Polar
The formula:
\int_{\alpha}^{\beta}2\pir(\theta)sin(\theta)\sqrt(r^2+r'^2)d\theta
is one formula in polar form for this quantity.
What is Surface Area?
The definition of a sequence.
What is:
A list of numbers, either following an iterative formula, or a recursive one.
The defintion of the related function of a sequence.
What is:
The function f \text( where ) f(n) = a_n
The rational equality to this expression:
\sum_{n=1}^{\infty}c*r^n
Assuming c is constant, and
|r|<1
What is:
a/(1-r)
Whether or not the following statement is true or false:
text(If ) lim_{n \to \infty} a_n = 0, text(then ) \suma_n text( converges)
What is:
False
The formula for arc length of a polar curve is given as this formula.
(Write me the formula for an arc length)
What is:
\int_{\alpha}^{\beta}sqrt(r^2 + r'^2)d\theta
The definition of a sequence's limit to limit L.
What is :
a_n approaches L iff for large n, a_n approaches L
The value of the limit:
lim_{n \to \infty} {1/n}
What is 0?
The type of series that this is:
\sum_{n=1}^{\infty}(1/n - 1/(n+1))
What is a telescoping series?
What values of x allow the following series to converge:
sum_{n=1}^{infty}x^n
What is:
|x|<1
(Note: This is called the interval of convergence, and becomes very important in the later part of the unit)
Team Problem: The arc length of the following polar graph:
r(theta)=2sin(theta) \forall theta in[0, 2pi]
What is:
4pi
Team Problem: List out the first 6 terms of the sequence:
{a_n} = {1/n^2}
What is:
{a_n} = {1, 1/4, 1/9, 1/16, 1/25, 1/36, ...}
Team Problem: Find whether the limit converges or diverges. If it converges, say what it converges to.
lim_{n \to infty}e^(-1/n)
What is 1?
Team Problem: Find the sum of the following series:
\sum_{n=1}^{infty}1/(2n+n^2)
What is:
3/4
Team Problem: Determine whether the series converges or diverges.
sum_{n=1}^{infty}1^n
What is:
Diverges
Team Problem: The area of the following polar portion:
r(theta)=3-cos^3(theta) \forall theta \in [0,pi]
Team Problem: Find a limit L if the sequence converges, otherwise find if it diverges.
{a_n} = {((-1)^(n+1)*n)/(sqrt(n^3 + 1))}
What is 0?
Team Problem: The formal value given to the limit of the sequence:
{a_n} = {(1+x/n)^n}
What is:
e^x
Team Problem: Determine what value, if any, the series converges to. Otherwise, say the series diverges.
\sum_{n=1}^{infty} ln(n/(n^2-1))
What is:
Diverges
Team Problem: Determine whether the series converges or diverges. If it converges, find it's value.
sum_{n=1}^{infty}(1/(n^2+n) - 1/2^(n+1))
What is:
Converges to 0.
Challenge Problem: Show me the reasoning why the surface area of a polar graph rotated around the x-axis is:
S = \int_\alpha^\beta2pirsinthetasqrt(r^2+r'^2)d\theta
What is:
Same a Parametric Equations one, but now y=rsintheta . We use y as the radius of a small dA for a strip at some x value.
Challenge Problem: What mathematical induction is.
What is:
A way of proving mathematical statements by:
1) Showing a base case is meets the criteria
2) Showing that cases continue to meet the criteria for general terms
Challenge Problem: The formal definition of a limit for a sequence a_n.
What is:
text(For ) \epsilon > 0 text( there is some ) N \text( such that if ) n > N, |a_n - L| < \epsilon
Challenge Problem: Give me the value of the series (NOTE: you cannot use our ways of solving series to figure it out. You either know it or you don't).
\sum_{n=1}^{infty}1/n^2
What is:
pi^2/6
(Look up: Basel Problem)
Challenge Problem: Turn the following fraction into a series. Assume |x| < 1.
1/(1-x^2)
(Hint: Use the definition of a geometric series, but now backwards)
What is:
sum_{n=0}^{infty}x^(2n)
(Note: This idea of using fractions to series will be very important later in the unit!)