Maclaurin Series
1/(1-x)
Σ_(n=0)^∞ x^n = 1 + x + x^2 +...
R = 1
I = (-1, 1)
State four conversions between Cartesian and Polar coordinates.
BONUS 100 POINTS: State a conversion between Cartesian and Cylindrical coordinates which is not present between Cartesian and Polar coordinates
x = rcosθ
y = rsinθ
r^2 = x^2 + y^2
tanθ = y/x
Bonus: z = z
Spiral
r = aθ
Ellipsoid
x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
Trace: Ellipse
Integral of 1/(1+x^2)
arctan(x) + C
e^x
Σ_(n=0)^∞ x^n/n! = 1 + x/1! + x^2/2! + x^3/3! + ...
R = ∞
I = (-∞, ∞)
If x = 0 such that tanθ = y/x, what value of θ should be used?
θ = pi/2
Name the three types of circles described by polar coordinates.
1) r = a (circle centered at (0, 0))
2) r = asinθ (circle sitting on the x-axis)
3) r = asinθ (circle sitting on the y-axis)
Elliptic Paraboloid
z/c = x^2/a^2 + y^2/b^2
The following two traces are valid:
Vertical trace - Parabola
Horizontal trace - Ellipse
Maclaurin series of arctan(x)
Σ_(n=0)^∞(-1)^n * x^(2n+1)/2n+1 = x - x^3/3 + x^5/5 - x^7/7 + ...
R = 1
I = [-1, 1]
cos(x)
Σ_(n=0)^∞ (-1)^n x^(2n)/(2n)! = 1 - x^2/2! + x^4/4! - x^6/6! + ...
R = ∞
I = (-∞, ∞)
Name the three conversions for going from Spherical to Cartesian coordinates.
x = psinφcosθ
y = psinφsinθ
z = pcosφ
State the general form of a limacon and its four forms.
r = a + bsinθ (right-side up); r = a + bcosθ (90 degree CW rotation)
1) a < b limacon with inner loop (heart with hole)
2) a = b cardioid (upside down heart)
3) a > b dimpled limacon (heart bottom not touching origin)
4) a ≥ 2b convex limacon (deflated circle)
Hyperbolic Paraboloid
z/c = x^2/a^2 - y^2/b^2
The following two traces are valid:
Vertical trace - Parabola
Horizontal trace - Hyperbola
Equation of Hyperboloid of Two Sheets
-x^2/a^2 - y^2/b^2 + z^2/c^2 = 1
The following two traces are valid:
Vertical trace - Parabola
Horizontal trace - Ellipse
sin(x)
Σ_(n=0)^∞ (-1)^n x^(2n+1)/(2n+1)! = x - x^3/3! + x^5/5! - x^7/7! = ...
R = ∞
I = (-∞, ∞)
Name the three conversions for going from Cartesian to Spherical coordinates.
p = sqrt(x^2 + y^2 + z^2)
tanθ = y/x
cosφ = z/p
What is the general formula and rules for Polar curve roses?
r = acos(nθ)
n-leaves if n is odd
2n-leaves if n is even
r = asin(nθ) gives a 90 degrees CW rotation
Cone
z^2/c^2 = x^2/a^2 + y^2/b^2
The following two traces are valid:
Vertical trace - Hyperbola
Horizontal trace - Ellipse
Maclaurin series of (1 + x)^k
(1 + x)^k = Σ_(n=0)^∞ (k n) x^n
= 1 + kx + k(k-1)/2! x^2 + k(k-1)(k-2)/3! x^3 + ...
R = 1
ln(1+x)
Σ_(n=0)^∞ (-1)^n * x^(n+1)/(n+1)
OR
Σ_(n=1)^∞ (-1)^(n-1) * x^n/n = x - x^2/2 + x^3/3 - x^4/4 + ...
R = 1
I = (-1, 1]
What are the restrictions for φ and θ in Spherical coordinates?
0 ≤ φ ≤ pi
0 ≤ θ ≤ 2pi
What are the two forms of a lemniscate?
r^2 = a^2sin(2θ) (symmetric about y = x)
r^2 = a^2cos(2θ) (symmetric about the Cartesian plane)
Hyperboloid of One Sheet
x^2/a^2 + y^2/b^2 - z^2/c^2 = 1
The following two traces are valid:
Vertical trace - Hyperbola
Horizontal trace - Ellipse
Let ax + by + cz + d = 0 be a plane and let P = (x0, y0, z0) be a point.
What is the formula for the distance from P to the plane?
BONUS 300 POINTS: Given two planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0, what is the formula for distance between the two planes?
1) |ax0 + by0 + cz0 + d| / sqrt(a^2 + b^2 + c^2)
2) |d1 - d2| / sqrt(a^2 + b^2 + c^2)