During these COVID-19 times, we often hear the word "exponential". The word “exponent" refersto the n in k xx", and an exponential function is one where that exponent n is the variable, insteadof r. A very famous example (which has appeared before in this course) is called the exponentialfunction, and it can be defined in many different ways:exp(x) = e" ,where e is a very special number, called "Euler's number", with a value of about 2.71828,exp(x) = 1++... = 1+x+2+1 x 21 x 2 x 31x 2 × 3 x 4+241and72exp(x) = lim (1+)" .The exponential function also has an important interpretation (meaning): "The rate of change ofthe exponential function is equal to its value", which we can write in mathematical form asexp(x + h) — expр (2)lim— еxp(»),h0hand this, with exp(0) = 1, is also one of the exponential function's many definitions. function can be related to the real world. It is also used in mathematical models which informthe decisions of governments wanting to control the spread of the new coronavirus.Question 1: Please approximate the value of the exponential function at r = 0.01 using the firstthree definitions given above, and describe how you did it in each case.

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Description: During these COVID-19 times, we often hear the word "exponential". The word “exponent" refersto the n in k xx", and an exponential function is one where that exponent n is the variable, insteadof r. A very famous example (which has appeared before i

Answered: Question 1: Please approximate the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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During these COVID-19 times, we often hear the word "exponential". The word “exponent" refers
to the n in k xx", and an exponential function is one where that exponent n is the variable, instead
of r. A very famous example (which has appeared before in this course) is called the exponential
function, and it can be defined in many different ways:
exp(x) = e" ,
where e is a very special number, called "Euler's number", with a value of about 2.71828,
exp(x) = 1+
+... = 1+x+
2
+
1 x 2
1 x 2 x 31x 2 × 3 x 4
+
24
1
and
72
exp(x) = lim (1+)" .
The exponential function also has an important interpretation (meaning): "The rate of change of
the exponential function is equal to its value", which we can write in mathematical form as
exp(x + h) — expр (2)
lim
— еxp(»),
h0
h
and this, with exp(0) = 1, is also one of the exponential function's many definitions.

function can be related to the real world. It is also used in mathematical models which inform
the decisions of governments wanting to control the spread of the new coronavirus.
Question 1: Please approximate the value of the exponential function at r = 0.01 using the first
three definitions given above, and describe how you did it in each case.

Expert Solution

Step 1

The objective here is to approximate the value of the exponential function at x=0.01.

Step 2

According to the first given definition exponential function is given by:

$\begin{matrix} \exp (x) = e^{x} \\ e^{x} = 1 + \frac{x^{1}}{1} + \frac{x^{2}}{1 \times 2} + \frac{x^{3}}{1 \times 2 \times 3} + \frac{x^{4}}{1 \times 2 \times 3 \times 4} + .... \\ = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \frac{x^{4}}{24} + ... \end{matrix}$

Step 3

Now, to approximate the value of the exponential function at x=0.01, put x=0.01 above

$\begin{matrix} \exp (0.01) = e^{0.01} \\ e^{(0.01)} = 1 + 0.01 + \frac{{(0.01)}^{2}}{2} + \frac{{(0.01)}^{3}}{6} + \frac{{(0.01)}^{4}}{24} + ... \end{matrix}$

Now, use the given definition

$\begin{matrix} \exp (x) = \lim_{n \to \infty} {(1 + \frac{x}{n})}^{n} \\ Putting x = 0.01 \\ \exp (0.01) = \lim_{n \to \infty} {(1 + \frac{0.01}{n})}^{n} \end{matrix}$

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