Studied by 2 people
11.8, 11.9, 11.10, 10.1, 10.2, 10.3, 10.4
What is the MacLaurin series for 1/(1-x) and what x does it converge for?
Σn=0 ∞ (x^n) for |x|<1
What is the MacLaurin series for e^x and what x does it converge for?
Σn=0 ∞ (x^n/n!) for all x
What is the MacLaurin series for sinx and what x does it converge for?
Σn=0 ∞ [(-1)^n * x^(2n+1)/(2n+1)!] for all x
What is the MacLaurin series for cosx and what x does it converge for?
Σn=0 ∞ [(-1)^n * x^(2n)/(2n)!] for all x
What is the MacLaurin series for arctanx and what x does it converge for?
Σn=0 ∞ [(-1)^n * x^(2n+1)/(2n+1)] for -1<=x<=1
What is the MacLaurin series for ln(1+x) and what x does it converge for?
Σn=1 ∞ [(-1)^(n-1) * (x^n)/n] for -1<x<=1
What is the form of a power series and its radius of convergence R?
Σ Cn (x-a)^n for |x-a| < R where Cn is a coefficient
What is Cn if f has a power series representation at a (f(x) = Σ Cn (x-a)^n)?
Cn = f(n)(a)/n!
What is the form of a Taylor series and its radius of convergence R?
Σn=0∞ f(n)(a)/n! * (x-a)^n for |x-a| < R
What is the form of a MacLaurin series?
Σn=0∞ f(n)(0)/n! * (x)^n or a Taylor series where a=0
Describe how to represent a function f(x) as a geometric power series
rewrite f(x) in the form 1/(1-x); may have to factor, derive, or integrate to do so
rewrite 1/(1-x) as Σn=0∞ x^n
What are the only 3 convergence outcomes of a power series?
Converges at x=a
Converges for all x
Converges for |x-a| < R and diverges for |x-a| > R
What is the radius of convergence R if a power series converges at only one point (x=a)?
R = 0
What is the radius of convergence R if a power series converges for all x?
R = infinity
What is the radius of convergence R if a power series converges for a finite interval of x?
Can find R in the form |x-a| < R or |x-a| > R
What is the interval of convergence I if a power series converges at only one point (x=a)?
I = {a}
What is the interval of convergence I if a power series converges for all x?
I = (-infinity, infinity)
What is the interval of convergence I if a power series converges for a finite interval of x?
a+R, a-R
* must plug x = a+R and x = a - R into the series to determine if it converges at those points
Describe how to find a Cartesian equation in the form y=f(x) given the parametric equations x=f(t) and y=g(t)
eliminate the parameter t through algebraic manipulation
strategies include:
solving for t in one equation and plugging it into the other
using trig identities such as sin²t + cos²t = 1
taking the ln of an equation containing e^t
What is the formula for dy/dx given the parametric equations x=f(t) and y=g(t)?
dy/dx = (dy/dt)/(dx/dt)
What is the formula for arc length on a<=t<=b given the parametric equations x=f(t) and y=g(t)?
L = ∫ab √[(dy/dt)²+{dx/dt)²] dt
What is the formula for arc length on a<=x<=b given the equation y=f(x)?
L = ∫ab √[1+(dy/dx)²] dt
What is the formula for arc length on a<=x<=b given the equation x=g(y)?
L = ∫ab √[1+(dx/dy)²] dt
Name the formulas for converting polar coordinates (r, θ) to Cartesian coordinates (x, y)
x = rcos(θ)
y = rsin(θ)
Name the formulas for converting Cartesian coordinates (x, y) to polar coordinates (r, θ)
r²=x²+y² or r=sqrt(x²+y²)
tanθ = y/x or θ = arctan(y/x)
what is the area of the region under a polar curve?
A = ∫ab ½ r² dθ where a<=θ<=b
what is the area of the region inside the polar curve r1 and outside the polar curve r2?
A = ∫ab ½ (r1²-r2)² dθ where a<=θ<=b
what is the length of a polar curve?
L = ∫ab √[r² + (dr/dθ)²] dθ where a<=θ<=b
how do you find the values of θ where there is a horizontal tangent given r=f(θ)?
plug r into y=rsinθ or simply use dy/dθ = dr/dθ * sinθ + rcosθ
find the values of θ in the domain that satisfy dy/dθ = 0 and dx/dθ ≠ 0
how do you find the values of θ where there is a vertical tangent given r=f(θ)?
plug r into x=rcosθ or simply use dx/dθ = dr/dθ * cosθ - rsinθ
find the values of θ in the domain that satisfy dx/dθ = 0 and dy/dθ ≠ 0
how do you know if there is a vertical or horizontal tangent at values of θ where dx/dθ and dy/dθ both = 0?
evaluate limθ→the value (dy/dx)
if it goes to 0, there’s a horizontal tangent
if it goes to infinity, there’s a vertical tangent
how do you find dy/dx given r=f(θ)?
dy/dx = (dy/dθ)/(dx/dθ)
or
dy/dx = (dr/dθ * sinθ + rcosθ)/(dr/dθ * cosθ - rsinθ)
what is dy/dθ given r=f(θ)?
dy/dθ = dr/dθ * sinθ + rcosθ
what is dx/dθ given r=f(θ)?
dx/dθ = dr/dθ * cosθ - rsinθ
