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42 Cards in this Set
- Front
- Back
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2 real distinct
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y = c1e^(m1x) + c2e^(m2x)
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1 real (double root)
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y = c1e^(mx) + c2*x*e^(mx)
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2 complex (A and B)
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y = e^(Ax)*(c1cos(Bx) + c2sin(Bx))
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Aux equation for
ay'' - by' + cy = 0 |
am^2 + bm + c = 0
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Spring:
m* x''(t) + kx = 0 (no damping factor) |
x(t) = c1cosWt + c2sinWt
W (omega) = sqrt(k/m) |
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Parabola form with vertex (h, k) and axis parallel to x-axis
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(y-k)^2 = 4p(x - h) (opens right)
(y-k)^2 = -4p(x - h) (opens left) |
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Parabola form with vertex (h, k) and axis parallel to y-axis
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(x-h)^2 = 4p(y - k) (opens up)
(x-h)^2 = -4p(y - k) (opens down) |
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Ellipse form with center (h, k)
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(x-h)^2 + (y-k)^2 = 1
--------- -------- a^2 b^2 a > b: short and fat a < b: tall and skinny a = b: circle |
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Hyperbola form with center (h, k)
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+- (x-h)^2 +- (y-k)^2 = 1
--------- -------- a^2 b^2 x term negative: Mirrors about horizontal (no y-intercept) y term negative: Mirrors about vertical (no x-intercept) |
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Eccentricity of parabola
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e = 1
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Eccentricity of ellipse
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e < 1
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Eccentricity of circle
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e = 0
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Eccentricity of hyperbola
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e > 1
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cos(2x) (double-angle ID)
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cos^2(x) - sin^2(x)
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sin(2x) (double-angle ID)
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2*sinx*cosx
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Ellipse: how to find c?
Hyperbola: how to find c? |
c^2 = a^2 - b^2 (a>b)
c^2 = a^2 + b^2 (a>b) |
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Foci/vertices of a horizontal ellipse?
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Foci: (h+-c, k)
Vert: (h+-a, k) a > b |
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Foci/vertices of a vertical ellipse?
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Foci: (h, k+-c)
Vert: (h, k+-a) a > b |
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Foci/Vertices/asymptotes of hyperbola mirrored about a vertical? (y term negative)
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Foci: (h+-c, k)
Vert: (h+-a, k) Asym: y = +- (b/a)x a > b |
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Foci/Vertices/asymptotes of hyperbola mirrored about a horizontal? (x term negative)
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Foci: (h, k+-c)
Vert: (h, k+-a) Asym: y = +- (a/b)x |
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Focus of vertical parabola?
Directrix? Vertex? |
Focus: (h, k+p)
Directrix: y = k - p Vertex: (h, k) |
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Focus of horizontal parabola?
Directrix? Vertex? |
Focus: (h+p, k)
Directrix: x = h - p Vertex: (h, k) |
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Polar equation of a conic with COSINE in the denominator
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Major axis is HORIZONTAL
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Polar equation of a conic with SINE in the denominator
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Major axis is VERTICAL
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Polar equation of a conic with ADDITION in the denominator
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Directrix equation is:
y = +d (for vertical major axis) x = +d (for horizontal major axis) |
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Polar equation of a conic with SUBTRACTION in the denominator
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Directrix equation is:
y = -d (for vertical major axis) x = -d (for horizontal major axis) |
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polar form conic equation
r = ? |
de
-------- 1 +- e * (sine/cosine theta) |
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Arclength of parametric curve?
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L = integral of
sqrt( dx/dt^2 + dy/dt^2) with respect to t from t1 to t2 |
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Arclength of polar curve?
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L = integral of
sqrt( r^2 + dr/dQ^2 ) in respect to Q (theta) From Q1 to Q2 |
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r = sin(2Q)
What is it? (Q = theta) |
Rose curve with 4 petals and a period of 2pi
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r = sin(3Q)
What is it? (Q = theta) |
Rose curve with 3 petals and a period of pi
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r = 1 + 2sin(Q)
What is it? (Q = theta) |
Limacon. Period = 2pi.
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Parametrics:
vertical velocity = ? |
dy/dt
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Parametrics:
horizontal velocity = ? |
dx/dt
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Parametrics:
Mtan = ? |
dy/dx = dy/dt / dx/dt
Slope of the line, not a velocity |
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Parametrics:
y'' = ? |
(dy/dx)' / dx/dt
Function of t, describes concavity of the path y(x) at time t |
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Polar equations:
x = ? |
r*cos(theta)
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Polar equations:
y = ? |
r*sin(theta)
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Polar equations:
r^2 = ? |
x^2 + y^2
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Polar equations:
Theta = ? |
tan inverse (y/x)
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How to find the slope at the pole?
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Solve for r = 0 to find the angles
m = tan(theta). Only works at the pole. |
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e = ?
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c/a
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