M438 Semester 2 Collective Formulas

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Calculus A

Calculus

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66 Terms

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log(xy)
logx + logy

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**note : logx + logy ≠ log(xy)**
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log(x/y)
logx - logy

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**note : lox - logy ≠ log(x/y)**
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logx^y
y(logx)
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To convert rectangular (Cartesian) equation into Parametric
Do x = t

Take given equation and substitute x for t

Give appropriate range for t
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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
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Point Slope Form
Y - Y1 = m(X - X1)
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Finding the inverse of an equation
Swap instances of x with y and vice versa, solve for the appropriate variable.
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lim 𝑥→c+ f(x) and lim 𝑥→c- f(x) are what type of limits?
one sided limits
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what type of limit is lim 𝑥→c f(x)
two - sided limit
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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
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if lim 𝑥→c+ f(x) **≠** lim 𝑥→c- f(x)
then lim 𝑥→c f(x) DOES NOT EXIST 
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Power Rule for Limits
lim 𝑥→c \[f(x)\]^r/s = \[lim 𝑥→c f(x)\]^r/s
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Constant Multiple Rule for Limits
lim 𝑥→c \[k \* f(x)\] = k \* lim 𝑥→c \[f(x)\]
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lim 𝑥→0 (sinx/x) = ?
1
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lim 𝑥→0 (tanx/x) = ?
1 (remember tan = sin/cos, so sin answer applies)
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lim 𝑥→0 ((cosx-1)/x) = ?
0 (remember c”o”s has a “0”
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lim 𝑥→0 (sinNx/ Nx) = ?
1
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lim 𝑥→0 (x/sinx) = ?
1
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lim 𝑥→0 (x/tanx) = ?
1
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lim 𝑥→0 (x/(cosx-1)) = ?
DOES NOT EQUAL ZERO
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lim 𝑥→ ∞ (sinx/x) = ?
0
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Jump Discontinuity
Piecewise functions, step functions and (|x|/x)

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NON- removable
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Point Discontinuity
Holes

ONLY REMOVABLE DC
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Infinite Discontinuity
Asymptotes

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NON - removable
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Definition of Derivative
m = lim (f(a+h)−f(a))/h

h→0
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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.
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Alternate Definition of Derivative
m = lim (f(x)−f(a))/(x-a)

h→0
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If function is NOT differentiable and one sided limits are not equal then there is a ….
corner
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If function is NOT differentiable and one sided limits = ∞ then there is a ….
Vertical Tangent
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If function is NOT differentiable and lim 𝑥→c+ =  ∞ and lim 𝑥→c- = -∞ then there is a ….
cusp
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INFORMAL Recognition

odd root and even power

y = x^ 2/3
CUSP
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INFORMAL Recognition 

odd root and odd power

y = x^ 1/3
VERTICAL TANGENT
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Power Rule ✨
f(x) = x^n

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f’(x) = n\*x^(n-1)
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Sum/ Difference Rule
𝑑/𝑑𝑥(u ± v) = 𝑑/𝑑𝑥(u) ± 𝑑/𝑑𝑥(v)
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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.\*\*\*
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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.\*\*\*
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Velocity
s’(t)

first derivative

e.g : meters/sec (change in position)
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Speed
|v(t)| (always positive)
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Acceleration
a(t) = v’(t) = s’’(t)

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

second derivative
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Average Velocity
s(t1)-s(t0)/ (t1 - t0)

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AROC

**** if you use v(t) then it becomes average acceleration** \*\*
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Instantaneous Velocity
find derivative of s(t)

s’(t) = v(t)
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𝑑/𝑑𝑥(sinx)
cosx
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𝑑/𝑑𝑥(cosx)
\-sinx
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𝑑/𝑑𝑥(tanx)
sec^2x
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𝑑/𝑑𝑥(cscx)
\-cscxcotx
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𝑑/𝑑𝑥(secx)
secxtanx
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𝑑/𝑑𝑥(cotx)
\-csc^2x
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𝑑/𝑑𝑥(arcsin)
1/√1- x^2
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𝑑/𝑑𝑥(arctan)
1/(x^2 + 1)
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𝑑/𝑑𝑥(arcsec)
1/ |x|\*√x^2-1
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𝑑/𝑑𝑥(arccos)
\-1/√1- x^2
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𝑑/𝑑𝑥(arccot)
\-1/(x^2 + 1)
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𝑑/𝑑𝑥(arccsc)
\- 1/ |x|\*√x^2-1
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𝑑/𝑑𝑥 e^x
e^x
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𝑑/𝑑𝑥 a^x
a^x \* ln a
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𝑑/𝑑𝑥 ln x
1/ x
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𝑑/𝑑𝑥 logₐx
1/ xlna
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Chain Rule
Work Outside- In

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Take derivative of outside

take derivative of inside

multiply together
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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
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Absolute Extremas
absolute max (HIGHEST point)

absolute min (LOWEST point)

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

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if “\[“ or “\]” then abs extrema, if “(“ or “)” then NO abs extrema
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Relative Extremas 
relative max (HIGH point)

relative min (LOW point)

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

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must be continuous and defined at that point
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Mean Value Theorem
f’(x) = AROC
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Anti- derivative
f’(x) = ax^n

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

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n cannot equal -1
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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
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Newton’s Method
Xₙ₊₁ = Xₙ - (f(xₙ)/f’(Xₙ))
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Related Rates and dy (how to solve)
Use implicit Differentiation