f(-x)=f(x)

y-axis symmetry

f(-x)=-f(x)

origin symmetry

d/dx sin^-1x

(arcsinx)

1

d/dx tan^-1x

(arctanx)

1

1

Chain Rule

d/dx f(g(x))

(c,f(c)) = Max if f'(x) changes from + to -

(c,f(c)) = Min if f'(x) changes from - to +

(c, f(c)) = Max if f''(c) = neg.

(c, f(c)) = Min if f''(c) = pos.

f''(x)=0 or dne

f''(x) changes sign

If f(x) is continuous on [a,b] and differentiable on (a,b), there exists a number "c", a<c<b such that:

f'(c)=

f(b)-f(a)

∫ 1 dx

∫ 1 dx

ln|secx| + C

-ln|cosx| + C

Trapezoidal Rule

(n is # of subdivisions)

_a∫^bf(x) dx =

(1/2)·(b-a/n)·(f(c)+2f(x₁)+2f(x₂)+...+f(b))

∏_x=a∫^x=b(f(x))²dx

or

∏ _y=c∫^y=d(f(y))²dy

∏_x=a∫^x=b[(f(x))² - (g(x))²]dx

or

∏_y=c∫^y=d[(f(y))² - (g(y))²]dx

Avg. Value on Function

on [a,b]

Position

Velocity

Acceleration

s(t)

s'(t)

s''(t)

If lim f(x)/g(x) = 0/0 or ∞/∞

then

lim f(x)/g(x) = lim f'(x)/g'(x)

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

d²y/dx² = (d/dt(dy/dx))/(dx/dt)

<x(t), y(t)>

x(t)i + y(t)j

<x'(t), y'(t)>

x'(t)i + y'(t)j

<x''(t), y''(t)>

x''(t)i + y''(t)j

_a∫^bSpeed

_a∫^b√(x'(t))² + (y'(t))² dt

x = rcosθ

y = rsinθ

R_n(x) = (fⁿ⁺¹(z)/(n+1)!)(x-c)ⁿ⁺¹

where z is some number between c and the value of x being evaluated

Σ arⁿ converges if |r|<1

sum = (a/1-r)

∑(1/n^p) converges if p>1

diverges if p<1

lim a_n+1

<1 ∑a_n converges

>1 ∑a_n diverges

=1 Test FAILS

∑(-1)ⁿa_n converges if

lim a_n=0 AND a_n+1 ≤ a_n