A rose garden is formed by joining a rectangle and semicircular, as shown below. The rectangle is 24ft long and 15 ft wide. If t
he gardener wants to build a fence around the garden, how many feet of fence are required? (Use the value 3.14 for pi, and do not round your answer. Be sure to include the correct unit in your answer).
1 answer:
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Answer:
a) (x³ -x² +x +2) +2/(x+1)
b) (x² +2x -5) +6/(x+3)
Step-by-step explanation:
Polynomial long division is virtually identical to numerical long division, except that the quotient term does not require any guessing. It is simply the ratio of the leading terms of the dividend and divisor. As with numerical long division, the product of the quotient term and the divisor is subtracted from the dividend to form the new dividend for the next step.
The process stops when the dividend is of lower degree than the divisor.
In part (a), you need to make sure the dividend expression has all of the powers of x present. This means terms 0x³ and 0x² must be added as placeholders in the given dividend. They will become important as the work progresses.
Assuming the series is
The series will converge if
We have
So the series will certainly converge if
, but we also need to check the endpoints of the interval.
If
, then the series is a scaled harmonic series, which we know diverges.
On the other hand, if
, by the alternating series test we can show that the series converges, since
and is strictly decreasing.
So, the interval of convergence for the series is
.
The series will converge if
We have
So the series will certainly converge if
If
On the other hand, if
and is strictly decreasing.
So, the interval of convergence for the series is