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Unit 8 - Coggle Diagram
Unit 8
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,π/2, π,3π/2, 2π, observe the movement of x direction and y direction, decide which one to flip
θ 0 π/2 π 3π/2 2π
x(θ)
y(θ)
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 )⋅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 (向量): magnitude and direction
Draw a vector from a starting point
Dot product
a=(a_x, a_y ), b=(b_x, b_y ), a ⋅b=a_x×b_x+a_y×b_y
Angle between vectors
a⋅b=|a| × |b| × cos(θ)
cos(θ)=(a⋅b)/(|a| × |b|)
Perpendicular vectors
a⋅b=0
Parallel vectors
a=k × b
b=k × a
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 −x_1, y_2 −〖 y〗_1 ) (t−t_1)/(t_2−t
1 ), 〖 t〗
!≤t≤t_2
Lesson 7: Problem solving with 3D vectors
Dot product
a=(a_x, a_y,a_z ), b=(b_x, b_y,b_z ), a ⋅b=a_x×b_x+a_y×b_y+a_z×b_z
Angle between vectors
a⋅b=|a| × |b| × cos(θ)
cos(θ)=(a⋅b)/(|a| × |b|)
Perpendicular vectors
a⋅b=0
Parallel vectors
a=k × b
b=k × a
Lesson 3: Modeling vertical motion
Vertical motion (in feet)
h(t)=−16t^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)=−32t+v_0
Acceleration: rate of change of velocity
Is the derivative of v(t) aka a = v'(t)
a=−32 feet per sequare second
Lesson 6: Applications of non-linear parametric equations
PreCalc Index