Category: Lessons

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 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 .

Exponential Functions and their Graphs

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) $y=(-0.2) \cdot 2^{x}$

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.

Best of luck,
Prof. Reitz

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 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.

Introduction to polynomial inequalities

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.

Here are more video resources if you’d like to see additional examples.

ASSIGNMENT: Watch videos, try webwork. Ask questions in comments below, or using the “Ask for Help” button in WeBWorK. Good luck, and stay safe!