Generation X

Examples of Generation X in the following topics:

• The Natural Logarithmic Function: Differentiation and Integration

  • The natural logarithm, generally written as $\ln(x)$, is the logarithm with the base e, where e is an irrational and transcendental constant approximately equal to $2.718281828$.
  • for $\left | x \right | \leq 1$ (unless $x = -1$).
  • Substituting $x − 1$ for $x$, we obtain an alternative form for $\ln(x)$ itself:
  • $\ln(x) = (x - 1) - \dfrac{(x - 1)^{2}}{2} + \dfrac{(x - 1)^{3}}{3} - \cdots$ 
  • for $\left | x -1 \right | \leq 1$ (unless $x = 0$).

• Reflections

  • In this general equation, all $y$ values are switched to their negative counterparts while the $x$ values remain the same.
  • The result is that the curve becomes flipped over the $x$-axis.  
  • In this general equation, all $x$ values are switched to their negative counterparts while the y values remain the same.
  • Let's look at the case involving the line $y=x$.  
  • Calculate the reflection of a function over the $x$-axis, $y$-axis, or the line $y=x$

• Trigonometric Integrals

  • Generally, if the function, $\sin(x)$, is any trigonometric function, and $\cos(x)$ is its derivative, then
  • \\ \int x\sin^2 {ax}\;\mathrm{d}x = \frac{x^2}{4} - \frac{x}{4a} \sin 2ax - \frac{1}{8a^2} \cos 2ax +C\!
  • \\ \int x^2\sin^2 {ax}\;\mathrm{d}x = \frac{x^3}{6} - \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax - \frac{x}{4a^2} \cos 2ax +C\!}
  • Two simple examples of such integrals are $\int \sin^k x \cos x \; \mathrm d x$ and $\int \cos^k x \sin x\; \mathrm d x$ , which can be solved used the substitutions $u = \sin x$ and $u = \cos x$ , respectively.
  • We now consider the more general case of $\int \sin^n x \cos^m x\; \mathrm d x$ , where $n$ and $m$ are positive integers.

• Introduction to Exponential and Logarithmic Functions

  • Thus $log_{b}b^{x}=x $ and $b^{log_{b}x}=x $.
  • Another way of saying this is that if $f(x)=log_{b}x $, then the inverse function is given by $f^{-1}(x)=b^{x} $, and vice versa.
  • Lastly, as with all inverse functions, if we graph $f(x)=log_{b}x $  and $f^{-1}(x)=b^{x}$ on the same plane, the graphs will be symmetric across the line $y=x$.
  • Another way of thinking about this is that if we generate points on the curve of $f(x)=log_{b}x$ we can find the points on the curve of  $f^{-1}(x)=b^{x}$ by interchanging the $x$ and $y$ coordinates of the points.
  • That is, $e^{lnx}=lne^x=x$.

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • where $P$ and $Q$ are polynomial functions of $x$ and $Q(x) \neq 0$.
  • However, the adjective "irrational" is not generally used for functions.
  • The domain of a rational function $f(x) = \frac{P(x)}{Q(x)}$ is the set of all values of $x$ for which the denominator $Q(x)$ is not zero.
  • However, for $x^2 + 2=0$ , $x^2$ would need to equal $-2$.

• Finding Polynomials with Given Zeros

  • One type of problem is to generate a polynomial from given zeros.
  • If $x_1, x_2, \ldots x_n$ are the zeros of $f(x)$ and the leading coefficient of $f(x)$ is $1$, then $f(x)$ factorizes as
  • For any nonzero constant $a$, we have that $(af)(x)=af(x)$ factorizes as
  • Thus for given zeros $x_1, x_2, \ldots, x_n$ we find infinitely many solutions
  • Given zeros $0$, $1$, and $2$, our general solution is of the form

• Partial Fractions

  • The denominators of the terms of this summation, $g_{j}(x)$, are polynomials that are factors of $g(x)$, and in general are of lower degree.
  • $\displaystyle R(x) = \frac{f(x)}{(x - a_1)(x - a_2)\cdots (x - a_p)}$
  • $g(x)=\frac{8x^2 + 3x - 21}{x^3 - 7x - 6}=\frac{c_1}{(x+2)} + \frac{c_2}{(x-3)}+ \frac{c_3}{(x+1)}$
  • $g(x)=\frac{8x^2 + 3x - 21}{x^3 - 7x - 6}=\frac{1}{(x+2)} + \frac{4}{(x-3)}+ \frac{4}{(x+1)}$
  • Dividing through by $g(x)$ gives $\frac{f(x)}{g(x)}=E(x)+\frac{h(x)}{g(x)}$, which you can then perform the decomposition on $\frac{h(x)}{g(x)}$.

• Introduction to Least Squares

  • $\left( \begin{array}{c} 1 \\ 1 \\ 1 \\ \vdots \\ 1 \end{array} \right) x = \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{array} \right)$
  • $\displaystyle{\min _x \sum _ {i=1} ^ n \left( x - x_i\right)^2. }$
  • The matrix $(A^T A)^{-1} A^T$ is an example of what is called a generalized inverse of $A$$A$ .
  • In the even that $A$ is not invertible in the usual sense, this provides a reasonable generalization (not the only one) of the ordinary inverse.
  • The generalized inverse projects the data onto the column space of $A$ .

• Integration By Parts

  • $\displaystyle{\int u(x) v'(x) \, dx = u(x) v(x) - \int u'(x) v(x) \ dx}$
  • $dv = \cos(x)\,dx \\ \therefore v = \int\cos(x)\,dx = \sin x$
  • $\begin{aligned} \int x\cos (x) \,dx & = \int u \, dv \\ & = uv - \int v \, du \\ & = x\sin (x) - \int \sin (x) \,dx \\ & = x\sin (x) + \cos (x) + C \end{aligned}$
  • Similarly, the area of the red region is $A_2=\int_{x_1}^{x_2}y(x)dx$.
  • Assuming the curve is smooth within a neighborhood, this generalizes to indefinite integrals $\int xdy + \int y dx = xy$, which can be rearranged to the form of the theorem: $\int xdy = xy - \int y dx$.

• The Natural Exponential Function: Differentiation and Integration

  • Now that we have derived a specific case, let us extend things to the general case of exponential function.
  • Since we have already determined the derivative of $e^{x}$, we will attempt to rewrite $a^{x}$ in that form.
  • Note that the exponential function $y = e^{x}$ is defined as the inverse of $\ln(x)$.
  • Therefore $\ln(e^x) = x$ and $e^{\ln x} = x$.
  • Since $e^{x} = (e^{x})'$ we can integrate both sides to get: