AP Calculus BC Review Flashcards

Flashcards covering key Calculus BC concepts and theorems.

What is the Taylor Series generated by f(x) at x = 0 also known as?

Maclaurin Series

What is the formula for the Taylor series generated by f(x) at x = 0?

f(x) = Σ [f^(k)(0) / k!] * x^k from k=0 to infinity

What is the formula for the Taylor series generated by f(x) at x = a?

f(x) = Σ [f^(k)(a) / k!] * (x-a)^k from k=0 to infinity

What is the Maclaurin series for 1/(1-x)?

Σ x^n from n=0 to infinity, valid for -1 < x < 1

What is the Maclaurin series for e^x?

Σ x^n / n! from n=0 to infinity, valid for all real x

What is the Maclaurin series for sin(x)?

Σ (-1)^n * [x^(2n+1) / (2n+1)!] from n=0 to infinity, valid for all real x

What is the Maclaurin series for cos(x)?

Σ (-1)^n * [x^(2n) / (2n)!] from n=0 to infinity, valid for all real x

How is error maximized in an alternating series?

The error is maximized in the next unused term evaluated at the difference between the center of the interval of convergence and the x-coordinate being evaluated.

What is the formula tied to the next unused term when using Lagrange Error Bound?

Error < [f^(n+1)(z) / (n+1)!] * (x-a)^(n+1)

What is the formula for Euler's Method?

xnew = xold + Δx, ynew = yold + dy/dx * Δx = y_old + f(x, y) * Δx

What is the general solution for the exponential growth and decay model dy/dt = ky?

y = Ce^(kt), where C is the initial quantity

What is the differential equation that represents logistic growth where the rate of change is jointly proportional to the size of the quantity and the difference between the quantity and the carrying capacity M?

dy/dt = ky(M-y)

Given parametric equations x(t) and y(t), how do you find dy/dx?

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

Given parametric equations x(t) and y(t), how do you find d²y/dx²?

d²y/dx² = (d/dt (dy/dx))/(dx/dt)

How do you calculate arc length for a parametric function?

Arc length = ∫√((dx/dt)² + (dy/dt)²) dt

What are the relationships between rectangular and polar coordinates?

x = r cos θ, y = r sin θ

How do you find dy/dx in polar coordinates?

dy/dx = (dy/dθ) / (dx/dθ)

What is the formula for the area inside a polar curve?

Area = ∫ (1/2) * [r(θ)]² dθ

What must be true for a Geometric Series to converge?

|r| < 1

When does the Alternating Series Test converge?

If lim an = 0 and an is decreasing.

When does the Integral Test converge?

If ∫f(x) dx converges, where f(x) is continuous, positive, and decreasing.

When does a p-series converge?

When p > 1.

Using the Direct Comparison Test, when does Σa_n converge?

If 0 < an < bn and Σb_n converges.

Using the Limit Comparison Test, when does Σa_n converge?

If lim (an / bn) = L > 0 and Σb_n converges.

What does absolute convergence imply?

Convergence

When can't the ratio test be used?

If lim |a(n+1)/an| = 1

What is the formula for point-slope form?

y - y₁ = m(x - x₁)

State the 3 conditions for continuity at a point c.

1. f(c) exists, 2. lim x->c f(x) exists, 3. lim x->c f(x) = f(c)

What is the average rate of change on the interval [a, b]?

(f(b) - f(a)) / (b - a)

What is the definition of the derivative, f'(x)?

f'(x) = lim h->0 (f(x+h) - f(x)) / h

What is the equation of the tangent line to f(x) at x=a?

y - f(a) = f'(a)(x - a)

State the chain rule.

d/dx [f(g(x))] = f'(g(x)) * g'(x)

State the product rule.

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

State the quotient rule.

d/dx (u/v) = (vu' - uv') / v²

How do you find a critical value?

Set f'(x) = 0 or undefined.

How do you justify that y is increasing?

Establish that y' > 0

How do you justify that y has a local minimum?

Establish that y' changes from - to +

How do you justify that y is concave up?

Establish that y'' > 0

State the Intermediate Value Theorem.

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

State the Extreme Value Theorem.

If f(x) is continuous on [a, b], then f has both an absolute maximum and an absolute minimum value in that interval.

State the Mean Value Theorem.

If f(x) is continuous on [a, b] AND differentiable on (a, b), then there exists at least one value x = c in (a, b) where f'(c) = (f(b)-f(a))/(b-a).

State Rolle's Theorem.

If f(x) is continuous on [a, b] AND differentiable on (a, b) AND f(a) = f(b), then there exists at least one value x = c in (a, b) where f'(c) = 0.

State L'Hopital's Rule.

If lim x->a f(x)/g(x) = 0/0 or ∞/∞, then lim x->a f(x)/g(x) = lim x->a f'(x)/g'(x).