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 \((-\infty,\infty)\)
    • Click here to see the graphs of some exponential functions
    • 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.

Application: radioactive decay

  • 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 \(x=0\).
    • Click here for an interactive plot.
    • 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: \[ \sin(a+b)=\cos(a)\sin(b)+\cos(b)\sin(a). \]