Calculus Review Flashcards

Flashcards covering key concepts from calculus lecture notes, including trigonometric functions, properties of logarithms, limits, derivatives, integrals, theorems, differential equations, growth rates, series, Taylor series, and applications of integration.

lim (x->0) sin(x)/x = ?

sin(x)/x approaches 1 as x approaches 0

lim (h->0) (cos(h)-1)/h = ?

cos(h)-1 all over h approaches 0 as h approaches 0

lim (h->0) (e^h - 1) / h = ?

(e^h - 1) / h approaches 1 as h approaches 0.

d/dx sin(x) = ?

cos(x)

d/dx cos(x) = ?

-sin(x)

d/dx tan(x) = ?

sec^2(x)

d/dx cot(x) = ?

-csc^2(x)

d/dx sec(x) = ?

sec(x)tan(x)

d/dx csc(x) = ?

-csc(x)cot(x)

d/dx arcsin(x) = ?

1 / sqrt(1 - x^2)

d/dx arccos(x) = ?

-1 / sqrt(1 - x^2)

d/dx arctan(x) = ?

1 / (x^2 + 1)

d/dx arccot(x) = ?

-1 / (x^2 + 1)

d/dx arcsec(x) = ?

1 / (|x| * sqrt(x^2 - 1))

d/dx arccsc(x) = ?

-1 / (|x| * sqrt(x^2 - 1))

d/dx e^x = ?

e^x

d/dx a^x = ?

a^x * ln(a)

d/dx ln(x) = ?

1/x

d/dx log_b(x) = ?

1 / (x * ln(b))

Power Rule

n*x^(n-1)

Product Rule

u'v + uv'

Quotient Rule

(u'v - uv') / v^2

Chain Rule

f'(g(x)) * g'(x)

Integral of 1/(a^2 + x^2) dx = ?

(1/a)Arctan(x/a) + C

Integral of 1/sqrt(a^2 - x^2) dx = ?

Arcsin(x/a) + C

Integral of 1/(x*sqrt(x^2 - a^2)) dx = ?

(1/a)Arcsec(|x|/a) + C

Mean Value Theorem

f'(c) = (f(b) - f(a)) / (b - a)

Fundamental Theorem of Calculus Part 2

Integral from a to b of f(x)dx = F(b) - F(a)

dy/dt = ky

y = Ae^(kt)

Integration by Parts

∫udv = uv - ∫vdu

Logistic Differential Equation

P = M / (1 + Ae^(-kt))

Parametric Arc Length

Total distance traveled, integral of speed

Polar Equations

x = rcos(θ), y = rsin(θ)

Slope of the Curve in Polar

dy/dx = (dy/dθ) / (dx/dθ)

p-series

Converges if p > 1, diverges if p ≤ 1

nth Term Test for Divergence

If lim (n->inf) a_n != 0, then the series diverges

Taylor Series

f(a) + f'(a)(x-a) + f''(a)/2! (x-a)^2 + …

Disk Method

π∫[R(x)]^2 dx

Washer Method

π∫([R(x)]^2 - [r(x)]^2)dx

Shell Method

2π∫r(x)H(x)dx

Volumes of Known Cross Sections

∫ A(x)dx