PreCalc
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Sequence and Series
Background Knowledge
Probability
Limits
Polynomial Functions and Optimization
Exponential Functions`
Arithmetic Sequence and Series
Common difference d
General Term a_n
Explicit rule 
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Problems in Motion
Geometric Sequence and Seriese
Arithmetic Series
5.1 Quadratic functions
Definition
vertex
axis of symmetry
x-ints
y-int
domain
range
Forms
Standard form
Intercept form
Vertex form
Parabola and imaginary roots
Complex number
Higher degree Poly and Intermediate Value Theorem
Turning Points
End behavior
Graph of polynomal
Intermediate Value Threoem
Long division
Steps for long division
Equivalence of notions
Root
x-corssing
P(a) = 0
factor
remainder of division
Remainder theorem
Factor theorem
Operation of complex numbers
Real part vs. imaginary part
Standard form
Rational Functions
Complex conjugate
Limits
f(x) = N(x)/D(x) where N(x) and D(x) are polynomial functions and D(x) != 0
(a^2 - b^2) = (a+b)(a-b)
Complex roots
Comlex Zeros Theroem
Polynormal Inequalities and optimization
The limit of π(π₯)as π₯approaches π from the left is denoted by π₯π’π¦πβπ!π(π)
The limit of π(π₯) as π₯ approaches π from the right is denoted by π₯π’π¦πβπ!π(π)
A function π(π₯) will have a vertical asymptote at π₯=π if:
β’π(π)is undefined, and
β’π₯π’π¦πβπ!π(π)=Β±β or π₯π’π¦πβπ!π(π)=Β±β
A function π(π₯)will have a horizontal asymptote at π¦=π when:
π₯π’π¦πβ β π(π)=π
π₯π’π¦πβ -β π(π)=π
Unit 8: Problems in Motion
Lesson 1: Vector and parametric equations for a line
The velocity vector (a, b)
Speed
Parametric equations of a line with parameter t
x=k_1 t+b_1
y=k_2 t+b_2
Vector equations of a line
(x,y)=(b_1, b_2 )+(k_1,k_2 )β t
Eliminating the parameter
Get rid of the parameter "t" through substitution or elimination
Write in rectangular form aka in terms of x and y
Lesson 2: The Dot product and angles between vectors
Vector (ει): magnitude and direction
Draw a vector from a starting point
Dot product
a=(a_x, a_y ), b=(b_x, b_y ), a β b=a_xΓb_x+a_yΓb_y
Angle between vectors
aβ b=|a| Γ |b| Γ cosβ‘(ΞΈ)
cosβ‘(ΞΈ)=(aβ b)/(|a| Γ |b|)
Perpendicular vectors
aβ b=0
Parallel vectors
a=k Γ b
b=k Γ a
Lesson 3: Modeling vertical motion
Vertical motion (in feet)
h(t)=β16t^2+v_0 t+s_0
Velocity: rate of change of displacement
Is the derivative of h(t) aka v(t) = h'(t)
v(t)=β32t+v_0
Acceleration: rate of change of velocity
Is the derivative of v(t) aka a = v'(t)
a=β32 feet per second squared
Lesson 4: Piecewise Parametric Equations
t_1 (x_1,y_1 ) t_2 (x_2,y_2)
(x,y)=(x_1,y_1 )+(x_2 βx_1, y_2 βγ yγ_1 ) (tβt_1)/(t_2βt1 ), γ tγ!β€tβ€t_2
Lesson 5: Parametric equations for parabolas, circles, and ellipses
Parametric equation for circle
Parametric equation for parabolas
Parametric equation for ellipses
Simplify to rectangular form
Change moving direction in parametric form
List a table of 0,Ο/2, Ο,3Ο/2, 2Ο, observe the movement of x direction and y direction, decide which one to flip
Lesson 6: Applications of non-linear parametric equations
Lesson 7: Problem solving with 3D vectors