Calculus 2 Final (power series, taylor/maclaurin series, polar coordinates

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 C_n (x-a)^n for |x-a| < R where C_n is a coefficient

What is C_n if f has a power series representation at a (f(x) = Σ C_n (x-a)^n)?

C_n = f⁽ⁿ⁾(a)/n!

What is the form of a Taylor series and its radius of convergence R?

Σ_n=0∞ f⁽ⁿ⁾(a)/n! * (x-a)^n for |x-a| < R

What is the form of a MacLaurin series?

Σ_n=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

1. rewrite f(x) in the form 1/(1-x); may have to factor, derive, or integrate to do so

2. rewrite 1/(1-x) as Σ_n=0∞ x^n

What are the only 3 convergence outcomes of a power series?

1. Converges at x=a

2. Converges for all x

3. 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 = ∫_a^b √[(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 = ∫_a^b √[1+(dy/dx)²] dt

What is the formula for arc length on a<=x<=b given the equation x=g(y)?

L = ∫_a^b √[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 = ∫_a^b 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 = ∫_a^b ½ (r₁²-r₂)² dθ where a<=θ<=b

what is the length of a polar curve?

L = ∫_a^b √[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