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education/math/MATH1210 (calc 1)/Derivatives.md

@@ -0,0 +1,21 @@
+A derivative can be used to describe the rate of change at a single point, or the *instantaneous velocity*.
+
+The formula used to calculate the average rate of change looks like this:
+$$ \dfrac{f(b) - f(a)}{b - a} $$
+Interpreting it, this can be described as the change in $y$ over the change in $x$.
+
+- Speed is always positive
+- Velocity is directional
+
+As the distance between the two points $a$ and $b$ grow smaller, we get closer and closer to the instantaneous velocity of a point. Limits are suited to describing the behavior of a function as it approaches a point.
+
+If we have the coordinate pair $(a, f(a))$, and the value $h$ is the distance between $a$ and another $x$ value, the coordinates of that point can be described as ($(a + h, f(a + h))$. With this info:
+- The slope of the secant line can be described as $\dfrac{f(a + h) - f(a)}{a + h - a}$, which simplifies to $\dfrac{f(a + h) - f(a)}{h}$. 
+- The slope of the *tangent line* or the *instantaneous velocity* can be found by taking the limit of the above function as the distance ($h$) approaches zero:
+$$\lim_{h \to 0}\dfrac{f(a + h) - f(a)}{h}$$
+The above formula can be used to find the *derivative*. This may also be referred to as the *instantaneous velocity*, or the *instantaneous rate of change*.
+# Line Types
+## Secant Line
+A **Secant Line** connects two points on a graph.
+
+A **Tangent Line** represents the rate of change or slope at a single point on the graph.

education/math/MATH1210 (calc 1)/Limits.md

@@ -17,6 +17,59 @@ The standard notation for a limit is:
 $$ \lim_{x \to a} f(x) = L $$
 - As $x$ approaches $a$, the output of $f(x)$ draws closer to $L$. In the above notation, $x$ and $a$ are not necessarily equal.
 - When plotted, the hole is located at $(a, L)$.
+# Indeterminate Limits
+If they have a limit of the form $lim_{x \to a} \frac{f(x)}{g(x)}$ where both $f(x) \to 0$ and $g(x) \to 0$ as $x \to a$ then this limit **may or may not** exist and is said to be an indeterminate form of type $\dfrac{0}{0}$.
+To find this limit if it exists we must perform some mathematical manipulations on the quotient in order to change the form of the function. Some of the manipulations that can be tried are:
+- Factor or Foil polynomials and try dividing out a common factor.
+- Multiply numerator and denominator by the conjugate of a radical expression
+- Combine fractions in the numerator or denominator of a complex fraction
+
+# Limits of the Form $\frac{k}{0}, k \ne 0$
+If we have a one sided limit of the form $\lim_{x \to a^*} \frac{f(x)}{g(x)}$ $f(x) \to k (k \ne 0)$ and $g(x) \to 0$ as $x \to a$ then:
+$$ \lim_{x \to a^*} \frac{f(x)}{g(x)} = \infty \space or \space \lim_{x \to a^*} \frac{f(x)}{g(x)} = -\infty $$
+# Limits of the Form $\frac{\infty}{\infty}$
+If we have a limit of the form $\lim_{x \to a} \frac{f(x)}{g(x)}$ where both $f(x) \to \infty$ and $g(x) \to \infty$ as $x \to a$ then the limit may or may not exist and is said to be an indeterminate form of type $\frac{\infty}{\infty}$.
+
+To find the limit if it exists we must perform some algebraic manipulations on the quotient in order to change the form of the function.
+
+If $f(x)$ and $g(x)$ are polynomials, then we can multiply the numerator and denominator by $\dfrac{1}{x^n}$, where $n$ is the degree of the polynomial in the denominator. 
+# Continuity
+A function is continuous if their graph can be traced with a pencil without lifting the pencil from the page.
+
+Formally, a function $f$ is continuous at a point $a$ if:
+- $f(a)$ is defined
+- $\lim_{x \to a} f(x)$  exists
+- $\lim_{x \to a} = f(a)$
+
+- A function is continuous on the open interval $(a, b)$ if it is continuous at all points between $a$ and $b$
+- A function is continuous on the closed interval $[a, b]$ if it is continuous at all points between $a$ and $b$
+
+# Elementary Functions
+An elementary function is any function that is defined using:
+- Polynomial functions
+- Rational functions
+- Root functions
+- Trig functions
+- Inverse trig functions
+- Exponential functions
+- Logarithmic functions
+- Operations of:
+	- Addition
+	- Subtraction
+	- Multiplication
+	- Division
+	- Composition
+
+A piece-wise function is *not*  considered an elementary function
+
+- If $f$ and $g$ are continuous at a point $x = a$ and $c$ is a constant then the following functions are also continuous at $x = a$
+- If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f(g(a))$ is continuous at $a$
+- If $f$ is an elementary function and if $a$ is in the domain of $f$, then $f$ is continuous at $a$
+Together, the above theorems tell us that if $a$ is in the domain of an elementary function, then $\lim_{x \to a} f(x) = f(a)$.
+
+# Intermediate Value Theorem
+Let $f$ be a continuous function on the interval $[a, b]$ and let $N$ be any number strictly between $f(a)$ and $f(b)$. Then there exists a number $c$ in $(a, b)$ such that $f(c) = N$.
+
 # Definitions
 
 | Term                  | Definition                                                                    |