Calculus on Polar Curves
Geometric and Power Series
Convergence Tests
Taylor Series
Vectors
100
The area inside one petal of the curve r=sin(2theta).
What is Pi/4?
100
A closed form function equal to x^2 + x^4 + x^6 + ....
What is x^2/(1-x^2)?
100
The series of 1/(n^2 + n).
What is converges by comparison?
100
The Taylor series for e^(2x) at x=0.
What is 1 + 2x + 4 x^2 /2!+ 8 x^3/3! ....?
100

The dot product of <0,5, 2> and <4,-6,-22>

What is -74?
200
The area inside the curve r=1+1/3 cos(theta).
What is 19 Pi / 9?
200
A closed form function equal to the sum 1 - e^x + e^(2x) - e^(3x) + ....
What is 1/(1+e^x)?
200
The series 1/(sqrt(n)+1).
What is diverges by comparison or integral comparison?
200
The Taylor series of 1/2*( e^x - e^(-x)).
What is x - x^3/3! + x^5/5! + ....?
200

The projection of <1,-1,1> onto <3,0,0>

What is <1,0,0>?

300
An integral giving the area inside r = theta^2 in the first segment of the first quadrant.
What is the integral theta^4 dtheta for theta=0 to Pi/2?
300
A geometric series expression that shows 1.999999.... = 2.
What is (1-9) + 9 + 9/10 + 9/100 + ....?
300
The series n! 2^n/ (2n)!
What is converges by the ratio test?
300
The radius of convergence of the Taylor Series of 1/(1 - 4x^2) centered at x=0.
What is 1/2?
300

The vector perpendicular to <1,2,3> and <4,5,6>

What is <-3,6,-3>?

400
An equation giving the value of theta for which the tangent line to r=cos(3 theta) is horizontal.
What is cos(3theta) sin(theta) + cos(theta)sin(3theta)=0?
400
A closed form function equal to the sum 1 + 2x + 3x^2 + 4 x^3 + .....
What is 1/(1-x)^2?
400
The series (-1)^(n+1) / n^2.
What is absolutely converges?
400
A closed form function giving by (x-1) -1/2(x-1)^2 + 1/3(x-1)^3 - ...
What is ln(x)?
400

The tangent vector at t = 2 for the vector function:

v(t) = <ln(t), e^t, t>

What is <1/2, e^2, 1>?

500
An integral giving the area inside the circle r=10 and beyond the line r cos(theta)=6.
What is the integral (10^2 - 6^2 / cos^2(theta) ) dtheta for theta = -arccos(3/5) to arcos(3/5)?
500
A value of x for which 1 - x/2 + x^2/4 - x^3/8 + .... = 4
What is -3/2?
500
The series (-1)^(n+1) n/(1 - n^2)
What is converges conditionally by the alternating series test?
500
The Taylor series of sqrt(x) at x=1 can be used to approximate the sqrt(1.5). The approximation from the first five terms.
What is 1 + 1/2 0.5 -1/4 0.5^2 /2 + 3/8 0.5^3 / 3! - 15/16 0.5^4 / 4! = 1.22412
500

The unit tangent vector at t = 1 on the vector valued function:

v(t) = <t,t^2,t^3>

What is <2, 2, 3>/sqrt(17)?

M
e
n
u