AP Calculus BC Review

pls get a 5

nth - Term Test

Series will diverge when the limit of the terms does not approach zero as n approaches infinity.

A geometric series will converge when…

0 < absolute value of r < 1

A geometric series will diverge when…

the absolute value of r is greater than or equal to 1.

How do you find the sum of a geometric series?

S = a / (1 - r)

Alternating Series Test

An alternating series converges if the absolute value of |a_n+1| is less than |a_n| term AND the limit of a_n equals 0 as n approaches infinity.

What criteria must be met to use the integral test on a series?

The integral test can be applied if the function is continuous, positive, and decreasing on the interval being considered.

When will a p-series converge?

A p-series converges if the p-value is greater than 1.

When will a p-series diverge?

A p-series diverges if the p-value is less than or equal to 1.

How do you use the direct comparison test?

The direct comparison test involves comparing a given series to a known benchmark series to determine convergence or divergence.

When using the limit comparison test and we are given a series a_n, how do we find b_n?

Choose a positive series—using the direct comparison test—that resembles a_n

When will a series converge using the Limit Comparison Test?

If the limit as n approaches infinity of the ratio of a_n to b_n is positive and finite AND b_n converges.

When will a series diverge using the Limit Comparison Test?

If the limit as n approaches infinity of the ratio of a_n to b_n is positive and finite AND b_n diverges.

In regards to Absolute Convergence, the series a_n will absolutely converge if…

the series |a_n| converges.

How do we test for conditional convergence?

The series a_n converges but the series |a_n| diverges.

In regards to the Ratio Test, when will a series converge?

The series converges if the limit of the ratio of consecutive terms, |a_n+1/a_n|, is less than 1 as n approaches infinity.

In regards to the Ratio Test, when will a series diverge?

The series diverges if the limit of the ratio of consecutive terms, |a_n+1/a_n|, is greater than 1 or undefined as n approaches infinity.

When will the Ratio Test be inconclusive?

The Ratio Test is inconclusive if the limit of the ratio of consecutive terms, |a_n+1/a_n|, equals 1 as n approaches infinity.

When will a Power Series converge for only at c?

A power series will only converge at c IF lim_n→∞ |a_n+1/a_n| = 0

When will a power series converge absolutely for all x?

A power series will only converge at c IF lim_n→∞ |a_n+1/a_n| = infinity

What is the Error Bound for an Alternating Series?

|S - S_n| < |a_n+1|

What is the equation to find the total distance traveled of a particle given x(t) and y(t)?

Integral of sqrt( x’(t)² + y’(t)² )dt over the interval [a, b].

How do you find the position vector?

( x(t), y(t) )

How do you find delta x when using Euler’s method?

(x₂ - x₁) / n, where n is the amount of steps.

How do you find delta y when using Euler’s method?

(dy/dx) times delta x.

In regards to Euler’s method, how do you find the new coordinate after each step?

(x + delta x, y + delta y)

What is the Intermediate Value Theorem and what criteria must be met for it to apply?

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on values f(a) and f(b), then there must exist at least one value c in the interval (a, b) such that f(c) equals any value between f(a) and f(b).

What is the Mean Value Theorem and what criteria must be met for it to apply?

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that the derivative at that point is equal to the average rate of change of the function over the interval.

What is the Extreme Value Theorem and what criteria must be met for it to apply?

The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], then it attains both a maximum and a minimum value at least once within that interval.

What is Rolle’s Theorem and what criteria must be met for it to apply?

Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) where the derivative f'(c) equals zero.

Given a function f, how do you find its arc length from the interval (a, b)?

The integral of the square root of 1 plus the derivative squared, from a to b.

Given a parametric equation, how do you find its arc length from the interval (a, b)?

The arc length is calculated using the integral of the square root of the sum of the squares of the derivatives of the parametric equations, from a to b.

Particle position vector

r(t) = ( x(t), y(t) )

Magnitude of Position Vector

L = sqrt( x(t)² + y(t)² )

Particle Velocity Vector

( x’(t), y’(t) )

Speed of Velocity Vector

sqrt( (dx/dt)², (dy/dt)² )

Acceleration Vector

(x''(t), y''(t))

Displacement of a particle vector from t = a to t = b

( integral( x’(t) ) from a to b, integral( y’(t) ) from a to b)

Total distance traveled by position vector

Integral of the speed of the velocity vector from a to b.

Arc length of a smooth curve from [a,b]

L = integral( sqrt( 1 + f’(x)²) ) from a to b

dy/dx of a parametric equation

(dy/dt) / (dx/dt)

Second derivative of a parametric equation

[ (d/dt)(dy/dx) ]/(dx/dt)

Arc length in parametric form

Same as total distance traveled by position vector