Theorems
This rule helps by solving limits of indeterminate forms. Original, correct spelling only!!!
L'Hôpital's rule
\intxe^xdx
e^x(x-1)+C
A spring has a constant k=6N/m, and it's already stretched 2 meters out. How much work is needed to stretch it an additional 2 meters?
36 Joules
\lim_{x->0}(1-cos(4x))/x^2
8
Series of this form are called what?
\sum_{n=0}^\inftyf^((n))(0)/{n!}x^n
Taylor series
Name this theorem. Consider a_n . If lim|a_n/b_n|=L\ne0 , then \sum_{n=1}^\inftya_n converges if and only if \sum_{n=1}^\inftyb_n converges, and vice versa for divergence.
Generalized Limit Comparison Test
\int(x+2)/{x^2-1}dx
3/2ln|x-1|-1/2ln|x+1|+C
A guy is standing on the floor while pushing against a wall with a 62.5 lbs of force. Gravity is pulling down at a rate of 9.8m/s^2 . The wall is made of concrete with density of 82 firkins per cubic inch, while the floor is made of solid uranium. How much work does the guy do after 8.2 seconds given that he never comes to office hours and he failed the first exam?
0 ft-lbs
\lim_{x->\infty}(6sin3x)/tan(6x+1)
DNE
Find the numerical answer to
\sum_{n=1}^\infty(3^n+4^n)/12^n
5/6
What are the 3 requirements of a_n by the Alternating Series Test to say that the series \sum_{n=1}^{\infty}a_n converges?
1. a_n is alternating
2. a_n is decreasing
3. \lim_{n->\infty}a_n=0
\intsin^5xcos^3xdx
sin^6(x)/6-sin^8(x)/8+C
Find the arclength of the function described by
x=2cost,y=2sint
where
t\in[0,4\pi]
8\pi
lim_{x->\infty}(1+1/x)^{2x}
e^2
Converge or diverge?
\sum_{n=1}^{\infty}(-1)^nsin^3(n)/cos^2(n)
Diverges
Name this theorem/test. \sum_{n=1}^\inftya_n and \int_1^\inftya_ndn either both converge or both diverge.
The Integral Test
\int_-1^1x^2sin(x)dx
0
Find the volume of the shape where the cross-sections perpendicular to the x-axis are circles with radius \sqrt(a^2-x^2) from x=-a to x=a where a is a constant
4/3\pia^3
\lim_{x->\infty}(2tan^3x)/(5sin^3x)
2/5
Converge or diverge?
\sum_{n=1}^\infty1/{nlnn}
Diverges
Name this theorem. If \sum|a_n| converges, then \suma_n converges.
Absolute Convergence Theorem
\intx^5\sin(x)\cos^5(x)dy
x^5\sin(x)\cos^5(x)y+C
Suppose we have a pool filled with liquid molten rock whose walls are formed by revolving ln(x) with domain 0<x<1 around the y-axis. How much work is required to drain the pool given that liquid molten rock has force density of 180{lbs}/{ft^3} and we lift the liquid rock to 1 foot above the top of the pool.
\infty ft-lbs
\lim_{x->\infty}x(\sqrt(x^2+1)-x)
1/2
What is the Taylor series expansion of
f(x)=xe^{x^2}
x+x^3+x^5/{2!}+x^7/{3!}+...=\sum_{n=0}^\inftyx^{2n+1}/{n!}