CALC BC MEMORIZATION!!

i'm cooked

critical point

dy/dx = 0 or undefined

point of inflection

d²y/d²x changes sign, concavity changes

chain rule

d/dx[f(u)] = f’(u)(du/dx)

product rule

(derivative of first × second) + (first × derivative of second)

quotient rule

((derivative of top × bottom) - (top × derivative of bottom)) / (bottom²)

power rule

d/dx[xⁿ] = nxⁿ⁻¹

d/dx(sinx)

cosx

d/dx(cosx)

-sinx

d/dx(tanx)

sec²x

d/dx(cotx)

-csc²x

d/dx(secx)

secx tanx

d/dx(cscx)

-cscx cotx

d/dx(lnx)

1/x

d/dx(e^x)

e^x

d/dx(sin⁻¹x)

1/√1-x²

d/dx(cos⁻¹x)

-1/√1-x²

d/dx(tan⁻¹x)

1/(1+x²)

d/dx(csc⁻¹x)

-1/(|x|√x²-1)

d/dx(sec⁻¹x)

1/(|x|√x²-1)

d/dx(cot⁻¹x)

-1/(1+x²)

d/dx(a^x)

(a^x)(lna)

d/dx(log base a of x)

fundamental theorem of calculus

∫f(x) dx = F(b) - F(a)

intermediate value theorem

if a function is continuous over a closed interval [a, b], it encompasses every value between f(a) and f(b) within that range

mean value theorem

the average of some function f(x) is equal to 1 divided by the width of the region (if my region goes from a to b, that's 1/(b - a)) times the integral from a to b of f(x)dx

extreme value theorem

If f is continuous on a closed interval [a,b], then f has both an absolute maximum value and an absolute minimum value in [a,b]

arc length

L = ∫(a to b) √1+[f’(x)]² dx

L = ∫(a to b) √[x’(t)]² + [y’(t)]² dt

velocity

derivative of position

acceleration

derivative of velocity

velocity vector

<dx/dt, dy/dt>

speed

|velocity|

√(x’)² + (y’)²

displacement

∫(a to b) v(t) dt

distance

∫(initial time to final time) |v| dt

∫(a to b) √(x’)² + (y’)² dt

average velocity

(final position - initial position) / (total time)

l’Hopital’s Rule

if f(a)/g(b) = 0/0 or ∞/∞, then lim f(x)/g(x) = limf’(x)/g’(x)

slope of parametric equation

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

euler’s method

x(new) = x(old) + ∆x

y(new) = y(old) + dy/dx(x old to x new) ∆x

polar coordinates

x = r cosθ

y = r sinθ

area inside leaf of polar

∫½[r(θ)]² dθ (bounds are first two times that r=0)

slope of r(θ) at given θ

dy/dx = (dy/dθ) ÷ (dx/dθ)

integration by parts

uv - ∫v du

integral of log

x lnx - x +C

ratio test

if lim(n→∞) |n + 1th term/nth term| < 1, then series converges

taylor series

(fⁿ(a)/n!)(x-a)ⁿ

lagrange error bound

|error| ≤ |(M(x-c)ⁿ⁺¹)÷(n+1)!|

M= maximum of fⁿ⁺¹

alternating series error bound

|error| ≤ |next term|

geometric series convergence

arⁿ⁻¹ diverges if |r| ≥ 1, converges to a/(1-r) if |r| < 1

pythagorean trig identities

sin²x + cos²x =1

1 + tan²x = sec²x

cot²x+1 = csc²x

odd-even trig identities

sin(-x) = -sinx

cos(-x) = cosx

∫tanx dx

ln|secx| +c

-ln|cosx| +c

∫secx dx

ln|secx + tanx| +c

limit comparison test

if lim(n→∞) aₙ/bₙ = positive constant, then both aₙ and bₙ converge or diverge

if it approaches 0 and bₙ converges, then aₙ converges

if it approaches ∞ and bₙ diverges, then aₙ diverges

integral test

if aₙ is positive and decreasing, then does whatever ∫(1 to infinity) aₙ dn does

alternating series test

does NOT test for absolute convergence

if non-alternating part is positive, decreasing, and lim as n→∞ is 0, then series converges