Infinite Sequences and Series (BC Only)

$p < 0$

$p > 1$

$0 < p \leq 1$

$p = 0$

A circle centered at the pole with radius 4

A line passing through the pole at an angle of 4 radians

An ellipse centered at the pole with major axis length of 4

A spiral that passes through points that are distance 4 from the pole

A larger error bound guarantees a more accurate approximation.

The accuracy of the approximation solely depends on the degree of the Taylor polynomial.

A smaller error bound guarantees a more accurate approximation.

The error bound does not provide information about the accuracy of the approximation.

$\frac{(0.5)^4}{4!} \times e^0$

$\frac{(0.5)^5}{5!} \times e^{0.5}$

$\frac{(0.5)^6}{6!} \times e^{0.5}$

$\frac{(0.5)^3}{3!} \times e^0$

The convergence radius of the Taylor series centered at a and its relation to the interval from a to c.

The absolute maxima and minima values of f itself within the neighborhood of a and c.

The exact bounds on f's $(n + l)th$ derivative over the interval of interest.

The values of all derivatives of f up to the nth degree specifically at points c and a, but nothing else.

When revolving region around axis lies parallel plane containing area making discs washers impractical due thin cylindrical slices created

If revolved region contains hole center yielding toroidal shape where internal external radii need considered computation volumes

Should object symmetry rotational exists about any arbitrary angle making sectorial cuts necessary evaluate component forces shapes formed

When revolution takes place about horizontal vertical axis crossing through section area allowing simple radial measurements disks washers application simpler

$\frac{6}{3!}(3-2)^3$

$\frac{6}{5!}(3-2)^5$

$\frac{6}{4!}(3-2)^4$

$\frac{30}{5!}(3-2)^5$

$\frac{6}{5!} \cdot (0.5)^5$

2. $\frac{6}{3!} \cdot (0.5)^3$

3. $\frac{6}{4!} \cdot (1.5)^4$

1. $\frac{6}{4!} \cdot (0.5)^4$

Magnitude gap distance between b and a multiplied by the function's maximum curvature essentially restrains potential error scope tunings.

Utilizing the exact same appropriate degree relationship across all applicable derivations effectively caps the maximum error extending possibilities.

Largest value of successive derivative function between b and a governs potential error in estimations.

Our chosen derivative order plus one evaluated somewhere within interval b-a sets limits upon error sizing possibilities.

$(\pi/4)^5/5!$

$(\pi/4)^5/4!$

$(\pi/4)^5/4!$

$(\pi/4)^4/4!$