Math 161

Extra Credit Problems

Spring/Fall 2000

 

  1. Give at least two examples of even functions such that for any value of , the integral
  2. Let
be a differentiable function. Show or give a counterexample to the following statements:
  1. If is odd, then is even.
  2. If is even, then is odd.
  3. If is even, then any antiderivative of is odd.
  4. If is odd, then any antiderivative of
is even.

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3. Show that any regular pyramid (by regular, I mean that the shape of any cross-section is the same shape as the bottom, only proportionally smaller) has volume given by the formula:

V=(1/3)Ah

 

Where A is the area of the base and h is the height of the pyramid.

 

 

 

 

 

 

                                         

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4. Find a formula for the volume for a 4-dimensional sphere (if you are ambitious, see if you can extend this to find a formula for the volume of an n-dimensional sphere).


 

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6. Consider the following picture:

 

 

                                                                                 (0,a)

 

 


                                       y=mx+a

 

 


                                                                                                    

                                                                                               (a,0)

 

 

 

 

 

Find the volume obtained when the shaded triangle is spun about the line y=mx+a. What happens as m goes to infinity?

 

 

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7. Consider the function . Show that when this function (for x1) is revolved about the x-axis, the volume is finite, but the surface area is infinite. So we can “fill” this thing with a bucket of paint, but we cannot paint the side? Resolve this “paradox”.

 

8. Find a function f(x) and an interval [a,b] such that for some n, the trapezoid and/or the midpoint rule is a genuinely better approximation than Simpson’s rule for the integral or show that this cannot be done.

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9. Show that the series  converges.

10. Find .

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11. Consider the infinite series . Show that if I give you any real number, then you can rearrange this series in such a way that it adds up to my given number.

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12. Give an example of an alternating series such that , but the series does not converge. Prove your answer.

 

13. Consider the series , does this converge? Does it converge absolutely?

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14. For what values of  does the series converge?

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15. Let , show that:

 

            (hint: use the power series representations for the exponential, sine and cosine).

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16. Give and example of an infinitely differentiable function that is not equal to its Taylor series on any interval.

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17. What is wrong with the following argument? Consider Since we have an  indeterminate form, we apply L’Hospital’s Rule and obtain  which does not exist.

18. A RATIO/ROOT TEST FOR SEQUENCES: Prove that if   is a sequence and  (or respectively ) then