APPC - Midterm

Synthetic Division

When dividing you must descend in powers, if power is missing then you have to put a place holder 0, i.e. 5x^4 + 3x² the top row of synthetic division 5 0 3 0 0, then for the divisor, solve for x, if you are dividing by x-3, then synthetically divide by 3

Odd multiplicity

Cross

Synthetic Division

When dividing you must descend in powers, if power is missing then you have to put a place holder 0, i.e. 5x^4 + 3x² the top row of synthetic division 5 0 3 0 0, then for the divisor, solve for x, if you are dividing by x-3, then synthetically divide by 3

Odd multiplicity

Cross

Even multiplicity

Touch & Turn

Odd End Behavior

x→ -infinity y→-infinity

x→+infinity y→+infinity

Even end behavior

x→-infinity y→+infinity

x→-infinity y→+infinity

Rule of Signs

The number of sign changes indicates the number of ± solutions

observe,

x⁵ -2x⁴ -3x³ +4x² -x -1

3 signs changes →3 or 1 positive real solutions

then multiply everything by negative (even powers are unaffected and so are constants)

-x⁵ -2x⁴ +3x³ +4x² +x -1

2 signs changes →2 or 0 negative real solutions

Rational Zero Test

(Factors of Constants) /

——————————-

(Factors of leading coefficient)

4x^5 -3x^4 +8x³ +2x² - 5x + 12
±12, ±6 , ±4 , ±2 , ±1

Divided by

±4,±2,±1

These are all the possible rational zeros

Difference of Squares

(a² - b²) = (a+b)(a-b)

Addition of Squares

(a² + b² ) = a² +2ab + b²

Maximum turns

n-1 , Where n is the degree of the polynomial

Vertical Asymptotes

Set Denominator to 0, solve for x

Horizontal Asymptote

Big Bottom HA→y=0

Big Top HA→None

Slant Asymptote

Top is 1 greater than the bottom

divide top by bottom

U subsitution

Can be used with even powers

x^4 + x² +16 = 0

u = x²

u²+u+16=0

Can be used with e^x

e²^x -5e^x +6=0

u=e^x

u² - 5u +6

Compounding Interest

A=P91+ r/n)^nt

P → initial

r → rate (decimal)

n → how many times per year

Continuously Compounding Interest

A= Pe^rt

P→ Initial

e → the constant

r→rate (decimal)

t → years [time]

Half-life formula

N(t) = N₀(1/2)^{t / (half-life)}

N(t) = Remaining Substance

t → time

Half-life → is the amount of time for half of the substance to decay

Sum of Arithmetic Sequence

Sn = (n/2)(a₁ + a_n )

Sum of Geometric Sequence

Sn = a(1-r^n ) / (1-r)

a→ first term

Sum of infinite Geometric Series

S = a /(1-r)

nCr also written as

(n!) / (r!(n-r)!)

Error

Predicted - Actual = Error

Residual

Actual - Predicted = Residual

Residual Plot

Equal number above and below

No pattern for the model to be accurate

Above → the model under estimates

Below → The model over estimates

Vertex

-b/2a

Horizontal Asymptote for Exponentials Functions

The “b” term in the equation

y=1+10^-2x

HA y = 1