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 base10. 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) = P_{0} 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 P_{0} 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^{x1} = 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.)
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