Limits, I.F., L'Hopital's Rule
When and how is L'Hopital's Rule used?
L'Hopital's Rule is used when taking the limit of something in indeterminate form. You find the derivative of the numerator and denominator (independently).
What is the population growth equation, and what do the different variables mean?
P(t) = P(0)ekt
P(t) = population at time t
P(0) = initial population at t=0
k = relative growth rate (constant)
t = time
Name the 4 main integration techniques.
U-substitution, integration by parts, partial fraction decomposition, and trig substitution.
Series from n=0 to +inf [9(1/3)n]
Converges
Convert to cartesian: r=2cos(theta)
Convert to polar: x2+y2=9
(x-1)2+y2=1 (circle centered at (1,0), radius of 1)
r=3
If f(1) = 4, g(4) = 2, h= g◦f, and h′(1) = 3, then find (h-1)′(2).
(h-1)'(2) = 1/3
Show that the function y= sin(2x) is a solution to the differential equation d4y/dx4−16y= 0
Define improper type I and type II.
Type 1: either one or both bound of the integral is +/- infinity.
Type 2: there is a discontinuity within the bounds of the integral
Write the MacLaurin series for sin(x)
(-1)n x2n+1/(2n+1)!
Find the arc length of x=cos(t) and y=sin(t), [0,pi]
L = pi
List all 7 indeterminate forms.
What is... 0/0, +-inf/+-inf, 0 x inf, inf - inf, 00, inf0, 1inf
Consider the curves y= sin(x) and y= cos(x), 0≤x≤pi/2. Find the area between the curves in quadrant 1.
2(root 2) - 2
Solve the integral [1/x2+5x-14]dx
= 1/9 ln(|x-2 / x+7|) + C
Find the radius and interval of convergence for:
the series from n=1 to inf [(-1)n(x+1)n / n2]
interval [-2,0]
Given x=t3-t and y=t2+1, find dy/dx
dy/dx = 2t/(3t2-1)
Indeterminate difference. The limit equals -infinity.
dy/dx= y2xsin(x) + y2x2 ; y(0) = 1
-1/y = -xcos(x) - sin(x) + x3/3 - 1
Integral from 1 to infinity of [1/(x(lnx)2)]dx.
Equals +infinity, so the integral diverges.
Determine if the series is convergent, divergent, or absolutely convergent.
N=1 to +infinity [n!/(23nn4]
Divergent
Given x=t2+2t and y=t3-t, find the equation of the tangent line at t=1
y = 1/2(x-3)
Solve the limit as x --> 1+ [x1/(x-1)]
Indeterminate power. The limit equals e.
The area under the curve y = sin(x); 0 < x < pi/2 is rotated about the x-axis. Find the volume of the resulting solid of revolution.
= pi2/4
Use the Comparison Test to determine if the following integral converges or diverges.
Integral from 1 to +inf [cos2(x) / x2]dx
Converges
Find the radius and interval of convergence of the series
n=1 to infinity [(x-3)n/(n)(2n)]
R = 2
[1,5)
Find the area inside one petal of r=sin(3theta)
A = pi/12