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Lesson 6: Operations on functions
Topic. This lesson covers Session 6: Operations on functions
Learning Outcomes.
WeBWorK. There is 1 WeBWorK assignment on today’s material:
Additional Video Resources.
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Lesson 5: Basic functions and transformations
Topic. This lesson covers Session 5: Basic functions and transformations
Learning Outcomes.
WeBWorK. There are two WeBWorK assignments on today’s material:
Additional Video Resources.
Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.
Session 4: Introduction to the TI-84
Topic. This lesson covers Session 4: Introduction to the TI-84
Learning Outcomes.
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Additional Video Resources.
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Lesson 2: Lines and Functions
Topic. This lesson covers Session 2: Lines and Functions
Learning Outcomes.
WeBWorK. There is one WeBWorK assignment on today’s material:
Additional Video Resources.
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Lesson 1: The Absolute Value
Topic. This lesson covers Session 1: The Absolute Value.
Learning Outcomes.
WeBWorK. There are two WeBWorK assignments on today’s material:
Additional Video Resources.
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Lesson 20: Trigonometric Equations
Topic. This lesson covers Chapter 20: Trigonometric Equations.
WeBWorK. There is one WeBWorK assignment on today’s material:
Trigonometry - Equations
Lesson Notes. (download pdf)
The lessons below follow the course outline for the SECOND EDITION of the textbook. An update of these lessons that follows the outline of the third edition is currently UNDER CONSTRUCTION.
Lesson 3: Functions by formulas and graphs
Lesson 4: Introduction to the TI-84
Lesson 5: Basic functions and transformations
Lesson 6: Operations on functions
Lesson 7: The inverse of a function
Lesson 8: Dividing polynomials
Lesson 9: Graphing polynomials
The following lessons are in development (will be completed soon):
The remaining lessons below were created in Spring 2020 – the content is complete but they need to be cleaned up (work in progress – will be completed soon). NOTE: The URLs for these lessons may change slightly as they are finalized – You can always check this page for the latest link.
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Lesson 24: The geometric series
Topic: This lesson covers Chapter 24: The geometric series
WeBWorK: There are two WeBWorK assignments on today’s material:
Sequences – Geometric
Series – Geometric
Question of the day: Can we add up infinitely many numbers?
This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.
Today we look at a new kind of sequence, called a geometric sequence, and the corresponding series, geometric series.
A geometric sequence is a sequence for which we multiply by a constant number to get from one term to the next, for example:
Definition 24.1. A sequence $\left\{a_{n}\right\}$ is called a geometric sequence, if any two consecutive terms have a common ratio $r$. The geometric sequence is determined by $r$ and the first value $a_{1}$. This can be written recursively as:
$$a_{n}=a_{n-1} \cdot r \quad \text { for } n \geq 2$$
Alternatively, we have the general formula for the $n$ th term of the geometric sequence:
$$a_{n}=a_{1} \cdot r^{n-1}$$
Example 24.2. Determine if the sequence is a geometric or arithmetic sequence, or neither or both. If it is a geometric or arithmetic sequence, then find the general formula.
