)]²) dt

When does a p-series converge?

Σ 1/n^p converges if p > 1, diverges if p ≤ 1.

What is the sum formula for a geometric series?

Sum = a / (1 - r), converges only if |r| < 1.

What does the n-th term test for divergence state?

If lim (as n → ∞) a_n ≠ 0, then the series diverges.

What are the conditions for the Alternating Series Test (AST)?

1. b_n is positive, 2. b_n is decreasing, 3. lim (as n → ∞) b_n = 0.

What is the Lagrange Error Bound?

Error |R_n(x)| ≤ [max|f^(n+1)(z)| / (n+1)!] * |x - c|^(n+1)

What are the conditions for L'Hospital's Rule?

Use when lim f(x)/g(x) results in 0/0 or ±∞/∞. Must state f and g are differentiable and show individual limits approach 0 or ±∞.

What is the limit definition of a derivative?

f'(x) = lim (as h → 0) [f(x+h) - f(x)] / h

How is the average value of a function calculated?

f_avg = [1 / (b - a)] * ∫ (from a to b) f(x) dx

What does the Intermediate Value Theorem state?

If f is continuous on [a, b], then for any value y between f(a) and f(b), there exists at least one c in [a, b] such that f(c) = y.

What is the formula for velocity in parametric motion?

v(t) =

What is the formula for acceleration in parametric motion?

a(t) =

How is speed defined in terms of velocity?

|v(t)| = √([x'(t)]² + [y'(t)]²)

How is total distance traveled calculated in parametric motion?

Distance = ∫ (from a to b) √([x'(t)]² + [y'(t)]²) dt

What is the formula for the volume using the Disk Method?

V = π ∫ [R(x)]² dx

What is the formula for the volume using the Washer Method?

V = π ∫ ([R(x)]² - [r(x)]²) dx

How is volume by cross-sections perpendicular to the x-axis calculated?

V = ∫ (from a to b) A(x) dx (A = area of the specific shape, e.g., s² for squares).

What is the integral of 1 / (a² + u²)?

∫ 1 / (a² + u²) du = (1/a) arctan(u/a) + C

What defines absolute convergence?

Σ |a_n| converges.

What defines conditional convergence?

Σ a_n converges, but Σ |a_n| diverges.

What is the approach for improper integrals?

Rewrite with limits! ∫ (from a to ∞) f(x) dx = lim (as b → ∞) ∫ (from a to b) f(x) dx.

Integration of 1 / (a² + u²)

∫ 1 / (a² + u²) du = (1/a) arctan(u/a) + C

Absolute vs. Conditional Convergence

Absolute: Σ |a_n| converges. Conditional: Σ a_n converges, but Σ |a_n| diverges.

Improper Integrals

Rewrite with limits! ∫ (from a to ∞) f(x) dx = lim (as b → ∞) ∫ (from a to b) f(x) dx.

Inverse Trig Derivatives (arcsin and arctan)

d/dx(arcsin u) = u' / √(1 - u²) and d/dx(arctan u) = u' / (1 + u²)

Alternating Series Error Bound

|Error| ≤ |a_{n+1}| (The absolute value of the first omitted term).

Geometric Power Series (1 / (1 - x))

1 / (1 - x) = 1 + x + x² + x³ + ... = Σ xⁿ for |x| < 1.

Speeding Up vs. Slowing Down

Speeding Up: v(t) and a(t) have the SAME sign. Slowing Down: v(t) and a(t) have DIFFERENT signs.

Volume of a Sphere

V = (4/3)πr³

Volume of a Cone

V = (1/3)πr²h