Math 152 Project 2

Derivatives as Rates of Change

You need to use Maple to do this project. To run maple, click start-programs-Maple V Release 1 - Maple V release 1.

Be sure that you type carefully and include semicolons at the end of each line!.

where t is the time (measured in minutes) elapsed after having a drink and ( a,b ) are constants depending on the amount of alcohol ingested and bodily characteristics of the patient.

1. Suppose the contents of alcohol measured at times t₁= 5 minutes and t₂=15 minutes are A(t₁)= 0.0003 and
   A(t₂)= 0.0008, respectively. Compute the constants a and b.

   To do this task using Maple you type:
   solve({5*a/(1+b*5^2)=.0003,15*a/(1+b*15^2)=.0008},{a,b});

   Maple will give you a value for a and b.

   To set those values as a and b in Maple type:
   a: = #1; b:=#2;
   Where #1 and #2 are the values maple gave you for a and b respectively




2. Use Maple to plot the function f over the interval from 0 to 240 minutes.

To do this you must first define the function A by typing:
A:=t->a*t/(1+b*t^2);
where a and b are the values you got in 1.

Then plot by typing:
plot(A, 0..240);
Maple gives you a plot of the function.

3. One possible dosage scheme for Soberol is to administer it to patients at the time ( t₀ ) when blood alcohol content is at its maximum value. Calculate this time using the values for a and b you found in 1.

You can do this using Maple by calculating the derivative and solving for the t where it equals 0. To calculate the derivative of the function and name it Aprime you type:
Aprime :=D(A)


You can see a simplified version of the derivative by typing
simplify(Aprime);



Then find the time for which this is 0 by using the solve function, only this time we are only solving one equation for one variable:

solve(Aprime(t)=0,t);

4. Calculate A(t₀), the maximum blood alcohol content.
   

   Maple does this if you type:
   A(tmax);
   Where tmax is the value you calculated in step 3.

5. Suppose that for another patient the contents of alcohol measured at times t₁= 5 minutes and t₂=15 minutes are A(t₁)= 0.0004 and A(t₂ )= 0.001, respectively. Calculate the corresponding constants a and b. Use the same procedure as in step 1.
6. Suppose this time we want to administer Soberol at the time when the blood alcohol content is decreasing the fastest. To find this time, calculate the first derivative A'(t)and the second derivative A''(t). Then use the Maple's solve function to find the time when the first derivative is the smallest. This is a lot like step 3.
7. Plot the function A(t), its first derivative A'(t) and its second derivative A''(t) over the interval from 0 to 240.
8. On each plot label any critical points and inflection points. Do these three plots seem to make sense together?