M438 Semester 2 Collective Formulas

log(xy)

logx + logy

note : logx + logy ≠ log(xy)

log(x/y)

logx - logy

note : lox - logy ≠ log(x/y)

logx^y

y(logx)

To convert rectangular (Cartesian) equation into Parametric

Do x = t

Take given equation and substitute x for t

Give appropriate range for t

Convert Parametric equation into Cartesian Equation

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

Point Slope Form

Y - Y1 = m(X - X1)

Finding the inverse of an equation

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

lim 𝑥→c+ f(x) and lim 𝑥→c- f(x) are what type of limits?

one sided limits

what type of limit is lim 𝑥→c f(x)

two - sided limit

if lim 𝑥→c+ f(x) = L and lim 𝑥→c- f(x) = L

then lim 𝑥→c f(x) = L

vice versa is also true

if lim 𝑥→c+ f(x) ≠ lim 𝑥→c- f(x)

then lim 𝑥→c f(x) DOES NOT EXIST

Power Rule for Limits

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

Constant Multiple Rule for Limits

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

lim 𝑥→0 (sinx/x) = ?

1

lim 𝑥→0 (tanx/x) = ?

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

lim 𝑥→0 ((cosx-1)/x) = ?

0 (remember c”o”s has a “0”

lim 𝑥→0 (sinNx/ Nx) = ?

1

lim 𝑥→0 (x/sinx) = ?

1

lim 𝑥→0 (x/tanx) = ?

1

lim 𝑥→0 (x/(cosx-1)) = ?

DOES NOT EQUAL ZERO

lim 𝑥→ ∞ (sinx/x) = ?

0

Jump Discontinuity

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

NON- removable

Point Discontinuity

Holes

ONLY REMOVABLE DC

Infinite Discontinuity

Asymptotes

NON - removable

Definition of Derivative

m = lim (f(a+h)−f(a))/h

h→0

How to find equation for normal line?

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

Alternate Definition of Derivative

m = lim (f(x)−f(a))/(x-a)

h→0

If function is NOT differentiable and one sided limits are not equal then there is a ….

corner

If function is NOT differentiable and one sided limits = ∞ then there is a ….

Vertical Tangent

If function is NOT differentiable and lim 𝑥→c+ =  ∞ and lim 𝑥→c- = -∞ then there is a ….

cusp

INFORMAL Recognition

odd root and even power

y = x^ 2/3

CUSP

INFORMAL Recognition

odd root and odd power

y = x^ 1/3

VERTICAL TANGENT

Power Rule ✨

f(x) = x^n

f’(x) = n*x^(n-1)

Sum/ Difference Rule

𝑑/𝑑𝑥(u ± v) = 𝑑/𝑑𝑥(u) ± 𝑑/𝑑𝑥(v)

Product Rule

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.***

Quotient Rule

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.***

Velocity

s’(t)

first derivative

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

Speed

|v(t)| (always positive)

Acceleration

a(t) = v’(t) = s’’(t)

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

second derivative

Average Velocity

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

AROC

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

Instantaneous Velocity

find derivative of s(t)

s’(t) = v(t)

𝑑/𝑑𝑥(sinx)

cosx

𝑑/𝑑𝑥(cosx)

-sinx

𝑑/𝑑𝑥(tanx)

sec^2x

𝑑/𝑑𝑥(cscx)

-cscxcotx

𝑑/𝑑𝑥(secx)

secxtanx

𝑑/𝑑𝑥(cotx)

-csc^2x

𝑑/𝑑𝑥(arcsin)

1/√1- x^2

𝑑/𝑑𝑥(arctan)

1/(x^2 + 1)

𝑑/𝑑𝑥(arcsec)

1/ |x|*√x^2-1

𝑑/𝑑𝑥(arccos)

-1/√1- x^2

𝑑/𝑑𝑥(arccot)

-1/(x^2 + 1)

𝑑/𝑑𝑥(arccsc)

- 1/ |x|*√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 logₐx

1/ xlna

Chain Rule

Work Outside- In

Take derivative of outside

take derivative of inside

multiply together

Implicit Differentiation

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 Extremas

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 Extremas

relative max (HIGH point)

relative min (LOW point)

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

must be continuous and defined at that point

Mean Value Theorem

f’(x) = AROC

Anti- derivative

f’(x) = ax^n

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

n cannot equal -1

Optimization

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

Newton’s Method

Xₙ₊₁ = Xₙ - (f(xₙ)/f’(Xₙ))

Related Rates and dy (how to solve)

Use implicit Differentiation