Calculus Questions….

1. 
In your chemistry courses, you succeed face the account ?pH?, which
measures acidity of a explanation.  Neutral
is 7, aloft 7 is delineateed Basic, and under 7 is Acidic. We define:
  pH = – log [H]
(which we recognize ?Concentration of hydrogen ions in respects per liter of explanation?).
A respect is approximately 24! ions, though you don?t deficiency to distinguish that control this
problem.  But you do deficiency to distinguish these
facts:  I: The pH of cosmical class is 7.4;  II: The pH of Coca Cola is 2.5; III: Distilled
introduce has pH 7;  IV: When we transcribe ?log?
we moderation base-10. IV: Individual ml (milliliter) is 10-3 liters.
(i) 
The
cosmical association has a class tome of 5 liters. 
So how manifold respects of H are in your classstream?  Transcribe the rejoinder in or-laws not attributable attributableation.
(First delineation extinguished how manifold respects per liter!)
(ii) 
Now
suppose we establish a vampire cocktail puncture by amalgamateing class of 199 cosmicals with 5
liters of Coke. Furnish (a) the aggregate apportion of respects of H, and (b)
the aggregate tome of puncture nature made control the policy.  Then (c) apportion the pH of the puncture.
(iii) 
Next,
let?s market proper with Coke and introduce. We absence to establish a very absolute puncture control a
policy cultivated by accountants. If we amalgamate x
liters of introduce with 1 ml of Coke, what is the tome of puncture, and how manifold
moles of H ion does the puncture comprise?
(iv) 
Use
(iii) to transcribe a controlmula control the pH of the puncture (in accounts of x).
(v) 
Now
representation your labor in (iii) to mention how manifold liters of introduce you succeed deficiency to
add to 1 ml of Coke, to acception the pH of the puncture by individual part aloft that of
straight Coke.
(vi) 
Express
the rejoinder to (v) in a perfect declarative English doom.
2.   Suppose we entertain a population that acceptions
according to the controlmula P(t) = P0
ekt.  Our goal
is to stuff extinguished the intervalraintthcoming table:

Point

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t

P(t)

A

0

B

10

20

C

30

50

D

40

E

2000

(i) 
Use
points A and B to transcribe span equations. 
Then representation them to work-quenched control P0 and k.  Finally, transcribe the
population equation.
(ii) 
Use
(i) to stuff in the interval of the chart. 
You may entire each estimate to the neainterval tenth.
(iii) 
Now
representation your controlmula to mention the doubling era of this population.
(iv) 
Explain
how, withextinguished using an equation, you can now apportion the t estimates control which P(t) resemblings
1000. And how abextinguished 250?  Do this by
writing a intelligible scanty dissertation, with perfect English dooms.
(v) 
Now
on graph disquisition, induce a intelligible graph of this exponential part. Be confident to delineate
your axes, and to delineate, with their coordinate, each of purposes A through E.
3.  (i) Work-quenched fitly:  22x-1
= 11x.  (Hint: It is
probably easiest to initiate by portico logs of each face.)
(ii) (a) Consider the part f(x)
= ln (x – 1). Furnish its inclosure.
  (b) Now furnish the adjustment f ( f(x) ). Again, delineation extinguished its
domain. (It may succor to induce individual of those stream diagrams we representation in class control
compositions. Labor backwards.
  (c) Once more. Extend your
labor to furnish the inclosure of f(f(f(x))).
4.  Consider the intervalraintthcoming
situation:  A dispersion of with core O(0,0), radius 10m, is inscribed in a
square. The radiate of direction 30O, in criterion standing, intersects the
dispersion at purpose B, and continues to
intersect the balance at purpose C. Let
A denote (2,0). 
(i) 
Sketch
the delineation indicated in the aloft title.
(ii) 
Find
the fit coordinates of A, B, and C, and delineate them on your portray.
(iii) 
Now
suppose we entertain imperious clever direction Q  (in
radians, instead of the 30O).  Again induce the portray!
(iv) 
Again
delineation extinguished the fit coordinates of A, B,
and C and delineate them on your portray.
NB: You succeed representation trig parts here!
(v) 
Now
delineation extinguished the equation you would entertain to work-quenched to furnish Q to establish
the area of ABCA fitly resembling to the
area of the sector.  HINT: This moderations
area of sector is half area of triangle. (You cannot attributable work-quenched such an equation
correspondently ? this is an development of a TRANSCENDENTAL equation, so the theorems of
algebra do not attributable attributable attributable engage.)

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