Locations

Source Code

GitHub.com/colbyn/colbyns-calculus-notes

Online Edition

Colbyn.GitHub.io/colbyns-calculus-notes

New home: https://github.com/colbyn/colbyns-math-notes

Limits

Limits of Trigonometric Functions

General reference:

\[\begin{equation} \begin{split} \lim_{x \to a} \sin(x) &= \sin(a) \\ \lim_{x \to a} \cos(x) &= \cos(a) \\ \lim_{x \to a} \tan(x) &= \tan(a) \\ \lim_{x \to a} \csc(x) &= \csc(a) \\ \lim_{x \to a} \sec(x) &= \sec(a) \\ \lim_{x \to a} \cot(x) &= \cot(a) \end{split} \end{equation}\]

Others:

\[\begin{equation} \begin{split} \lim_{x \to 0} \frac{\sin(x)}{x} &= 1 \\ \lim_{x \to 0} \frac{1 - \cos(x)}{x} &= 0 \\ \end{split} \end{equation}\]

Derivatives

\[f^\prime(x) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = \lim_{x \to a} \frac{f(a) - f(x)}{a - x} = \frac{dy}{dx}f(x) = \frac{d}{dx}f(x)\]

Formal Limit Definition of A Derivative

Derivative of \(f(x)=|x|\) using the formal limit definition of a derivative:

\[\begin{equation} \begin{split} f(x) & = |x| \\ f^\prime(a) & = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \\ & = \lim_{h \to 0} \frac{|a + h| - |a|}{h} \cdot \frac{|a + h| + |a|}{|a + h| + |a|} \\ & = \lim_{h \to 0} \frac{|a + h|^2 - |a|^2}{h \cdot (|a + h| + |a|)} \\ & = \lim_{h \to 0} \frac{(a + h)^2 - a^2}{h (|a + h| + |a|)} \\ & = \lim_{h \to 0} \frac{a^2 + 2ah + h^2 - a^2}{h (|a + h| + |a|)} \\ & = \lim_{h \to 0} \frac{2ah + h^2}{h (|a + h| + |a|)} \\ & = \lim_{h \to 0} \frac{h \cdot (2a + h)}{h \cdot (|a + h| + |a|)} \\ & = \lim_{h \to 0} \frac{2a + h}{|a + h| + |a|} \\ & = \frac{2a + (0)}{|a + (0)| + |a|} \\ & = \frac{2a}{|a| + |a|} \\ & = \frac{2a}{2|a|} \\ & = \frac{2a}{2|a|} \cdot \frac{|a|}{|a|} \\ & = \frac{2a|a|}{2|a|^2} \\ & = \frac{2a|a|}{2a^2} \\ & = \frac{|a|}{a} \\ f^\prime(x) &= \frac{|x|}{x} \end{split} \end{equation}\]

Derivative of \(f(x)=|x|\) using an alternative limit definition of a derivative:

\[\begin{equation} \begin{split} f(x) & = |x| \\ f^\prime(a) & = \lim_{x \to a} \frac{f(a) - f(x)}{a - x} \\ & = \lim_{x \to a} \frac{|a| - |x|}{a - x} \cdot \frac{|a| + |x|}{|a| + |x|} \\ & = \lim_{x \to a} \frac{|a|^2 - |x|^2}{(a - x) \cdot (|a| + |x|)} \\ & = \lim_{x \to a} \frac{(a)^2 - (x)^2}{(a - x) \cdot (|a| + |x|)} \\ & = \lim_{x \to a} \frac{(a - x) \cdot (a + x)}{(a - x) \cdot (|a| + |x|)} \\ & = \lim_{x \to a} \frac{a + x}{|a| + |x|} \\ & = \frac{a + (a)}{|a| + |(a)|} \\ & = \frac{2a}{2|a|} \\ & = \frac{a}{|a|} \cdot \frac{|a|}{|a|} \\ & = \frac{a|a|}{|a|^2} \\ & = \frac{a|a|}{a^2} \\ & = \frac{|a|}{a} \\ f^\prime(x) &= \frac{|x|}{x} \end{split} \end{equation}\]

Derivative Continuity

Is a derivative continuous at \(x = a\)?

First we ensure:

\[\lim_{x \to a^{-}} f^\prime(x) = \lim_{x \to a^{+}} f^\prime(x) = \lim_{x \to a} f^\prime(x)\]

then:

\[\lim_{x \to a} f^\prime(x) = f^\prime(a)\]

therefore the derivative is continuous and differentiable at \(x = a\).

Continuous Piecewise Function

Example

Is this function differentiable at \(x = 2\)?

\[f(x) = \begin{cases} x^2 - 3 & \text{if $x \lt 2$} \\ 4x - 7 & \text{if $x \ge 2$} \end{cases}\]
\[f^\prime(x) = \begin{cases} 2x & \text{if $x \lt 2$} \\ 4 & \text{if $x \ge 2$} \end{cases}\]

To check:

\[\begin{equation} \begin{split} \lim_{x \to 2^{-}} f^\prime(x) &= 2(2) = 4 \\ \lim_{x \to 2^{+}} f^\prime(x) &= 4 \\ \text{therefore} \\ \lim_{x \to 2} f^\prime(x) &= 4 \\ \text{then we check} \\ \lim_{x \to 2} f^\prime(x) &= f^\prime(2) \\ \end{split} \end{equation}\]

Therefore the derivative is continuous and differentiable at \(x = 2\).

Lines

Tangent Line

\[\begin{equation} \begin{split} y - y_1 &= f^\prime(x_1)(x-x_1) \\ y &= mx + b \end{split} \end{equation}\]
More succinctly
\[\begin{equation} \begin{split} y &= f(a) + f^\prime(a)(x - a) \end{split} \end{equation}\]

At a given point \(x = a\).

Example

For example, given \(f(x) = x^3 + 3x^2\), and asked to find the tangent like at \(x = 1\):

\[\begin{equation} \begin{split} f(x) &= x^3 + 3x^2 \\ y &= f(1) + f^\prime(1)(x - 1) \\ &= 4 + 9(x - 1) \\ &= 9x - 5 \end{split} \end{equation}\]

Therefore, the tangent line at \(x = 1\) is \(y = 9x - 5\).

Alternative Overview

\[y = mx + b\]

Generally, there are three things you will need when given some \(f(x)\):

  • The value of \(x\) (usually provided to you)

  • The value of \(y\) or \(f(x)\) (if it’s not provided to you, just plug it in)

  • The value of m (i.e. the slope, such as the value from \(f^\prime(x)\))

  • Once you have \(m\), then solve for \(b\) by plugging in some \((x, y)\) into your \(y = mx + b\) equation.

Normal Lines

The slope of the normal line is the negative reciprocal of the slope of the tangent line. Everything else is the same as the tangent line.

\[\begin{equation} \begin{split} y - y_1 &= -\frac{1}{f^\prime(x_1)} (x-x_1) \\ y &= -\frac{1}{m}x + b \end{split} \end{equation}\]

Perpendicular Lines

The same as the Normal Lines. According to these Quora answers:

A normal makes an angle of 90° with a 2 dimensional SURFACE
while,
A perpendicular makes an angle of 90° with a one dimensional LINE.
Basically perpendicular relates to line and normal relates to a plane,but both make 90° with their respective counterparts. The later is a vector quantity, where the former is scalar.