Extension Flashcards
What is column vector notation?
Representation of a vector in the form (a
b)
When is maximum height reached?
When the projectile’s vertical velocity is 0
What is component form?
Representation of a vector in the form xi + yj
What proves if vectors are parallel?
u . v = +/- |u||v|
What is the horizontal equation of acceleration?
ẍ = 0
What is the horizontal equation of velocity?
ẋ = Vcosθ
What is the horizontal equation of displacement?
x = Vtcosθ
What is the vertical equation of acceleration?
ÿ = -g
What is the vertical equation of velocity?
ẏ = -gt + Vsinθ
What is the vertical equation of displacement?
y = -1/2gt^2 + Vtsinθ + h
What can (x,y) for vectors also be written as?
(Vcosθ, Vsinθ)
What is the equation for the acceleration vector?
a = -gj
What is the equation for the velocity vector?
v = Vcosθi + (-gt + Vsinθ)j
What is the equation for the displacement vector?
r = Vtcosθi + (-1/2gt^2 + Vtsinθ + h)j
How do you find time of flight?
Let vertical component (j) of r(t) equal 0 then solve for t
How do you find the range of flight?
Substitute time of flight into the horizontal component (i) of r(t)
How do you find the maximum height of a projectile using time of flight?
Half the time of flight and substitute into the j component of r(t)
How do you find the maximum height of a projectile using velocity?
Let the vertical component (j) of v(t) equal 0 then solve t and sub into r(t)
What is the formula to find a unit vector?
1/|a| x a
What is the projection of a onto b?
projba = a . b / |a|^2 X a
How do you find the Cartesian equation of vector displacement?
Rearrange x to get t on its own and then sub in y
Make sure final equation only has tan
How do you find the maximum range?
Change θ but not velocity
What degree gives the maximum range?
45
How do you find speed?
|V(t)|
What is the angle of projection for horizontal or waterfall?
0
What is the relationship between f(x) and g(x) when they are inverse?
f(x) = 1/g’(x)
What is the shape of cos^-1 graph?
Starts at (-1,π) passes through π/2 and ends at (1,0)
What is the shape of sin^-1 graph?
Starts at (-1,-π/2) passes through the origin and ends at (1,π/2)
What is the shape of tan^-1 graph?
Sideway tan function which starts in the negative y, passes through the origin and finishes in the positive asymptotes at y=+/-π/2
How do you calculate the volume of a graph rotated around the y axis?
π ∫x^2 dy
How do you calculate the volume of a graph rotated around the x axis?
π ∫y^2 dx
When do you permutations?
When order matters
What do you do if there are repetitions in a permutation?
n!/r!
n - number of items
r - number of repetition
How do you calculate a circle arrangement?
(n-1)!
How do you calculate a bracelet or necklace arrangement?
(n-1)!/2
When do you use combinations?
When order doesn’t matter
What is the pigeonhole principle statement?
By the pigeonhole principle…
What should be stated at the beginning of a pigeonhole principle question?
Let the … be pigeons and the … pigeonholes
What is the pigeonhole principle?
If there are n pigeons in k pigeonholes at least one of the holes contain n/k or more pigeons
How do you calculate the smallest number needed using the pigeonhole principle?
n=mk+1
Rules for square root function graphing?
- Does not exist in y < 0
- Has the same x intercepts
- New graph under old when y > 1
- New graph intercepts at y = 1
- New graph above the old when y < 1
Rules for reciprocal functions graphing?
- x intercepts give vertical asymptotes
- Intersect at y=+/- 1
- When y > 1 inverse is <1
- When y < 1 inverse is >1
- When original increases inverse decreases and vice versa
Rules for y =|f(x)| graphing?
- When f(x) > 0 graph is same as y=f(x)
- When f(x) < 0 y-values are made positive
Rules for y = f(|x|) graphing?
- When x > 0 the graph is the same
- When x < 0 graph on right hand side reflected across the y-axis
Rules for y^2 function graphing?
Same as square root but also reflected in the x-axis
Rules for y = f(|x|) graphing?
- When x > 0 the graph is the same
- When x < 0 graph on right hand side reflected across the y-axis
What is E(p^)?
p
What is Var(p^)?
p(1-p)/n
a^2+b^2+g^2
(a+b+g)^2-(ab+ag+bg)
What is an even function?
f(x) = f(-x)
What is an odd function?
-f(x) = f(-x)
Rules for y^2 function graphing?
Same as square root but also reflected in the x-axis
Rules for square root function graphing?
- Does not exist in y < 0
- Has the same x intercepts
- New graph under old when y > 1
- New graph intercepts at y = 1
- New graph above the old when y < 1
Rules for reciprocal functions graphing?
- x intercepts give vertical asymptotes
- Intersect at y=+/- 1
- When y > 1 inverse is <1
- When y < 1 inverse is >1
- When original increases inverse decreases and vice versa
Rules for y =|f(x)| graphing?
- When f(x) > 0 graph is same as y=f(x)
- When f(x) < 0 y-values are made positive
What is the formula for the dm/dt for water concentration?
Concentration pumped in x volume rate pump in - concentration pump out x volume rate pumped out
kg/L x L/h - m/V x L/h
