Alien from another Universe-ity

A retirement account is opened with an initial deposit of $8,500 and earns 8.12% interest compounded monthly. What will the account be worth in 20 years? What if the deposit was calculated using simple interest? Could you see the situation in a graph? From what point one is better than the other?

To calculate the worth of the retirement account in 20 years with compound interest, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = final amount

P = principal amount (initial deposit)

r = annual interest rate (8.12% or 0.0812 as a decimal)

n = number of times interest is compounded per year (monthly, so n = 12)

Reflect on the concepts of linear and non-linear systems. What concepts (only the names) did you need to accommodate the concept of linear and non-linear systems in your mind? What are the simplest linear system and non-linear system you can imagine? In your day to day, is there any occurring fact that can be interpreted as linear systems and non-linear systems? What strategy are you using to get the graph of linear systems and non-linear systems?

Linear and non-linear systems are fundamental concepts in mathematics and have applications in various fields such as physics, engineering, economics, and computer science. These concepts require an understanding of certain terms and principles to accommodate them in our minds.

To comprehend linear systems, I must be familiar with terms such as linear equations, variables, coefficients, constants, and the concept of linearity. A linear system consists of linear equations, where each equation represents a straight line on a graph (Abramson, 2021). The simplest linear system can be represented by a single equation with two variables, such as y = mx + b, where m is the slope and b is the y-intercept. This equation represents a straight line on a graph.

The population of a culture of bacteria is modeled by the logistic equation P(t)= \frac{14,250}{1+29e^{-0.62t}. To the nearest tenth, how many days will it take the culture to reach 75% of its carrying capacity? What is the carrying capacity? What is the initial population for the model? Why a model like P(t)=P_0 \ e^{Kt} , where P_0 is the initial population, would not be plausible? What are the virtues of the logistic model?

1. To find out when the culture will reach 75% of its carrying capacity, we’ll use the same equation as before:

P(t) = 0.75 * Carrying Capacity

The carrying capacity is still 14,250 bacteria. Now, let’s solve for t:

0.75 * 14,250 = 14,250 / (1 + 29 * e^(-0.62 * t))

After some number crunching, we get:

e^(-0.62 * t) = 0.25 / 29

1.What can be said about the domain of the function f ∘ g where f(y) = 4/(y-2) and g(x) = 5/(3x-1)? Express it in terms of a union of intervals of real numbers. Go to www.desmos.com/calculator and obtain the graph of f, g, and f ∘ g .

To determine the domain of the composite function f ∘ g, we need to consider the restrictions imposed by both functions f and g.

Let’s start with the function g(x) = 5/(3x – 1). The only restriction here is that the denominator (3x – 1) cannot be equal to zero, as division by zero is undefined. So, we solve the equation 3x – 1 = 0 to find the value that makes the denominator zero:

3x – 1 = 0

3x = 1

x = 1/3

Therefore, x cannot be equal to 1/3. We can express the domain of g as the set of all real numbers except x = 1/3.

Reflect on the concept of composite and inverse functions. What concepts (only the names) did you need to accommodate these concepts in your mind? What are the simplest composite and inverse functions you can imagine? In your day to day, is there any occurring fact that can be interpreted as composite and inverse functions? What strategy are you using to get the graph of composite and inverse functions?

To understand the concepts of composite and universe functions, I had to familiarize myself with a few related ideas. First, I needed to understand the concept of a function itself. A function is a mathematical relationship between two sets of values, where each input value (the domain) corresponds to exactly one output value (the range) (Abramson, 2021). This understanding was crucial in grasping the concept of composite and inverse functions.

Next, I had to understand what it means for functions to be combined or composed. In the case of composite functions, it involves taking the output of one function and using it as the input for another function. The result is a new function that represents the composition of the two original functions. This idea of combining functions is similar to how we can combine different operations in mathematics to create more complex expressions.

What happens if we graph both ƒ and ƒ^(-1) on the same set of axes, using the x-axis for the input to both ƒ and ƒ^(-1)?

[Suggestion: go to www.desmos.com/calculator and type y = x^3 {-2 < x < 2}, y = x^(1/3) {–2 < x < 2}, and y = x {–2 < x < 2}, and describe the relationship between the three curves.] Then post your own example discussing the difficulty of graph both ƒ and ƒ^(-1) on the same set of axes.

Suppose ƒ: R → R is a function from the set of real numbers to the same set with ƒ(x) = x + 1. We write ƒ^2 to represent ƒ ∘ ƒ and ƒ^(n+1) = ƒ^n ∘ ƒ. Is it true that ƒ^2 ∘ ƒ = ƒ ∘ ƒ^2? Why? Is the set {g : R → R l g ∘ ƒ = ƒ ∘ g} infinite? Why?

If we graph both f and f^{-1} on the same set of axes, using the x-axis for the input to both f and f^{-1}, we can observe some interesting relationships between the curves. Let’s consider the suggested functions:

1. f(x) = x^3, where -2 < x < 2

2. f^{-1}(x) = x^{1/3}, where -2 < x < 2

3. g(x) = x, where -2 < x < 2

When we graph these three curves, we can see that f(x) = x^3 forms a curve that starts from the origin, passes through (-1, -1), (0, 0), and (1, 1), and then extends towards infinity in both positive and negative directions. This curve is symmetric about the origin.