horizontal tangents are at the points where y (not r) is a maximum. You
need, therefore, to maximize
Use the calculator to find that when θ = 0.7854. Therefore, y =
1.414,
so the desired rectangle has dimensions 6 × 2.828.
(b) Since the polar formula for the area is , the area of R (enclosed
by r) is , which is 14.137.
(a) Use the Ratio Test:
The radius of convergence is 1. At the endpoint .
Since
this series converges by the Alternating Series Test. Thus
converges for positive values 0 < x ≤ 1.
(b) Because satisfies the Alternating Series Test,
the error in approximation after n terms is less than the magnitude of the
next term.
The calculator shows that terms.
(c) is a negative series. Therefore the
error will be larger than the magnitude of the first omitted term, and thus
less accurate than the estimate for f(0.5).
