SOLUTION MTH 142 Practice Problems for Exam 2 - Spring 2000

Last changed: **March 30**
(a more complete answer No.15 was added since March 12)

NOTE: The last line of answer 10 was modified.
Also, please correct the following images:

problems 4,5,6,7,8,9, the Riemann sums should begin at j = 0

problem 12 b, change a_{0} = Pi/2

- a) Take in to get
- a) Direct calculation gives the polynomial
- a) The radius of convergence is given by
- a) By taking sections perpendicular to the axis of rotation, we get ``washers''. At the tickmark the washer has inner radius , outer radius , and thickness .
- a) By taking sections perpendicular to the axis of rotation, we get ``disks''. At the tickmark the radius and thickness . The Riemann sum that approximates the volumen is
- a)
- a)
- A cross-section of the cone (shown in the figure) is bounded by the lines and .
- a) A sketch of the dam is shown in the figure.
- The following table is helpful:
- a) The Taylor polynomial of degree 2 of f(x) is
- a)
- Solution 1: Note that behaves like when is large, so we suspect that the series diverges. The following inequalities are clearly valid:
- Solution 1: Note that . Therefore the series diverges.
- Solution 1:
Solution 2: Since the series is of the alternating (sign) type. To apply Leibnitz test we must first check two things. The first is,

and this is clearly true as each new term on the right hand side is obtained from the previous one upon multiplication by which is a number less than 1. Also, we must check that

One can see that the above statement is true once the limit is rewritten in this form:

(note that is less than 1, so raising this to a large power produces a small number). Because the two conditions to apply the test are satisfied, we have that, by Leibnitz test, the series is convergent.

This gives the answer in less time than actually calculating the coefficients one by one (which is ok too).

b) Replace the function by the first few terms of the series to get

b)

From the plot above we see that the gap between f ( x ) and p ( x )
is the largest at the endpoint x = 1.2.

Therefore the maximum error in the interval (0.7,1.2) is E = f ( 1.2 )
- P (1.2 ) = 0.20911

b) The interval

The Riemann sum that approximates the volumen is

The volume is obtained by taking limit as .
We have,

b) The volume is obtained by taking limit as .
We have,

b)

b)

c)

Introduce tick marks in the -axis. The slab at height is a disk with radius and thickness , so its volume is , and its weight is . The work involved in raising the slab a distance of to the top of the cone is

The total work is approximated by

The exact work is given by

Note that the equation of the right hand, non-horizontal side is . Introduce tick marks , in the y axis. At height , the slab has area , and the pressure at this height is . Therefore the force on the slab is

The total force is approximated by

The exact value of the total force is obtained by taking the limit
of the sum as :

day | amount before 24hrs are up |

Jan 2 | |

Jan 3 | |

Jan 4 | |

Jan 20 |

Then, the amount right before noon on January 20th is

so the absolute value of the error in approximating f(0.5) by P(0.5) is

b) The function

is decreasing on the interval (0,0.5), so it attains its maximum in
that interval at t=0. The maximum is

The error bound when approximating f(0.5) by P(0.5) is

b) we have that

For details, we calculate step by step below.

Therefore, the Fourier polynomial we seek is .

The term in the center simplifies to . Since diverges, so does . This is our original series with only the first term changed, so our original series also diverges.

Solution 2: We use the integral test. Set
for .
Now
Therefore, the series diverges too.

Solution 2: Using comparison, we have

Since is divergent, so is our original series.