a) $3,6,12,24,48, \dots$
b) $100,50,25,12.5, \ldots$
c) $700,-70,7,-0.7,0.07, \ldots$
d) $2,4,16,256, \dots$
e) $3,10,17,24, \ldots$
f) $\quad-3,-3,-3,-3,-3, \dots$
g) $a_{n}=\left(\frac{3}{7}\right)^{n}$
h) $a_{n}=n^{2}$
VIDEO: Introduction to geometric sequences, Example
Example 24.3. Find the general formula of a geometric sequence with the given property
a) $r=4,$ and $a_{5}=6400$
b) $a_{1}=\frac{2}{5},$ and $a_{4}=-\frac{27}{20}$
c) $a_{5}=216, a_{7}=24,$ and $r$ is positive
VIDEO: Finding the formula of a geometric sequence – Example 24.3
Example 24.4. Consider the geometric sequence $a_{n}=8 \cdot 5^{n-1},$ that is the sequence:
$$8,40,200,1000,5000,25000,125000, \ldots$$
Find the sum of the first 6 terms of this sequence
$$8+40+200+1000+5000+25000=31248$$
VIDEO: Sum of a geometric series – intro example
Observation 24.5. Let $\left\{a_{n}\right\}$ be a geometric sequence, whose $n$ th term is given by the formula $a_{n}=a_{1} \cdot r^{n-1} .$ We furthermore assume that $r \neq 1 .$ Then, the sum $a_{1}+a_{2}+\dots+a_{k-1}+a_{k}$ is given by
$$\sum_{i=1}^{k} a_{i}=a_{1} \cdot \frac{1-r^{k}}{1-r}$$
Example 24.6. Find the value of the geometric series.
a) Find the sum $\sum_{n=1}^{6} a_{n}$ for the geometric sequence $a_{n}=10 \cdot 3^{n-1}$
b) Determine the value of the geometric series: $\sum_{k=1}^{5}\left(-\frac{1}{2}\right)^{k}$
c) Find the sum of the first 12 terms of the geometric sequence
$$-3,-6,-12,-24, \dots$$
VIDEO: The sum of a finite geometric series, Example 24.6
Sometimes it makes sense to add up not just a finite number of terms in a sequence, but ALL the terms (infinitely many!).
Example 24.7. Consider the geometric sequence
$$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots$$
What is the initial term? What is the common ratio?
Let’s try adding up some of the terms. Try this by hand, and by using the formula for finite geometric series. What happens if we add up ALL the terms?
VIDEO: Infinite geometric series – intro example and formula
Definition 24.8. An infinite series is given by the formula
$$\sum_{i=1}^{\infty} a_{i}=a_{1}+a_{2}+a_{3}+\ldots$$
Observation 24.9. Let $\left{a_{n}\right}$ be a geometric sequence with $a_{n}=a_{1} \cdot r^{n-1}$ Then the infinite geometric series is defined whenever $-1<r<1$. In this case, we have:
$$\sum_{i=1}^{\infty} a_{i}=a_{1} \cdot \frac{1}{1-r}$$
$\quad$
Example 24.10. Find the value of the infinite geometric series.
a) $\sum_{j=1}^{\infty} a_{j},$ for $a_{j}=5 \cdot\left(\frac{1}{3}\right)^{j-1}$
b) $\sum_{n=1}^{\infty} 3 \cdot(0.71)^{n}$
c) $500-100+20-4+\ldots$
d) $3+6+12+24+48+\ldots$
Example 24.11. Consider the real number given by $0.555555\dots$. Rewrite this number as an infinite geometric series. Can you figure out what fraction it is equal to?
VIDEO: Infinite geometric series – examples
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Lesson 23: Sequences and series
Topic: This lesson covers Chapter 23: Sequences and series
WeBWorK: There are three WeBWorK assignments on today’s material:
Sequences – Introduction
Sequences – Arithmetic
Series – Finite Arithmetic
Question of the day: What is a sequence? What is a series?
Answer of the day: A sequence is just a list of numbers. A series is list of numbers, added up.
This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.
Today we will introduce two very powerful ideas (they are the building blocks of Calculus) – however, the ideas themselves are not complicated: sequences and series. They both have to do with lists of numbers, rather than individual numbers.
