POLAR EQUATIONS AND COORDINATES
What equation(s) could you use to find the cartesian coordinates given polar coordinates (r, θ)?
y=rsinθ and x=rcosθ
Determine if the series is convergent or divergent.
3/sqrt(n)
Divergent:
it is a p series with p<1.
integral of 1/(1+x2)dx
arctan(x)+C
What is the formula for volume when using the SHELL Method?
V= 2π * integral from a to b of [R(x)*h(x)]dx
u= <8, -5> , v= <-5, 3>
a) Find (8/17)u + (15/17)v
b) Find magnitude of part a
a) <-11/17, 5/17>
b) sqrt(146)/17
Find the cartesian coordinates of the point which is given in polar coordinates.
(2, 0)
(2, 0) = (r, θ)
x=rcosθ = 2cos(0) = 2
y=rsinθ = 2sin(0) = 0
Cartesian coordinate = (2, 0)
Write out the first 3 terms of the series and find its sum.
Rachel will write on the board
first 3 terms: -1/3, 1/9, -1/27
sum= -1/2
Solve the definite integral.
rachel will write it on the board
2/sqrt(3)
What is the formula for volume when using the DISK Method?
V= π * integral from a to b of R(x)2dx
v= 6i + 3j -2k u= 12i -3j
a) Find dot product
b) find magnitudes of v and u
c) find cosine of the angle between v and u
a) 63
b) |v|= 7, |u|=3*sqrt(17)
c) 3/sqrt(17)
Find the cartesian coordinate of the point which is given in polar coordinates.
(-2, π)
(-2, π) = (r, θ)
x=rcosθ = -2*cos(π) = 2
y=rsinθ = -2*sin(π) = 0
Cartesian coordinate = (2, 0)
Determine whether the series converges or diverges.
n18/18n
p = 1/18 < 1, so converges absolutely
Stand by while rachel writes the problem on the board
The improper integral converges to π/2.
A mountain climber is about to haul up 20m of hanging rope. How much work will it take if the rope weighs 0.4N/m?
80 Joules
u = 12i + 35j
v = 6i + 3j - 2k
Find the projection of u onto v.
(177/49) * 6i + 3j - 2k
Find a relation between x and y.
r2=4rcosθ
x2+y2=4x
Determine whether the series converges or diverges.
Rachel will write on board
4/9 <1, so the series is convergent by the ratio test
Determine whether the improper integral converges or diverges.
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The improper integral converges to π/30.
Rachel will write it on the board lol
V = 9π/2
u = -4i - 5j - 3k
v = 6i + 6j + 4k
a) Find u x v
b) Find the direction of u x v
a) -2i - 2j + 6k
b) -i/sqrt(11) - j/sqrt(11) + 3k/sqrt(11)
Find the slope of the curve at the given points.
r = -10 + 10cosθ
θ = π/2, -π/2
θ=π/2, dy/dx = -1
θ= -π/2, dy/dx = 1
Use the Alternating Series Test to determine whether the series converges or diverges.
Un= n/(7n7+1)
The series is 1) always positive, 2) non-increasing, and 3) The limit is 0.
The series converges by the AST.
Solve using integration by parts.
Dr. Shi-thead will write it on the board
x2ex-2xex+2ex+C
Find the length of the following curve:
x = y3/2/3 - y1/2 from y=4 to y=9
L = 22/3
a) Find the area of the triangle determined by P, Q, and R.
b)Then, find the unit vector perpendicular to the plane PQR.
P(2, 6, 1) , Q(2, 0, 1) , R(0, 3, 2)
a) A = 1/2 * sqrt(180)
b) -6/sqrt(180) i - 12/sqrt(180) k