Write Swift's Songs & approximate
What is the linear approximating polynomial of f(x) about x=c?
P1(x)=f(c)+f'(c)(x-c)
What function's first four terms in a Maclaurin Series are 1+x+x^2/2!+x^3/3!?
e^x
What the formula for Alternating Series Error bound?
|S∞-SN|≤|aN+1|?
What is the radius of convergence of a power series ∑cn(x−a)n?
The radius is the distance from x=a to the point where the series doesn't converge.
What is the quadratic approximating polynomial of f(x) about x=c?
P2(x)=f(c)+f'(c)(x-c)+(f''(c)/2!)(x-c)2
What function's first four terms in a Maclaurin Series are x-x^3/3!+x^5/5!-x^7/7!?
sinx
What is the formula of Taylor's inequality?
|Rn(x)|≤|(max[f(n+1)(z)]/(n+1)!)(x-c)n+1
In the series ∑cn(x−a)n, what does the center a represent?
The series converges for all x within radius R of that point.
According to the taylor polynomial of f(x), P2(x)=pi/4+(pi/2)(x-pi)+(x-pi)2, the value of f(x) at what x value is approximately equal to f(pi/2)?
x=pi
What function's first four terms in a Maclaurin series are 1-x+x^2-x^3?
1/(1+x)
What is the Alternating Series Error bound of the alternating series ∑∞n=1(-1)n+1/n for 3 terms?
Error is less than or equal to 1/4
For the geometric series ∑xn, identify its center, radius of convergencem and interval of convergence.
What is the exact approximation of sin2(pi/3) using a 2nd degree Taylor Polynomial centered about pi/2?
sin2(pi/2)+2cos(pi/2)(pi/3-pi/2)-(2sin(pi/2)/2!)(pi/3-pi/2)2 or anything equivalent to it.
Note: Answer of 0.726 is not accepted because a rounded decimal is not the EXACT approximation. A fractional answer like (36-pi2)/36 or expression like 1-pi2/36 is accepted.
What is the general term of the maclaurin series of arctan(x)?
((-1)nx2n+1)/(2n+1)
What is the maximum value of f''(x) on the interval [0,1] given that the error for the Maclaurin series approximation of f(1) with a degree of 1 is less than or equal to 0.1?
0.2
What is the center, radius, and interval of convergence ∑(x−2)n/3n?
The center is 2, the radius is 3, and the interval is (-1,5)
Because sin(x) is this type of function, all the degrees of each term of its Maclaurin Polynomial are odd, what is this function? On the other hand, because cos(x) is this other type of function, all the degrees of each term of its Maclaurin Polynomial are even, what is this function?
The first one is an odd symmetry function and the second is an even symmetry function.
Does the Maclaurin series (x+1)0 at converge at x=-1 and x=1? Note that radius of convergence of the Maclaurin series (x+1)0 is 1
It diverges at x=-1 and converges at x=1.
Use Taylor’s Inequality to find the smallest n so that the Maclaurin polynomial Pn(x) for sinx approximates sin(0.5) with error:
∣R(0.5)∣ ≤ 10-4
n=5
Does the series ∑(x-3)n/n converge at its endpoints, which are x=2 and x=4?
Converges at x=2 and diverges at x=4.