Hi everyone! Today is the last official class day before the final exam. I’ll be holding a final exam review session live on Zoom during our normal class time, 10:00 – 11:40am.
I’ll be using the time to answer your questions – either general questions, or questions from the review sheet, or questions from the practice final exams in WeBWorK. Please bring your questions!
Having trouble joining the meeting? Please comment below, I’ll do my best to check back regularly.
Hi everyone! Read through the material below, watch the videos, and send me your questions. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.
Lesson 24: The geometric series
Lesson Date: Tuesday, May 12th.
Topic: This lesson covers Chapter 24: The geometric series
WeBWorK: There are two WeBWorK assignments on today’s material, due in one week:
Sequences – Geometric
Series – Geometric
Question of the day: Can we add up infinitely many numbers?
Lesson NOtes (Notability – pdf):
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
Infinite geometric series
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?
Just a quick update to share two resources for preparing for the final exam:
Take a practice exam (or more than one!) on WeBWorK. The department has prepared a practice final exam – your actual final exam will be similar.
Where can I find the practice exam? The practice exams can be found in the “Final Exam WeBWorK” area (NOT our usual WeBWorK site), the same location where our actual final exam will take place
What is my login to the Final Exam WeBWorK area? You should have received an email sometime Friday, 5/8/20, with the title Practice Exams and Final Exam Login Information from the address WeBWorK Administrator <webwork@citytech.cuny.edu>. This email contains your exam WeBWorK account login and password, and a link to the WeBWorK final exam area for our class.
How many times can I take the Practice Final in WeBWorK? You can one practice final exam per day, from now until the actual finals begin. You will have 2 hours to complete the practice exam once you begin (just like a real exam).
Take a look at the official departmental Final Exam review sheet (follow the link and you will also find videos of the review problems worked out in their entirety).
Hi everyone! This is the second of our distance-learning lessons. Read through the material below, watch the videos, and send me your questions.
NEW THIS WEEK: Daily Quiz & Attendance. There will be a short quiz each class day. Complete the quiz before the end of the day to be marked present for today’s class (the quiz should appear below this post).
Lesson 13: Exponential and Logarithmic Functions
Lesson Date: Tuesday, March 24th.
Topic: This lesson covers Chapter 13 in the book, Exponential and Logarithmic Functions.
WeBWorK: There are two WeBWorK assignments on today’s material, due next Tuesday 3/31: Exponential Functions - Graphs and Logarithmic Functions - Graphs .
Lesson NOtes (Notability – pdf):
This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.
We’ve been living in the world of Polynomials and Rational Functions. We now turn to exponential functions. These functions are “very natural” – that is, they show up in the real world – but they are also more complicated than Polynomial and Rational functions (for example, an exponential function grows more quickly than any Polynomial)
The spread of coronavirus, like other infectious diseases, is modeled by exponential functions.
Definition. An exponential function is a function of the form $f(x)=c\cdot b^x$, where $b$ and $c$ are real numbers and $b$ is positive ($b$ is called the base, $x$ is the exponent).
Example 1 (Textbook 13.2): Graph the exponential functions $f(x)=2^x, g(x)=3^x, h(x)=10^x, k(x)=\left(\frac{1}{2}\right)^x, l(x)=\left(\frac{1}{10}\right)^x$.
Now let’s see what happens when we change the number $c$ in $y=c\cdot b^x$.
Example 2 (Textbook 13.6): Graph the exponential functions a) $y=2^{x}, \quad$ b) $y=3 \cdot 2^{x}, \quad$ c) $y=(-3) \cdot 2^{x}, \quad$ d) $y=0.2 \cdot 2^{x}, \quad$ e) $y=(-0.2) \cdot 2^{x}$
Example 3: The graph below shows an exponential function $f(x)$. Find a formula for $f(x)$.
Logarithmic Functions and their Graphs
Definition. If $b$ is a positive real number and $b\neq 1$, then the logarithm with base $b$ is defined: $y=\log_b(x) \iff b^y=x$
What does the definition of logarithm mean? The idea is that the logarithm is the inverse function of the exponential function. Let’s look at an example.
Question: Is an exponential function one-to-one? (What does one-to-one means).
Example 4. The graph below shows the function $y=\log_2(x)$ but shifted to the right 3 units. Find a formula for the function in the graph.
That’s it for now! Take a look at the WeBWorK assignment, leave your questions below (or use the Ask for Help button in WeBWorK, or send me an email) and don’t forget to complete Quiz#1.
For now, we are going to try out taking a quiz each class day, based on the material from the previous class. The quizzes will count towards the OpenLab portion of your grade, and they will also be used to track your attendance (to be marked PRESENT on Tuesday, 3/24, you must take the quiz before midnight).
Today’s lecture will be posted later this morning.
As always, let me know if you have any questions or problems. Stay healthy!
Prof. Reitz
Daily Quiz 1: 3/24/20
Our first daily quiz.
NOTE: This quiz MUST be taken on 3/24 to record your attendance.
Hi everyone! This is the first of our distance-learning lessons. Read through the material below, watch the videos, and send me your questions.
Lesson 12: Polynomial and Rational Inequalities
Lesson Date: Thursday, March 19th.
Topic: This lesson covers Chapter 12 in the book, Polynomial and Rational Inequalities.
WeBWorK: There are two WeBWorK assignments on today’s material, due next Thursday 3/26: Polynomials - Inequalities, and Rational Functions - Inequalities.
Lesson NOtes (Notability – pdf):
This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.
Definition. A polynomial inequality is an inequality (which means it uses one of these: $<,\leq,>,\geq$ instead of an equals sign) with a polynomial on each side
Example 1: $x^2-3x-4\geq 0$
We’re interested in solving these inequalities, which means answering the question: “For which real numbers x is the inequality true?”
Now let’s look at the same example, and see how to solve it without looking at the graph:
Example 2: Solve $x^{4}-x^{2}>5\left(x^{3}-x\right)$
Example 2, concluded:
Rational inequalities
What happens if we allow rational functions instead of just polynomials?
Example 3: Solve $\frac{x^{2}-5 x+6}{x^{2}-5 x} \geq 0$
Good job! You are now ready to practice on your own. Take a look at the WeBWorK assignment, and don’t forget to use the “Ask for Help” button if you get stuck.
