BC Calc Memory Quiz

definition of continuity

lim (x→c⁻) f(x)=lim(x→c⁺)f(x)=f(c) OR lim(x→c)f(x)=f(c)

Intermediate Value Theorem

(a) since f is continuous on [a,b] and

(b) f(a)<k<f(b),

(c) IVT guarantees at least one c in (a,b) such that f(c)=k

chain rule

(d/dx)(f(g(x)))=f’(g(x))*g’(x)

product rule

(d/dx)[f*g]= f’g+fg’

quotient rule

(d/dx)[f/g]= (f’g-fg’)/g²

(d/dx) sinu

cosu(u’)

(d/dx) cosu

-sinu(u’)

(d/dx) tanu

sec²u(u’)

(d/dx) cotu

-csc²u(u’)

(d/dx) secu

secutanu(u’)

(d/dx) cscu

-cscucotu(u’)

extreme value theorem

since f is a continuous function on [a,b] must have a maximum and minimum value within that interval.

relative/local extrema

a high/low point relative to the points around it; can only occur at a critical value.

Absolute Extremum (Min/Max)

the highest/lowest point on a given interval; can occur at a critical value OR an endpoint.

Mean Value Theorem(MVT)

Since f is continuous on [a, b] and differentiable on (a, b) MVT guarantees that there exists an x-value such that f’(x)=(f(b)-f(a))/(b-a)

∫(cosu)du

sin u +C

∫(sinu)du

-cos u +C

∫(sec²u)du

tan u +C

∫(csc²u)du

-cot u +C

∫(secutantu)du

sec u +C

∫(cscucotu)du

-csc u +C

1st FTC

∫(a to b) ƒ’(x)dx=f(b)-f(a)

2nd FTC

(d/dx)∫(u to v) g(t)dt=g(v)(v’)-g(u)(u’)

average value of a function on [a,b]

(1/(b-a))∫(a to b) f(x)dx

displacement

∫(a to b) v(t)dt

total distance

∫(a to b) |v(t)| dt

∫(tanu)du

-ln|cos u|+C

∫(cotu)du

ln|sin u| +C

∫(secu)du

ln|secu+tanu|+C

∫(cscu)du

-ln|cscu+cotu|+C

inverse derivative

(f⁻¹)’(x)=1/(f’(f⁻¹(x))

(d/dx)[eⁿ]

eⁿ+c

(d/dx)[aⁿ]

aⁿ(ln a) n’

(d/dx)ln u

u’/u

(d/dx) logₙu

u’/(u (ln n))

∫eⁿdn

eⁿ+C

∫aⁿdn

aⁿ/lna +C

∫(1/u)du

ln |u| +C

(d/dx)[arcsin u]

u’/√(1-u²)

(d/dx) [arctan u]

u’/(1+u²)

(d/dx)[arcsecu]

u’/(u√(u²-1))

(d/dx)[arccosu]

-u’/√(1-u²)

(d/dx)[arccotu]

-u’/(1+u²)

(d/dx) [arccsc u]

-u’/(u√(u²-1))

∫(1/√(a²-u²))du

arcsin (u/a) +C

∫(1/(a²+u²))du

(1/a) arctan (u/a) +C

exponential growth/decay

if dy/dt=ky, then y=c(e^kt)

logistics equation

if dy/dt =ky(1-(y/L)), then y=L/(1+C(e^-kt))

area between two curves

∫(a to b) (top-bottom)dx OR ∫(a to b) (right-left)dy

volume disk method

v=π(∫(a to b) R²dx

Volume washer method

v=π(∫(a to b) (R²-r²)dx

Volume shell method

v=2π(∫(a to b) (ph)dx

solids of known cross-section

v=(∫(a to b) A(x)dx

arc length

L=∫(a to b) √(1+[f’(x)]²)dx

Surface Area

S=∫(a to b) 2πr√(1+[f’(x)]²)dx

integration by parts

∫u dv = uv-∫v du

Taylor Series

f(x) = f(c)+f’(c)(x-c)+f’’(c)(x-c)²/2!+…+fⁿ(c)(n-c)ⁿ/n!

Taylor series for e^x (centered at 0)

1+x+x²/2!+x³/3!+…+xⁿ/n! (for all real numbers)

Taylor series for sin x (centered at 0)

x-x³/3!+x⁵/5!-x⁷/7!+…+(-1)ⁿx²ⁿ⁺¹/(2n+1)! (for all real numbers)

taylor series for cos x (centered at 0)

1-x²/2!+x⁴/4!-x⁶/6!+…+(-1)ⁿx²ⁿ/(2n)! (for all real numbers)

power series for 1/(1-x)

1+x+x²+x³+x⁴+…+xⁿ (for -1<x<1)

Alternating series error

error≤|Aₙ₊₁|

Lagrange error

error≤|fⁿ⁺¹(max)|/(n+1)!×(x-c)ⁿ

parametric equation slope

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

Parametric 2nd derivative

d²y/dx²=(d/dt)[dy/dx]/(dx/dt)

parametric speed

√[(dx/dt)²+(dy/dt)²]

parametric arc length (aka total distance

L=∫(a to b) √[(dx/dt)²+(dy/dt)²] dt

polar area

A=½∫(a to b) r²dθ

Polar parametrics

x=r cosθ and y=r sinθ