Math Is Fun Forum

http://mathplotter.lawrenceville.org/mathplotter/mathPage/index.htm

This pretty much has you set on any number of advanced math through higher level questions.

Hope it's Useful to everyone!!!

Cheers, Sven

There are 11 problems and each student was given 1 and a half hours to do it.  Ok, show all of your work ( aka post it and your answers ) and how long it took you to do them. Remember that this was given to Calc 1 students who haven't learned integration by parts yet.

1. A tank on the wing of a jet is formed by revolving the region bounded by the graph of

and the x-axis.  The axis of rotation is the x-axis. Find the tank's volume, given that x and y are measured in meters.

2. Set up the integral a, b, and c using the Shell Method, solving is not necessary. Woot!

     a. Revolving about the x-axis.
     b. Revolving about the y-axis.
     c. Revolving about y=1

3. Another JUST SETTING UP (Don't solve) using both the Washer and the Shell method.

4. Again just setting up (I don't think I need to say that it's not necessary to solve but I will again ) using any method.

5. Match each integral with the solid whose volume it represents.
Choices:     a. Right cylinder     b. Sphere     c. Cone

     i.

     ii.

     iii.

I know it's redundant to have

but that's how it's written.

6. Find the area of the zone of a sphere formed by revolving the graph of

about the y-axis.

7. Set up and evaluate with a calculator surface area generated by revolving the curve

about the x-axis on the interval [1,2]

8.A spring of natural length 10 in. stretches 1.5in. under a weight of 8lbs. Find the work done in stretching the spring:
     a. From it's natural length to a length of 14in.
     b. From a length of 11in. to a length of 13in.

9. A freight elevator weighing 3000lbs is supported by a 12ft. long cable that weighs 14lbs per foot. Approximate the work required to life the elevator 9ft. by winding the cable onto a winch.

10. A force of 25lbs is required to compress a spring of natural length .8tf to a length of .75ft. Find the work done in compressing the spring from its natural length to a length of .7ft.

11. A vertical cylindrical tank of diameter 3ft and a height 6 ft is full of water. Find the work required to pump all the water over the top of the tank.

When we did this review Monday we were scored out of 10 with the question we scored the least points on omitted. See how well you do!!
Good luck and remember to time yourself.

Prove that 2^(2n+1) + 3^(2n+1) is divisible by 5.

I completed the trivial first step and when n=1 the equation proves true.

So far I have:

f(n)=2^(2n+1) + 3^(2n+1), and since n=k and f(n)=f(k) then f(k) is divisible by 5 (or f(k)=5x, where x is an integer)

then:

2^(2k+1) + 3^(2k+1) for the second step

Inductive step:

2^(2(k+1)+1) + 3^(2(k+1)+1) = 2^(2k+3) + 3^(2k+3) = (4)2^(k+1)+(9)3^(k+1)

That's as far as I can get. I don't know what the next step is. PLEASE HELP!!!!

Thanks.