Calculus Questions….

1. 
In your chemistry courses, you will encounter the term ‘pH’, which
measures acidity of a solution.  Neutral
is 7, above 7 is labeled Basic, and below 7 is Acidic. We define:

  pH = – log [H⁺]
(which we read ‘Concentration of hydrogen ions in moles per liter of solution’).
A mole is approximately 24! ions, though you don’t need to know that for this
problem.  But you do need to know these
facts:  I: The pH of human blood is 7.4;  II: The pH of Coca Cola is 2.5; III: Distilled
water has pH 7;  IV: When we write ‘log’
we mean base-10. IV: One ml (milliliter) is 10^-3 liters.

(i) 
The
human body has a blood volume of 5 liters. 
So how many moles of H⁺ are in your bloodstream?  Write the answer in scientific notation.
(First figure out how many moles per liter!)

(ii) 
Now
suppose we make a vampire cocktail punch by mixing blood of 199 humans with 5
liters of Coke. Find (a) the total number of moles of H⁺, and (b)
the total volume of punch being made for the party.  Then (c) compute the pH of the punch.

(iii) 
Next,
let’s deal just with Coke and water. We want to make a very flat punch for a
party attended by accountants. If we mix x
liters of water with 1 ml of Coke, what is the volume of punch, and how many
moles of H⁺ ion does the punch contain?

(iv) 
Use
(iii) to write a formula for the pH of the punch (in terms of x).

(v) 
Now
use your work in (iii) to determine how many liters of water you will need to
add to 1 ml of Coke, to increase the pH of the punch by one unit above that of
straight Coke.

(vi) 
Express
the answer to (v) in a complete declarative English sentence.

2.   Suppose we have a population that increases
according to the formula P(t) = P₀
e^kt.  Our goal
is to fill out the following table:

Point	t	P(t)
A	0
B	10	20
C	30	50
D	40
E		2000

(i) 
Use
points A and B to write two equations. 
Then use them to solve for P₀ and k.  Finally, write the
population equation.

(ii) 
Use
(i) to fill in the rest of the chart. 
You may round each value to the nearest tenth.

(iii) 
Now
use your formula to determine the doubling time of this population.

(iv) 
Explain
how, without using an equation, you can now calculate the t values for which P(t) equals
1000. And how about 250?  Do this by
writing a clear brief essay, with complete English sentences.

(v) 
Now
on graph paper, draw a clear graph of this exponential function. Be sure to label
your axes, and to label, with their coordinate, each of points A through E.

3.  (i) Solve exactly:  22^x-1
= 11^x+1.  (Hint: It is
probably easiest to start by taking logs of each side.)

(ii) (a) Consider the function f(x)
= ln (x – 1). Find its domain.

  (b) Now find the composition f ( f(x) ). Again, figure out its
domain. (It may help to draw one of those flow diagrams we use in class for
compositions. Work backwards.

  (c) Once more. Extend your
work to find the domain of f(f(f(x))).

4.  Consider the following
situation:  A circle of with center O(0,0), radius 10m, is inscribed in a
square. The ray of angle 30^O, in standard position, intersects the
circle at point B, and continues to
intersect the square at point C. Let
A denote (2,0). 

(i) 
Sketch
the figure indicated in the above description.

(ii) 
Find
the exact coordinates of A, B, and C, and label them on your sketch.

(iii) 
Now
suppose we have arbitrary acute angle Q  (in
radians, instead of the 30^O).  Again draw the sketch!

(iv) 
Again
figure out the exact coordinates of A, B,
and C and label them on your sketch.
NB: You will use trig functions here!

(v) 
Now
figure out the equation you would have to solve to find Q to make
the area of ABCA exactly equal to the
area of the sector.  HINT: This means
area of sector is half area of triangle. (You cannot solve such an equation
exactly – this is an example of a TRANSCENDENTAL equation, so the theorems of
algebra do not apply.)