M438 Semester 2 Collective Formulas

logx + logy

note : logx + logy ≠ log(xy)

logx - logy

note : lox - logy ≠ log(x/y)

y(logx)

Do x = t

Take given equation and substitute x for t

Give appropriate range for t

Using given equation(s), take one and solve for t

Plug in t into the other equation to get everything in terms of x and y

Give appropriate limitations to x

Y - Y1 = m(X - X1)

Swap instances of x with y and vice versa, solve for the appropriate variable.

one sided limits

two - sided limit

then lim 𝑥→c f(x) = L

vice versa is also true

then lim 𝑥→c f(x) DOES NOT EXIST

lim 𝑥→c [f(x)]^r/s = [lim 𝑥→c f(x)]^r/s

lim 𝑥→c [k * f(x)] = k * lim 𝑥→c [f(x)]

1

1 (remember tan = sin/cos, so sin answer applies)

0 (remember c”o”s has a “0”

1

1

1

DOES NOT EQUAL ZERO

0

Piecewise functions, step functions and (|x|/x)

NON- removable

Holes

ONLY REMOVABLE DC

Asymptotes

NON - removable

m = lim (f(a+h)−f(a))/h

h→0

Use point-slope form like in tangent equation and use the opposite - reciprocal slope of the tangent slope.

m = lim (f(x)−f(a))/(x-a)

h→0

corner

Vertical Tangent

cusp

CUSP

VERTICAL TANGENT

f(x) = x^n

f’(x) = n*x^(n-1)

𝑑/𝑑𝑥(u ± v) = 𝑑/𝑑𝑥(u) ± 𝑑/𝑑𝑥(v)

𝑑/𝑑𝑥(u * v) = 𝑑v/𝑑𝑥(u) + 𝑑u/𝑑𝑥(v)

***** term 1 times derivative of term 2 plus term 2 times derivative of term 2.***

𝑑/𝑑𝑥(u / v) = (𝑑u/𝑑𝑥(v) + 𝑑v/𝑑𝑥(u)) / v^2

***** bottom times derivative of top minus top times derivative of bottom all over bottom squared.***

s’(t)

first derivative

e.g : meters/sec (change in position)

|v(t)| (always positive)

a(t) = v’(t) = s’’(t)

e.g : meters/sec^2 (change in velocity)

second derivative

s(t1)-s(t0)/ (t1 - t0)

AROC

**** if you use v(t) then it becomes average acceleration** **

find derivative of s(t)

s’(t) = v(t)

cosx

-sinx

sec^2x

-cscxcotx

secxtanx

-csc^2x

1/√1- x^2

1/(x^2 + 1)

1/ |x|*√x^2-1

-1/√1- x^2

-1/(x^2 + 1)

- 1/ |x|*√x^2-1

e^x

a^x * ln a

1/ x

1/ xlna

Work Outside- In

Take derivative of outside

take derivative of inside

multiply together

1. Take d/dx of both sides of equation

2. collect terms with dy/dx on one side

3. factor dy/dx

4. solve for dy/dx

absolute max (HIGHEST point)

absolute min (LOWEST point)

**if the point is not defined (HOLE) there is no abs min/max*

if “[“ or “]” then abs extrema, if “(“ or “)” then NO abs extrema

relative max (HIGH point)

relative min (LOW point)

****** NEVER AT ENDPOINTS ******

must be continuous and defined at that point

f’(x) = AROC

f’(x) = ax^n

f(x) = ax^n-1/ (n+1) + c

n cannot equal -1

Use given equations solve for one variable and plug it into the other equation

find derivative of that equation, use interval testing on the “zeros”

plug “zero” back in to find appropriate answers

Xₙ₊₁ = Xₙ - (f(xₙ)/f’(Xₙ))

Use implicit Differentiation