MATHEMATICAL PROBLEM PAGE Directed by EDWARD L. HAENISCH Montana State College, Bozeman, Montana
T
HE PAGE for this month will be devoted to a review of the simple essentials of calculus. The formulas for the most used diierentiations and integrations are assembled here for convenience. For their derivation and application, the reader is referred to any standard text or the references quoted below..
(c) Y = (x' (d) = xP (e)
(a)
(Uand v are functions oft) =
2. y
= X"
dy
dx
=
f (4~"
(4
,,&."-'
(4
j- (+- A)
l:$ ds
(i +
dx
$ = k du -;-
4. Y = ku
+st) ax
"
1.5
Q-e!dx + z du
3. y = , ' + "
y = log x
(b) j-=e=dx
z =o
constant
+
2. Carry out the integrations:
FORMULAS OF DIFFERENTIATION
1. y
+ x) + 2)* + 2%+ 1
(b) Y = X(X'
2)
(8)
3. Work = Sw. Calculate the amount of work done when 0.2 mole of a perfect gas (po = nRT) expands isothermallyat a temperature of 25'C. from a pressure of 10 atm. to a pressure of 2 atm. Express the answer in calories, cm.= atm., and joules. 4. The approximate form of the Clausius-Clapeyrou equation
a' A --T
IS-=-
2.
~
M
~
=
-f - s"ds =
~
xr+t
X
+
+c
4
5.
except for n = -1.
~
SoLmloNs To
c
+
~ k = ~ + ~
0 1
XI
3.181 3.158 3.159 3.180 3.152 3.182 3.152 3.158 3.162 3.161 3.155
2
More complicated integrations can be carried out by references to tables such as those in the "Handbook" or Peirce's "Tables."
3 4
5 8
REFERENCES
7
DANIELS: Chapters VII, VIII. XI. X N .
8
M ~ u o n : Chapters I. IV.
9
10
PROBLES
Au.
1. Find dy -: dx (a) y = s-' 5xt 3%' 2 *The symbol. In, is used for logarithms to the base "e" while "log" refers to logarithms to base 10. Thus, 1x1x = 2.303 log x.
-
+
+ +
--
3.159
0.002 0.001 0.000 0.001 0.007 0.003 0.007 0.001 0.003 0.002 0.004
-
Z(d)
average deviation of a single temp. = a-ge
144
lssm
PROBLEMS IN BxBuumy
1. (a) 1211.1 0.7641 f 1.12 = 1213.0 (b) 7516 X 1.31 X 0.2954 = 2.91 (c) (1.276 X 0.00056) - (1.2 X 0.0128) = 0.0019 2. Time Temperature Deviation (d)
J (udu + d v + +au) = J u d u + ~ a l v + J u n t w
6. c $ = I n 4
Integrate this equation assuming AH,
+
C
3. J e = ~ n x + c
'
+
FORMULAS OF INTEGRATION
1.
R'P
(the mold beat of vaporization) to be a constant. The vapor pressure of benzene at 40.O'C. is 181.1 mm.; at 70.0°C., 547.4 mm. Calculate the value of AH, for benzene. 5. The mold heat ca~aciev$constant oressure (c.1 . - ,of CO, - ~ is 7.0 0.0071T1 0 . b ~ 0 0 1 8 6 ~ ~ many calories of HOW heat will be required to beat 25 grams of C02 from 31°C. to 94%) 6. The volume of a gram of water is nearly 1 a($ - 4)2 where o is a constant = 8.38 X 10-O. Calculate the value of the coefficient of cubical expansion for water a t 25'C.
devktion of arith. mean =
-
(0.0023456 X d'
0 OW004 0.000001 0 0.000001 0.00004Q 0.000008 0.000049 0.000001 0.000009 0.000004 0.000016
--
-
0.031 Z(d')
0,004143 143 X lo-,
* 0.031 = * 0.0028 11
* 0'0028 -- fil
t
0.00084
.
mean square error of single temp. =
*
mean square error of arith. mean = probable error of a single temp. = probable ermr of arith. mean =
414:T-110*
=
t
0.0rn8 -- -
Jii
+
value of K, the higher the curve cuts the y ax*; i. c., the mare frequently the arithmetic mean occurs as a value in a set of observations. In Figure 2. the greater the value of h, the steeper the curves and the less frequent the occurrence of large errors. Thus h is the modulus of precision. (See MELLOR:pp. 512-3.)
* 0.0038
0.0014
* 0.6745 X 0.0038 = * 0.0026 -- = + 0.00078
+
0.0026
.
Jii -(Notice that errors are usually quoted to two significant figures.)
3. The smallest volume we can use to attain '/,rn precision is
t
20.00 (+0.02). 2 0 . 0 0 ~of ~ 0.1000nonnal . RnC130risequivalent to 20.00 X 0.1000 X 55.84 mg. of iron, or 111.6 mg. This corresponds to 1.116 g. of original sample. Notice that to attain the desired precision it would be necessary to weigh only to the closest mg. 4. The value 0.1032 seems to be in error.
h
-x
Dlainlinr
-
Av.
0.1042
The value 0.1032 deviates from the a w a g e by 0.0010, which is more than four times the average deviation and can therefore be discarded. The normality of the acid is 0.1042. 5. Figure 1shows a plot of y = Ke-*'r' against x with varying values of K. Figure 2 shows a similar plat with h variable; Figure 3 gives the probability curyes when both hand K are varied. In Figure 1, the larger the
-
0
+x
m o m ~2. (K, constant; h, variable).
Av. Dev.
0,OM)IS
,
-x FIOUR&
0 +x 1. (h, constant; K, vanable)
-%
FIG-
3.
0 (h and K variable).
+x
