AP Calculus BC Formulas and Terms

nth Term Test

applies to infinite series of real numbers

Geometric Series Test

Use if the ratio between each two consecutive terms of a series is constant

Sum of Covergant geometric series

a/1-r

P-series Test

Alternating series test

Integral Test

Only use if the function is continuous, positive, and decreasing.

Root Test

Ratio Test

Mostly used with X!

Direct Comparison Test

Limit Comparison Test

l’Hopital’s Rule

If f(x)/g(x) = 0/0 or ∞/∞ when evaluating a limit, you can apply L'Hôpital's Rule to find the limit by taking the derivative of the numerator and denominator separately.

ln 1

0

ln e^a

a

d/dx xⁿ

nx^n-1

derivative of sinx

cosx

derivative of cosx

-sinx

derivative of tanx

sec² x

derivative of cotx

-csc² x

derivative of secx

secxtanx

derivative of cscx

-cscxcotx

derivative of lnu

1/u du/dx

derivative of e^u

e^udu/dx

derivative of log_ax

1/xlna

d/dx a^x

a^xlna

d/dx f(u)

f’(u)du/dx

d/dx uv

udv/dx + vdu/dx

d/dx u/v

lodehi-hidelo/lo²

Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists a point in the interval where the derivative equals the average rate of change of the function over that interval. (Used for derivatives)

Rolle's Theorem

Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, and the function values at the endpoints are equal, then there exists at least one point in the interval where the derivative is zero.

Local Minimum

dy/dx goes (-,0,+)

Local Maximum

dy/dx goes (+,0,-)

Critical Point

dy/dx= 0 or und

Point of Inflection

2nd derivative goes (+,0,-),(-,0,+), (+,und,-), or (-,und,+)

Integral from a to b f(x)

f(b) - f(a)

Euler’s Method

Euler's Method is a numerical technique for solving ordinary differential equations. It approximates the solution by advancing in small steps based on the derivative at each point.

Logistics Curve

Integral of uⁿ

Integral of a^u

Integration by Parts

LIPET

ln, inverse trig, polynomial, exponential, trig

Arc length Formula (rectangular)

Arc length Formula (polar graph)

Lagrange Error Bound

x= remainder error

c= center

n= degree polynomial

fⁿ⁺¹(z)= some z between a and x (the biggest one)

Velocity Vector

Speed

Distance Traveled

Area of a Polar Curve

Slope of rθ

∑1/1-x

1 + x + x² + x³ + … xⁿ

∑1/1+x

1 - x + x² - x³ + … (-xⁿ)

∑e^x

1 + x + x²/2! … + xⁿ/n!

∑sinx

x - x³/3! +x⁵/5! + …. (-1)ⁿx²ⁿ⁺¹/(2n+1)!

∑cosx

1 - x²/2! + x⁴/4! +… (-1)ⁿx²ⁿ/(2n)!

∑ln(1+x)

x - x²/2 + x³/3 …. + (-1)^n-1xⁿ/n

tan^-1x

x - x³/3 + x⁵/5 + … (-1)ⁿx²ⁿ⁺¹/2n+1

How to find the first derivative formula

Intermediate Value Theorum

States that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between and at some point within the interval.

Population Density Formula

integral f(x) x population density