Definition 23.1. A sequence is an ordered list of numbers (we call each number in the list a term in the sequence). We write a sequence in order as follows: $a_{1}, a_{2}, a_{3}, a_{4},…$
In short we write the above sequence as $\left\{a_{n}\right\}$ or $\left\{a_{n}\right\}_{n \geq 1}$
Let’s look at examples of some sequences. As you look at this example, keep the following questions in mind:
Example 23.2. Here are some examples of sequences.
a) $4,6,8,10,12,14,16,18, \dots$
b) $1,3,9,27,81,243, \dots$
c) $+5,-5,+5,-5,+5,-5, \dots$
d) $1,1,2,3,5,8,13,21,34,55, \dots$
e) $5,8,-12,4,5.3,7,-3, \sqrt{2}, 18, \frac{2}{3}, 9, \dots$
Example 23.3. Consider the sequence $\left\{a_{n}\right\}$ with $a_{n}=4 n+3$. Calculate the first five terms of the sequence $a_1, a_2, a_3, a_4,$ and $a_5$. What is the $200$th term of the sequence?
VIDEO: Introduction to sequences – Examples 23.2 and 23.3
Example 23.4. Find the first 6 terms of each sequence.
a) $a_{n}=n^{2}$
b) $a_{n}=\frac{n}{n+1}$
c) $a_{n}=(-1)^{n}$
d) $a_{n}=(-1)^{n+1} \cdot 2^{n}$
Another way to describe a sequence is by giving a recursive formula for the $n$th term $a_{n}$ in terms of the lower terms. Here are some examples.
Example 23.6. Find the first $6$ terms in the sequence described below.
a) $a_{1}=4,$ and $a_{n}=a_{n-1}+5$ for $n>1$
b) $a_{1}=3,$ and $a_{n}=2 \cdot a_{n-1}$ for $n>1$
c) $a_{1}=1, a_{2}=1,$ and $a_{n}=a_{n-1}+a_{n-2}$ for $n>2$
VIDEO: Two ways of describing sequences -Examples 23.4 and 23.6
Definition 23.8. A series is a sum of terms in a sequence. We denote the sum of the first $k$ terms in a sequence with the following notation:
$\sum_{i=1}^{k} a_{i}=a_{1}+a_{2}+\cdots+a_{k}$
The summation symbol $\sum $ comes from the greek letter $\Sigma$, pronounced “sigma,” which is the greek letter for “S.”
Example 23.9. Find the sum.
a) $\sum_{i=1}^{4} a_{i},$ for $a_{i}=7 i+3$
b) $\sum_{j=1}^{6} a_{j},$ for $a_{n}=(-2)^{n}$
c) $\sum_{k=1}^{5}\left(4+k^{2}\right)$
VIDEO: Intro to series – Example 23.9
Definition 23.10. A sequence $\left{a_{n}\right}$ is called an arithmetic sequence if any two consecutive terms have a common difference $d$. The arithmetic sequence is determined by $d$ and the first value $a_{1}$. This can be written recursively as:
$a_{n}=a_{n-1}+d \quad \text { for } n \geq 2$
Alternatively, we have the general formula for the $n$th term of the arithmetic sequence
$a_{n}=a_{1}+d \cdot(n-1)$
Example 23.11. Determine if the sequence is an arithmetic sequence. If so, then find the general formula for $a_{n}$.
a) $7,13,19,25,31, \ldots$
b) $13,9,5,1,-3,-7, \dots$
c) $10,13,16,20,23, \dots$
d) $a_{n}=8 \cdot n+3$
VIDEO: Arithmetic sequences – Example 23.11
Example 23.12. Find the general formula of an arithmetic sequence with the given property.
a) $d=12,$ and $a_{6}=68$
b) $a_{1}=-5,$ and $a_{9}=27$
c) $a_{5}=38,$ and $a_{16}=115$
VIDEO: Arithmetic sequences – finding a general formula, Example 23.12
Example 23.13. Find the sum of the first 100 integers, starting from 1 . In other words, we want to find the sum of $1+2+3+\cdots+99+100$.
