Math 182                                                                     Name:                                                             

Test 3                                                                           April 19, 2013

                        You must show all work to receive credit!

Problems that ask for “justification” should be answered in complete English sentences

 

1.    For the differential equation  with initial condition y(0) =1:

a.      

8 pts

 
 Use Euler’s Method with step size ½ to approximate y(1.5).  Show all steps:

X

Y

Y’=-y+2x

Δx

Δy

0

1

-1+2(0)=-1

½

-1* ½ = -½

½

1– ½ = ½

-½ +2( ½ ) = ½

½

½ * ½  = ¼

1

½ + ¼ = ¾

-¾ +2(1)= 5/4

½

½ * 5/4 = 5/8

3/2

¾ +5/8 =11/8

 

 

 

 

 

b.     

4 pts

 
Repeat part (a), but this time use 10 steps of size 0.15 to get an improved approximation. 

You do not need to show any work – simply use your calculator program and copy the final answer.

 

1.5906

 

 

1        Verify that  satisfies the differential equation and initial condition from #1

8 pts

 
 


If ,  then  , and –y+2x =

So this satisfies the differential equation.

For the initial condition, y(0) = 3e0+2(0)-2=1

 

 

2       

10 pts

 
Find the general solution to the differential equation  

 

 

3       

6 pts each

 
A vat initially contains 300 gallons of pure water.  Sugar water containing 5 tablespoons of sugar per gallon enters the tank at a rate of 6 gallons per minute.  The vat is kept well mixed, and solution drains from the vat at the same rate of 6 gallons per minute.  Let S(t) denote the number of tablespoons of sugar after t minutes. 

a.  Set up – but do not solve -  a differential equation which models the value of S(t) in this mixing problem.

 

 

b.  Determine, with justification, the equilibrium solution for this problem. 

(Note:  the justification may be based on the wording of the problem itself or on your differential equation)

 

30-1/50 S  = 0 when S=1500. So the equilibrium value is when there are 1500 T of sugar in the vat. 

At this level, the concentration is 1500/300 = 5T/gal, so the process described would not change the amount of sugar in the tank.

4        A ball is dropped from a roof 12 feet high.  Each time it hits the ground, it bounces back to 2/3 of its previous peak height. 

a.      

3 pts

 
 Let a0 represent the initial height of the ball, and an the peak height after n bounces (so a0 =12 feet, a1 =8 feet, and so on).  What type of sequence is ?

 

Geometric

 

b.      Find an explicit formula for  

3 pts

 
 


  

 

 

c.       Find the total distance this ball travels before coming to rest.

8 pts

 
  (Hint: after the initial drop, the bounces up from the floor to the next peak height each time before falling back down) 

 

Falling down: 

Back up:  either reason that it is exactly the same, except for the initial 12 foot drop, so 36-12=24 feet, or

  Either way, the total is 36+24=60 feet.

 

 

 

 


 

5 pts each

 

 

 

 

 

b.    Determine with justification whether this sequence converges or diverges. 

 

 

 

 

 

7.     

6 pts each

 
 Determine, with justification, whether each of the following series converges or diverges.

 

Diverges by the divergence test – the sequence does not converge to 0.

 

 

 

 

 

 

For part c:  determine whether this series is converges conditionally, converges absolutely, or diverges.

 

This converges absolutely.  It can be shown to converge by the alternating series test, but then you’d still need to check if the convergence was conditional or absolute.  I’ll instead simply show that it is absolutely convergent, since we know that that implies convergences.   converges, which can be shown by the integral test, or by the result shown in class for p-series with p=4>1. 

The graph of the polar equation  is heart shaped (technically, it is called a cardioid).

8.     

8 pts

 
 Solve for the derivative  of this cardioid.

 

 

 

 

 

 

 

 

 

9.      Set up (but do not evaluate) an integral that when evaluated will give you the area inside this cardioid

8 pts

 
 


 

 

 

Extra Credit:  (5 points – answer either one of the two following)

a.       Solve the differential equation from #4 and determine how long it will take until the sugar has a concentration of 4 tablespoons per gallon

b.      Evaluate the integral from #9 to determine the exact amount of area inside the cardioid.

 

 a.

Initially S=0, so we solve for C=-1500, ie

When the concentration reaches the 4T/gal, there are 1200 T of sugar in the vat, so

 

 

b.