Arcs
Calculate the length of the curve described by
y = ln(cosx) for x ∈ [0, pi/3].
ln(2 + sqrt(3))
Find the area of the surface obtained by rotating the graph of y = x^(3/2)/3 for x ∈ [0, 12], about the y-axis.
pi/4(17sqrt(17) - 5sqrt(5))
Find the curvature of the curve given by r(t) = <sqrt(6)t^2, 2t, 2t^3>. Find the point on the curve such that the curvature equals sqrt(6)/8.
k(t) = sqrt(6)/(2(3t^2 + 1)^2)
Point(s): (sqrt(6)/3, 2/sqrt(3), 2/(3sqrt(3))),
(sqrt(6)/3, -2/sqrt(3), -2/(3sqrt(3)))
Besides the Pythagorean trig identity, what is another way to write (cosθ)^2 and (sinθ)^2?
(cosθ)^2 = 1/2(1 + cos(2θ))
(sinθ)^2 = 1/2(1 - cos(2θ))
(or any other equivalent form)
What can sinhx and coshx be written as?
sinhx = (e^x - e^(-x))/2
coshx = (e^x + e^(-x))/2
Calculate the length of the cardioid r = 1 + cosθ.
8
What are the equations for rotating the curve y = f(x) about
a) the x-axis
b) the y-axis
(Bounds are from some a to some b)
a) S = 2pi∫f(x)sqrt(1+(f'(x))^2)dx
b) S = 2pi∫xsqrt(1+(f'(x))^2)dx
What is the formula for determining curvature?
k(t) = |r'(t) x r''(t)|/|r'(t)|^3
What is the Pythagorean trig identity between tangent and secant?
(tanθ)^2 + 1 = (secθ)^2
(or any other equivalent form)
Integral of 1/x
ln|x| + C
Calculate the perimeter of the asteroid parametrized by r(θ) = <a(cosθ)^3, a(sinθ)^3> where a > 0 is a constant.
6a
Find the area of the surface obtained by rotating the curve x = a(cosθ)^3, y = a(sinθ)^3, θ ∈ [0, pi/2], about the x-axis.
S = 6pia^2/5
What are the formulas for determining the tangent, normal, and binormal vectors?
T(t) = r'(t)/|r'(t)|
N(t) = T'(t)/|T'(t)|
B(t) = T(t) x N(t)
Name all three cos(2θ) identities.
cos(2θ) = (cosθ)^2 - (sinθ)^2
cos(2θ) = 2(cosθ)^2 - 1
cos(2θ) = 1 - 2(sinθ)^2
Integral of secxtanx
secx + C
Consider the helix given by r(t) = <cost, sint, t>,
t ≥ 0. Reparametrize the helix with respect to arc length, measured from (1, 0, 0) in the direction of increasing t.
r(t(s)) = <cos(s/sqrt(2)), sin(s/sqrt(2)), s/sqrt(2)>
What are the equations for rotating the curve x = g(y) about
a) the y-axis
b) the x-axis
(Bounds are from some a to some b)
a) S = 2pi∫g(y)sqrt(1+(g'(y))^2)dy
b) S = 2pi∫ysqrt(1+(g'(y))^2)dy
What is the normal vector for the normal plane and the normal vector for the osculating plane?
Normal plane: T(t)
Osculating plane: B(t)
Integral of tanx and secx
∫tanx dx = ln|secx| + C
∫secx dx= ln|secx+ tanx| + C
Integral of 1/sqrt(1 - x^2)
arcsinx + C
Consider the curve given by r(t) = <e^tcos(t), e^tsin(t)>. Find the point(s) that are sqrt(2)/2 units away from the point P(1, 0) when measured on the curve.
(3/2cos(ln(3/2)), 3/2sin(ln(3/2)))
(1/2cos(ln(1/2)), 1/2sin(ln(1/2)))
What are the equations for rotating the parametric curve curve x = x(t), y = y(t) about
a) the x-axis
b) the y-axis
(Bounds are from some a to some b)
a) S = 2pi∫y(t)sqrt((x'(t))^2+(y'(t))^2)dt
iff y(t) ≥ 0
b) S = 2pi∫x(t)sqrt((x'(t))^2+(y'(t))^2)dt
iff x(t) ≥ 0
Consider the curve given by r(t) = <sin(2t), -cos(2t), 4t> and the point P(0, 1, 2pi). Find the TNB frame of this curve at P.
BONUS 500 POINTS: Find the normal plane and osculating plane at P.
T(pi/2) = 1/sqrt(5)<-1, 0, 2>
N(pi/2) = <0, -1, 0>
B(pi/2) = 1/sqrt(5)<2, 0, 1>
Bonus: Normal plane is x - 2z = -4pi, osculating plane is 2x + z = 2pi
jk this isn't a trig identity i ran out of important ones
lim(n->∞)(1 + a/n)^n
e^a
nya