VIDEO: Arithmetic series – intro example 23.13
Observation 23.14 . Let $\left\{a_{n}\right\}$ be an arithmetic sequence, whose $n$ th term is given by the formula $a_{n}=a_{1}+d(n-1)$. Then, the sum $a_{1}+a_{2}+\cdots+a_{k-1}+a_{k}$ is given by adding $\left(a_{1}+a_{k}\right)$ precisely $\frac{k}{2}$ times:
$\sum_{i=1}^{k} a_{i}=\frac{k}{2} \cdot\left(a_{1}+a_{k}\right)$
Example 23.15. Find the value of the arithmetic series.
a) Find the sum $a_{1}+\cdots+a_{60}$ for the arithmetic sequence $a_{n}=2+13(n-1)$
b) Determine the value of the sum: $\quad \sum_{j=1}^{1001}(5-6 j)$
c) Find the sum of the first 35 terms of the sequence
$4,3.5,3,2.5,2,1.5, \ldots$
VIDEO: Arithmetic series – finding the sum, Example 23.15
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Lesson 22: Vectors in the plane
Topic: This lesson covers Chapter 22: Vectors in the plane.
WeBWorK: There are four WeBWorK assignments on today’s material:
Vectors – Components
Vectors – Operations
Vectors – Magnitude and Direction
Vectors – Unit Vectors
This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.
Today we will be working with the plane $\mathbb{R}^2$, but looking at things in a slightly different way – instead of points (which have only a location), we will be focussing on vectors (which have a magnitude (size) and direction). This change in perspective is quite powerful, and brings to light many useful features of the plane – but in practice, you will find it similar to the work we did in the previous lesson on polar form of complex numbers.
Definition 22.1. A geometric vector in the plane is a geometric object in the plane $\mathbb{R}^{2}$ that is given by a direction (angle) and magnitude (size). We denote a vector by $\vec{v}$ (it is written by some authors as $v$ ), its magnitude is denoted by $||\vec{v}||,$ and its directional angle by $\theta$.
Vectors are often drawn as directed line segments $\vec{v}=\overrightarrow{P Q}$. Two such segments represent the same vector if they have the same magnitude and direction.
We can always represent a vector $\vec{v}$ by arranging the starting point of $\vec{v}$ to be the origin $O(0,0)$ (as in $\overrightarrow{O R}$ in the picture above). If $R$ has coordinates $R(a, b)$ then we also write for $\vec{v}=\overrightarrow{O R}$:
$\vec{v}=\langle a, b\rangle$, or $\vec{v}=\begin{bmatrix} a\\ b\end{bmatrix}$
Example 22.2. Graph the vectors $\vec{v}, \vec{w}, \vec{r}, \vec{s}, \vec{t}$ in the plane, where
$\vec{v}=\overrightarrow{P Q}$ with $P(6,3)$ and $Q(4,-2),$ and
$\vec{w}=\langle 3,-1\rangle, \quad \vec{r}=\langle-4,-2\rangle, \quad \vec{s}=\langle 0,2\rangle, \quad \vec{t}=\langle-5,3\rangle$
The formulas for magnitude and directional angle of a vector are the same as those for modulus (magnitude) and argument (angle) of a complex number:
Formulas for magnitude and angle of a vector: Suppose $\vec{v}=\langle{a,b}\rangle}$ is a vector in the plane $\mathbb{R}^2$. Then the magnitude and angle of $\vec{v}$ are given by:
$||\vec{v}||=\sqrt{a^2+b^2}$, $\tan(\theta)=\frac{b}{a}$
Conversely, we can obtain the coordinates of a vector from its magnitude and directional angle by:
$\vec{v}=\langle||\vec{v}|| \cdot \cos (\theta),||\vec{v}|| \cdot \sin (\theta)\rangle$
Example 22.4. Find the magnitude and directional angle of the given vectors:
a) $\langle-6,6\rangle$
b) $\langle 4,-3\rangle$
c) $\langle-2 \sqrt{3},-2\rangle$
d) $\langle 8,4 \sqrt{5}\rangle$
e) $\overrightarrow{P Q}$, where $P(9,2)$ and $Q(3,10)$
VIDEO: Intro to vectors, finding magnitude and direction – Example 22.4
There are two basic operations on vectors, scalar multiplication and vector addition.
