Elementary Calculus

Pre-calculus

Pythagorean theorem

Pythagorean theorem states that area of hypotenuse of right-angled triangle is equal to sum of area of legs.

$c^2=a^2+b^2$

This can be generalized into Law of cosines.

$c^2=a^2+b^2-2ab\cos \gamma$

Law of cosines can be applied to any triangle.

Law of sines states that,

$\frac{\sin \alpha }{a}=\frac{\sin \beta }{b}=\frac{\sin \gamma }{c}=\frac{1}{d}$

where d equals diameter of circumcircle (circle that touches all three vertices of triangle).

Ceva's theorem

Ceva's theorem provides criteria for determining when three lines are concurrent.

Ceva's theorem says that if perimeter of triangle constitutes AF and FB as one side, BD and DC as one side, and CE and EA as one side, then

$\frac{AF}{FB}.\frac{BD}{DC}.\frac{CE}{EA}=1$

if and only if three bisectors are concurrent (not neccesarily perpendicular to base of each side).

Ceva's theorem also says that product of ratios of sine of adjacent angles of sectored triangles is also 1.

Manelaus' theorem

Manelaus' theorem provides criteria for determing when three lines are collinear.

Thales theorem

Orthocenter is intersection of altitudes of a triangle. If orthocenter is located inside a triangle then it is acute. If orthocenter is located outside of a triangle then it is obtuse. If orthocenter lies on a side then it is a right triangle.

Special right triangles

Special right triangles have certain ratios of sides associated with them. The 45-45-90 triangle has side ratio of $1:1:\sqrt{2}$. The 30-60-90 triangle has side ratio of $1:\sqrt{3}:2$.

Functions

Inequalities

When solving inequality problems we must note that when dividing by a negative number, we must also invert sign of inequality. Inverting $<$ is $\ge$, for example. Other rules are quite straight foward and intuitively simple.

Solving,

$4x+3\le 7$

$(4x+3-3)/4\le (7-3)/4$

$x\le 1$

is the solution.

Domain and range

Domain is set of x elements of function. Range is set of y elements of function.

Domain can be written as $Dom(f(x))=\{x|x\in condition\}$. Range can be written as $Range(f(x))=\{y|y\in condition\}$.

Even and odd functions

Function is considered even if $f(x)=a$ and $f(-x)=a$.

Function is considered odd if $f(x)=a$ and $f(-x)=-a$.

Transformations

If $f(x)=a$, then $f(x)=a+c,c>0$ will translate previous $f(x)$ upwards by c units. Similarly, if $f(x)=a-c,c>0$ will translate original $f(x)$ downwards by c units.

$f(x-c)$ will translate f(x) right by c units. $f(x+c)$ will translate $f(x)$ left by c units.

$f(cx)$ will horizontally compress $f(x)$ by factor of c. $cf(x)$ will vertically stretch $f(x)$ by factor of c.

$-f(x)$ will reflect $f(x)$ about x axis. $f(-x)$ will reflect $f(x)$ about y axis.

Constant function

Constant function has a form $f(x)=c$ where c is a constant.

Polynomial function

Simplest polynomial function consists of polynomial multiplied by coefficient ($f(x)=c{x}^n$). Polynomial cannot be roots (eg. $\sqrt{x}$) and must be powers of natural numbers greater than or equal to 0.

Rational function

Simplest rational function is made of polynomial divided by another polynomial ($\frac{f(x)}{g(x)}$). Again we have to be careful that we determine $f(x)$ and $g(x)$ to be polynomials.

Algebraic function

Algebraic function can be roots of x and it can also be factors of x or powers of x. Simplest algebraic function may be $f(x)=x+c$ which is also polynomial function, since $x=x^1$ and $c=c{x}^0$.

Transcendental function

Transcendental functions are not algebraic. These include trigonometric, inverse trigonometric, logarithmic and exponential functions and many others.

Composite functions

$fog$ is read f ring g. It says that

$fog=f(g(x))$

$Dom(fog)=\{x|x\in Dom(g(x))\cap g(x)\in Dom(f(g(x))\}$