

# Real numbers and sequences



A sequence is an ordered collection of real numbers. Sequences can be used to model various phenomena in the real world, such as the growth of a population over time or the temperature of a room at different times of the day.

Consider the sequence of temperatures in a room over the course of a day:

$$
T(t) = 20 + 5t
$$

This sequence represents the temperature in degrees Celsius at time $t$ (in hours). The temperature starts at $20$ degrees and increases by $5$ degrees for every hour that passes.

## Exercise
Create a sequence that models the population of a city over the course of a year. The population starts at $100000$ and increases by $1000$ for every month that passes.

# Limit of a sequence

As we mentioned earlier, a sequence is an ordered collection of real numbers. The limit of a sequence is the value that the sequence approaches as the terms of the sequence get infinitely close to it.

To find the limit of a sequence, we need to determine the value that the sequence is converging to. This can be done by analyzing the behavior of the terms of the sequence as they get infinitely close to the limit.

Consider the sequence:

$$
a_n = \frac{1}{n}
$$

We can see that as $n$ gets larger, the terms of the sequence get infinitely close to $0$. Therefore, the limit of this sequence is $0$.

## Exercise
Find the limit of the following sequence:

$$
b_n = \frac{n^2 - 1}{n^2 + 1}
$$

# Continuity and its properties



A function is continuous at a point $x_0$ if the limit of the function as $x$ approaches $x_0$ is equal to the value of the function at $x_0$.

Consider the function $f(x) = x^2$. This function is continuous because the limit of $f(x)$ as $x$ approaches any value is equal to the value of $f(x)$ at that value.

## Exercise
Determine whether the function $g(x) = \frac{1}{x}$ is continuous at $x = 0$.

# Derivatives and their properties



The derivative of a function $f(x)$ is denoted as $f'(x)$ and is defined as the limit of the difference quotient as the difference between the points approaches $0$.

Consider the function $h(x) = x^3$. The derivative of this function is $h'(x) = 3x^2$.

## Exercise
Find the derivative of the function $j(x) = \sqrt{x}$.

# Integrals and their properties



The integral of a function $f(x)$ is denoted as $\int f(x) dx$ and is defined as the antiderivative of $f(x)$.

Consider the function $k(x) = x^2$. The integral of this function is $\int k(x) dx = \frac{x^3}{3} + C$.

## Exercise
Find the integral of the function $l(x) = \sqrt{x}$.

# Applications of continuity and derivatives in real-world scenarios



For example, continuity is used in physics to describe the behavior of objects that are in motion, while derivatives are used in engineering to analyze the stress and strain in materials.

## Exercise
Explain how continuity and derivatives can be applied in the field of economics.

# Techniques for evaluating integrals and derivatives



For example, we can use integration by substitution or the chain rule to evaluate derivatives, and we can use numerical methods or integration techniques to evaluate integrals.

## Exercise
Evaluate the following integral using numerical methods:

$$
\int_0^1 \sqrt{x} dx
$$

# Series and their properties



The sum of a series is denoted as $\sum_{n=1}^{\infty} a_n$ and is defined as the limit of the sequence of partial sums as the number of terms approaches infinity.

Consider the geometric series:

$$
\sum_{n=1}^{\infty} \frac{1}{2^n} = 1 + \frac{1}{2} + \frac{1}{2^2} + \cdots
$$

This series converges to $2$.

## Exercise
Determine whether the following series converges or diverges:

$$
\sum_{n=1}^{\infty} \frac{1}{n}
$$

# Convergence and divergence of series



A series converges if the sequence of partial sums approaches a finite value as the number of terms approaches infinity. A series diverges if the sequence of partial sums does not approach a finite value.

Consider the series:

$$
\sum_{n=1}^{\infty} \frac{1}{n^2}
$$

This series converges because the sequence of partial sums approaches a finite value as the number of terms approaches infinity.

## Exercise
Determine whether the following series converges or diverges:

$$
\sum_{n=1}^{\infty} \frac{1}{n}
$$