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AP Calculus BC Terms

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Section 1

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Harmonic series

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1 / 43

Calculus Math

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Mar 1, 2020

Cards (43)

Section 1

(43 cards)

Harmonic series

Front

∑ 1/n, diverges

Back

Direct comparison test

Front

0<aₙ≤bₙ AND bₙ converges is convergence 0<bₙ≤aₙ AND bₙ diverges is divergence *inconclusive if conditions not met

Back

Power Series for lnx

Front

(x-1)-(x-1)²/2+(x-1)³/3-(x-1)⁴/4+-...∑((-1)ⁿ-1(x-1)ⁿ)/n

Back

Euler's Method

Front

Series of linear approximations to find differential solutions. x(n)= x(n-1) + h y(n)=y(n-1) + hF(x(n-1), y(n-1))

Back

N-th term test

Front

limₙ→∞ ∑aₙ ≠ 0 is absolute divergence *CANNOT show convergence

Back

Mean Value Theorem for Integrals

Front

If f is continuous on [a, b], then there exists number c such that on any curve, somewhere between the inscribed and circumscribed rectangle, there exists another rectangle that exactly represents the region under the curve. b ∫ a f(x)dx = f(c)(b-a)

Back

Mean Value Theorem

Front

If f is continuous on [a,b] and differentiable on (a,b), then ∃c∈(a,b) such that f'(c) = [f(b)-f(a)]/(b-a)

Back

Power Series for arctan x

Front

x-x^3/3+x^5/5-x^7/7+-...

Back

Growth and Decay

Front

y=Ce^(kt) C=initial value t=time

Back

lim(x→0) (1-cosx)/x

Front

0

Back

Second Fundamental Theorem of Calculus

Front

Relationship between a derivative and its antiderivative. d/dx [b ∫ a f(x)dx]= f(t)

Back

Riemann Sum

Front

A way of approximating area under a curve by finding the area of respective rectangles

Back

Solids with Known Cross Sections

Front

V=∫A(x)dx A(x)= area of cross sections

Back

Power Series for cos x

Front

Back

Power series for 1/x

Front

1-(x-1)+(x-1)²-(x-1)³+(x-1)⁴-+....∑(-1)ⁿ(x-1)ⁿ

Back

Volume: Washer Method

Front

V=π∫[R(x)]^2 - [r(x)]^2 dx solid with a hole

Back

Logistic Differential Equation

Front

dy/dt = Ky(1-y/L) K=postive constant L=postive constant

Back

Limit is continuous if...

Front

...f(c) is defined, limit exists and lim(x→c)f(x)=f(c)

Back

Improper Integrals

Front

integrals with discontinuities/undefined points; take the limit of the integral

Back

Trapezoidal Rule

Front

b ∫ a f(x)dx ≈ (b-a)/2n [f(a) +2f(x₁) +2f(x₂)+ ...+f(b)]

Back

L'Hopital's Rule

Front

If lim x→a f(x)/g(x) is of an indeterminate form, then lim x→a f'(x)/g'(x) exists, then lim x→a f(x)/g(x) = lim x→a f'(x)/g'(x).

Back

Volume: Disk Method

Front

V=π∫[R(x)]^2 dx rectangles perpendicular of axis of rotation

Back

Taylor polynomial approximation

Front

f(c)+f'(c)(x-c)+f''(c)/2!(x-c)^2+f'''(c)/3!(x-c)^3....

Back

Difference Quotient

Front

lim(h→0) (f(x+h)-f(x))/h, h≠0

Back

Intermediate Value Theorem

Front

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

Back

Integral Test

Front

∫∞,n f(x) converges/diverges

Back

Area between Two Curves

Front

∫ f(x) - g(x) over interval a to b, where f(x) is top function and g(x) is bottom function

Back

Partial Fractions

Front

a/b +/- c/d = (ad+/-bc)/bd

Back

Alternating Series ∑(-1)ⁿaₙ

Front

0<a(n+1)≤aₙ AND limₙ→∞aₙ=0 is convergence

Back

Ratio test

Front

limₙ→∞|aₙ+1/aₙ| <1 is convergence limₙ→∞|aₙ+1/aₙ| >1 is divergence *inconclusive if =1

Back

Geometric Series ∑aₙⁿ

Front

|ⁿ|≥1 is divergence |ⁿ|<1 is convergence Sₙ=(starting value)/1-ⁿ

Back

Power Series for sin x

Front

Back

Fundamental Theorem of Calculus

Front

∫f(x)dx = F(b) - F(a)

Back

Rolle's Theorem

Front

If the function f(x) is continuous on [a,b], AND the first derivative exists on the interval (a,b), AND f(a)=f(b), then there is at least one number x=c in (a,b) such that f'(c)=0

Back

Power series for e^x

Front

1+x-x²/2+x³/3!+x⁴/4!+...∑xⁿ/n!

Back

Root Test ∑(aₙ)ⁿ

Front

limₙ→∞ n√(|aₙ|) <1 is convergence limₙ→∞ n√(|aₙ|) >1 is divergence *inconclusive if =1

Back

Arc Length

Front

s= ∫√(1+[f'(x)]^2) dx

Back

P-series ∑1/pⁿ

Front

ⁿ>1 is convergence ⁿ≤1 is divergence

Back

lim(x→0) sinx/x

Front

1

Back

Power series for 1 /(1−x)

Front

1+x+x²+x³+x⁴+x⁵...∑xⁿ

Back

Integration by parts

Front

∫udv=uv-∫vdu

Back

Volume: Shell Method

Front

V=2π∫p(x)h(x)dx rectangles parallel to axis and has hole

Back

Limit comparison test

Front

limₙ→∞(aₙ/bₙ) >0 AND bₙ converges is convergence limₙ→∞(aₙ/bₙ) <0 AND bₙ diverges is divergence

Back