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Differential Calculus MATH 22981

Assignment 1

Winter 2020

1. According to the kinetic theory of gases the temperature-variation of gaseous viscosity is given by:
𝜇 = 𝑎𝑇 𝑛
Where 𝜇 and 𝑇 are viscosity and absolute temperature, respectively. Also, 𝑎 snd 𝑛 are constants. Base on
the Boltzmann equation and Chapman–Enskog theory, 𝑛 = 0.5 for an ideal gas.
Using linear regression, find the values of 𝑎 and 𝑛 from the following experimental data and evaluate this
theory. [10 marks]
T

210

250

280

310

340

370

390

420

𝜇

0.0151

0.016

0.0172

0.018

0.0185

0.019

0.0195

0.0212

We plot the function
0.025

μ

0.02

0.015

y = 0.0013x0.456
0.01

0.005

0
0

50

100

150

200

250

300

T

The T is on the x-axis and 𝜇 on the y-axis.
We get the following values for a and n
𝑎 = 0.0013
𝑛 = 0.456
We got an n pretty close to 0.5, this could be the data for an ideal as.

350

400

450

2. The following limits deal with the behavior of a function at infinity. Evaluate the following limits,
providing with the necessary elaboration on your answers. Support your answers using graphs. (You
may use software.) [3×2 marks]
a)

lim

𝑥 3 −3𝑥 2 +5

x→∞ 2𝑥 2 +50000

We divide both the numerator and denominator by 𝑥 2 , and we get
5
2
𝑥
lim
50000
x→∞
2+
𝑥2
𝑥−3+

If we take the limit as 𝑥 tends to infinity we are left with
5
2
𝑥−3
𝑥
lim
=
50000
x→∞
2
2+
𝑥2
𝑥−3+

Notice that this is a line with a positive slope, as x increases so will...