Definition 22.5. The scalar multiplication of a real number $r$ with a vector $\vec{v}=\langle a, b\rangle$ is defined to be the vector given by multiplying each coordinate by $r$:
$r \langle a, b\rangle =\langle r \cdot a, r \cdot b\rangle$
Example 22.6. Multiply, and graph the vectors
a) $4 \cdot\langle-2,1\rangle$
b) $(-3)\cdot\langle-6,-2\rangle$
VIDEO: Scalar multiplication of vectors – Example 22.6
Observation. When we multiply a vector $\vec{v}$ by a positive real number $r>0$, the result will have the same angle as $\vec{v}$, while the magnitude will be stretched by a factor of $r$.
Definition 22.8. A vector $\vec{u}$ is called a unit vector if it has a magnitude of 1
$\vec{u}$ is a unit vector $\quad \Longleftrightarrow ||\vec{u}||=1$
There are two special unit vectors $\vec{i}$ and $\vec{j}$, which are the vectors pointing in the $x-$ and the $y$ -direction.
$\vec{i}=\langle 1,0\rangle \quad \text { and } \quad \vec{j}=\langle 0,1\rangle$
Example 22.9. Find a unit vector in the direction of $\vec{v}$
a) $\langle 8,6\rangle$
b) $\langle-2,3 \sqrt{7}\rangle$
VIDEO: Unit vectors – Example 22.9
The second operation on vectors is called vector addition.
Definition 22.10. Let $\vec{v}=\langle a, b\rangle$ and $\vec{w}=\langle c, d\rangle$ be two vectors. Then the vector addition $\vec{v}+\vec{w}$ is defined by component-wise addition:
$\langle a, b\rangle+\langle c, d\rangle:=\langle a+c, b+d\rangle$
In the plane, this corresponds to starting at the origin, following $\vec{v}$ and then $\vec{w}$ (or vice versa, following $\vec{w}$ and then $\vec{v}$). In the picture, note that whichever path you take from the origin you will still arrive at the same point in the upper right, $\vec{v}+\vec{w}$:
Example 22.11. Perform the vector addition and simplify as much as possible.
a) $\langle 3,-5\rangle+\langle 6,4\rangle$
b) $5 \cdot\langle-6,2\rangle-7 \cdot\langle 1,-3\rangle$
c) $4 \vec{i}+9 \vec{j}$
d) find $2 \vec{v}+3 \vec{w}$ for $\vec{v}=-6 \vec{i}-4 \vec{j}$ and $\vec{w}=10 \vec{i}-7 \vec{j}$
e) find $-3 \vec{v}+5 \vec{w}$ for $\vec{v}=\langle 8, \sqrt{3}\rangle$ and $\vec{w}=\langle 0,4 \sqrt{3}\rangle$
VIDEO: Vector addition – Example 22.11
Example 22.12. The forces $\vec{F}_{1}$ and $\vec{F}_{2}$ are applied to an object. Find the resulting total force $\vec{F}=\vec{F}_{1}+\vec{F}_{2}$. Determine the magnitude and directional angle of the total force $\vec{F}$. Approximate these values as necessary. Recall that the international system of units for force is the newton $\left[1 N=1 \frac{k g \cdot m}{s^{2}}\right]$
a) $\vec{F}_{1}$ has magnitude 3 newtons, and angle $\theta_{1}=45^{\circ}$
$\vec{F}_{2}$ has magnitude 5 newtons, and angle $\theta_{2}=135^{\circ}$
b) $\left|\vec{F}_{1}\right|=7$ newtons, and $\theta_{1}=\frac{\pi}{6},$ and $\left|\vec{F}_{2}\right|=4$ newtons, and $\theta_{2}=\frac{5 \pi}{3}$
VIDEO: Vector addition application (force) – Example 22.12
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