# Exponential functions 2

## Graphs of the exponential functions, and the Laws of exponents

• We have defined the exponents $$b^x$$ for all $$x$$, which implies that the exponential functions $$f(x)=b^x$$ have domain $$(-\infty,\infty)$$
• Notice on the graphs that there are 3 qualitatively different behaviours of the exponential functions $$f(x)=b^x$$, depening on the value of $$b$$:
• If $$b=1$$, then we have $$1^x=1$$, that is the function $$f(x)=1^x$$ has constant value 1, or in other words, its range is $$[1,1]$$.
• If $$b\ne1$$, then the range of $$f(x)=b^x$$ is $$(0,\infty)$$.
• If $$b>1$$, then $$f(x)=b^x$$ is strictly monotonously increasing.
• If $$b<1$$, then $$f(x)=b^x$$ is strictly monotonously decreasing.
• Note also that all the graphs go through the point $$(0,1)$$. This corresponds to that for all $$b>0$$, we have $$f(x=0)=b^0=1$$.
• The Laws of exponents are the following formulas. $b^{x+y}=b^xb^y,\quad b^{x-y}=\tfrac{b^x}{b^y},\quad b^{xy}=(b^x)^y,\quad (ab)^x=a^xb^x$

Exercises: 1.4:4,19,24

## Application: population modelling

• Exponential functions are used frequently in Mathematical models of populations. The textbook has examples about bacteria, humans and viruses, let's check out the human one.
• We are given the population (in millions) of the Earth at the years $$1900+t$$ for $$t=0,10,\dotsc,110$$.
• The method of least squares (which you can learn about in a Linear Algebra class) tells us which function of the form $$f(t)=ab^t$$ approximates the data the best.
• Mathematical models can be used for prediction. This model predicts that by 2020, the population of the Earth will be $$f(t=120)\approx7573.549$$ million.

• The half-life of an atom is the period of time during which half of any given quantity disintegrates
• For example, the half-life of strontium-90, $${}^{90}\mathrm{Sr}$$, is 25 years.
• Let $$m(t)$$ denote the mass of a sample of $${}^{90}\mathrm{Sr}$$, starting from $$m(t=0)=24$$ mg.
• By definition of half-life, we have $$m(t=25)=\frac1224$$, $$m(t=50)=\frac1424$$, etc.
• In general, we have $$m(t)=(\frac12)^{t/25}24=2^{-t/25}24$$.
• For example, after 40 years, the mass of the sample is $$m(t=40)=2^{-40/25}24\approx7.9$$ mg.
• Exercise: 1.4.34

## The number $$e$$

• Let $$b>0$$. Consider the tangent line to the graph of $$f(x)=b^x$$ at $$x=0$$.
• Notice that the tangent line is the graph of the function $$g(x)=1+f'(x=0)$$. We will start talking about derivatives two weeks later.
• You can see that the bigger $$b$$ is, the bigger the slope is.
• The number $$e$$ is defined to be the value $$b=e$$ such that the function $$f(x)=b^x$$ has slope 1 at $$x=0$$.
• The corresponding exponential function $$f(x)=e^x$$ is called the natural exponential function.

## Euler's formula and the Trigonometric addition formulas

• You don't have to know about complex numbers in this class, but I can't resist telling you about the following way of proving the Trigonometric addition formulas
• You can do complex arithmetic by introducing the imaginary number $$i$$. This number has the propety $$i^2=-1$$.
• Therefore, a complex number in general is of the form $$a+bi$$, for real numbers $$a,b$$. $$a$$ is called its real part, and $$b$$ is called its imaginary part
• Addition and subtraction is as usual: $$(a_1+b_1i)\pm(a_2+b_2i)=(a_1\pm a_2)+(b_1\pm b_2)i$$.
• Multiplication uses distributivity, and the magic property $$i^2=-1$$: $(a_1+b_1i)(a_2+b_2i)=a_1a_2+b_1b_2i^2+a_1b_2i+a_2b_1i=(a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i.$
• Two complex numbers $$a_1+b_1i$$ and $$a_2+b_2i$$ are equal precisely when both their real and imaginary parts agree, that is $$a_1=a_2$$ and $$b_1=b_2$$.

## Euler's formula and the Trigonometric addition formulas 2

• Euler's formula tells us how to take natural exponents with complex numbers: $e^{a+bi}=e^a(\cos(b)+\sin(b)i)$
• This means that if $$a=0$$, we get $$e^{bi}=\cos(b)+\sin(b)i$$.
• Let's apply this to the exponent law $$e^{(a+b)i}=e^{ai}e^{bi}$$: $\begin{multline*} \cos(a+b)+\sin(a+b)i=e^{(a+b)i}=e^{ai}e^{bi} \\ =(\cos(a)+\sin(a)i)(\cos(b)+\sin(b)i)=(\cos(a)\cos(b)-\sin(a)\sin(b))+(\cos(a)\sin(b)+\cos(b)\sin(a))i \end{multline*}$
• BAM! Equating the real parts gives the formula for cos: $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b),$ and equating the imaginary parts gives the formula for sin: $\sin(a+b)=\cos(a)\sin(b)+\cos(b)\sin(a).$