MATHEMATICS

1:

 

Problem 2:

 

Find the antiderivative

 

Problem 3:

 

Find the surface area when the line segment A in the figure below is rotated about the lines:

 

(a) y = 1

 

(b) x = -2

(a)       The line segment follows the function f (x) = x + 1. The integral for the surface area of revolution is:

(a)       The line segment follows the function f (y) = y – 1. The integral for the surface area of revolution is:

Problem 4:

 

A sphere of radius 2 foot is filled with 2000 pounds of liquid. How much work is done pumping the liquid to a point 5 feet above the top of the sphere?

Problem 5:

 

Find the integral

 

Problem 6:

 

Find the integral

 

Problem 7:

 

Use the definition of an improper integral to evaluate the given integral:

 

 

Problem 8:

 

Find the indefinite integrals and evaluate the definite integrals. A particular change of variable is suggested.

Problem 9:

 

Evaluate the integral

 

Problem 10:

 

Evaluate the integral

 

Problem 11:

 

Evaluate the integral

 

 

Problem 12:

 

Show that if m and n are integers, then . (Consider m = n and

m ≠ n.)

 

Problem 13:

 

Use derivatives to determine whether the sequence below is monotonic increasing, monotonic decreasing, or neither:

 

 

 

Problem 14:

 

Each special washing of a pair of overalls removes 80% of the radioactive particles attached to the overalls. Represent, as a sequence of numbers, the percent of the original radioactive particles that remain after each washing.

Problem 15:

 

Calculate the value of the partial sum for n = 4 and n = 5, and find a formula for sn. (The patterns may be more obvious if you do not simplify each term.)

Problem 16:

 

In the proof of the Integral Test, we derived an inequality bounding the values of the partial sums  between the values of two integrals:

 

Problem 17:

 

Use any of the methods learned from this MATH141 class to determine whether the given series converge or diverge. Give reasons for your answers.

 

Problem 18:

 

Determine whether the given series Converge Absolutely, Converge Conditionally, or Diverge, and give reasons for your conclusions.

 

Problem 19:

 

Find the interval of convergence for the series below. For x in the interval of convergence, find the sum of the series as a function of x. (Hint: You know how to find the sum of a geometric series.)

 

Problem 20:

 

Represent the integral as a numerical series:

 

 

Use the series representation of these functions to calculate the limits.

 

Determine how many terms of the Taylor series for f(x) are needed to approximate f to within the specified error on the given interval. (For each function use the center c = 0.)

 

 within 0.001 on [-1, 4].

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