

# Foundational concepts: limits and continuity

Limits and continuity are fundamental concepts in calculus. Understanding them is crucial for any further study in the field.

A limit is a concept that represents the value a function approaches as its input gets infinitely close to a certain point. In other words, it is the value a function takes when its input is very close to a given value.

Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$. We want to find the limit of this function as $x$ approaches 1.

$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1}$$

## Exercise
Find the limit of the following function as $x$ approaches 2.

$$\lim_{x \to 2} \frac{x^3 - 8}{x - 2}$$

Continuity is a property of functions that indicates whether they are smoothly connected at a certain point. A function is continuous at a point if the limit of the function as $x$ approaches that point exists and is equal to the value of the function at that point.

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

## Exercise
Is the function $f(x) = \frac{1}{x}$ continuous at $x = 0$?



# Derivatives: definition and applications

Derivatives are a fundamental concept in calculus that deal with the rate of change of a function. They are used to analyze the behavior of functions and to find their maximum and minimum points.

The derivative of a function is defined as the limit of the ratio of the difference in the function's output to the difference in its input, as the difference approaches zero.

Find the derivative of the function $f(x) = x^2$.

## Exercise
Instructions:

Find the derivative of the function $f(x) = x^2$.

### Solution
None

Derivatives have numerous applications in various fields, including physics, engineering, and economics. They are used to model the behavior of objects in motion, to find the maximum and minimum values of a function, and to analyze the growth rate of a function.

Use the derivative to find the maximum and minimum values of the function $f(x) = x^3$.

## Exercise
Instructions:

Find the maximum and minimum values of the function $f(x) = x^3$.

### Solution
None



# The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a fundamental concept in calculus that connects the concepts of integration and differentiation. It states that the integral of a function's derivative is equal to the function itself, plus a constant.

Consider the function $f(x) = x^3$. Use the Fundamental Theorem of Calculus to find the integral of $f'(x)$.

## Exercise
Instructions:

Use the Fundamental Theorem of Calculus to find the integral of $f'(x)$, where $f(x) = x^3$.

### Solution
None

The Fundamental Theorem of Calculus has numerous applications in various fields, including physics, engineering, and economics. It is used to find the area under a curve, to calculate the work done by a force, and to model the growth rate of a function.

Use the Fundamental Theorem of Calculus to find the area under the curve of the function $f(x) = x^2$ between $x = 0$ and $x = 2$.

## Exercise
Instructions:

Find the area under the curve of the function $f(x) = x^2$ between $x = 0$ and $x = 2$.

### Solution
None



# Integration techniques: substitution and integration by parts

Integration techniques are methods used to find the integral of a function. There are several integration techniques, including substitution and integration by parts.

Substitution is an integration technique that involves replacing the variable in the integral with a new variable. This method is useful for solving integrals that involve the product of a function and its derivative.

Use substitution to find the integral of the function $f(x) = x^2$.

## Exercise
Instructions:

Find the integral of the function $f(x) = x^2$.

### Solution
None

Integration by parts is an integration technique that involves the product of a function and its derivative. This method is useful for solving integrals that involve the product of a function and its derivative.

Use integration by parts to find the integral of the function $f(x) = x^3$.

## Exercise
Instructions:

Find the integral of the function $f(x) = x^3$.

### Solution
None



# Sequences and series: arithmetic, geometric, and harmonic

Sequences and series are mathematical objects that represent a collection of numbers. Sequences are ordered lists of numbers, while series are the sums of the terms in a sequence.

Arithmetic sequences are sequences where the difference between consecutive terms is constant. Geometric sequences are sequences where each term is the product of the previous term and a constant ratio. Harmonic sequences are sequences where each term is the reciprocal of the previous term.

Find the sum of the arithmetic sequence $a_n = 4n + 1$ from $n = 1$ to $n = 10$.

## Exercise
Instructions:

Find the sum of the arithmetic sequence $a_n = 4n + 1$ from $n = 1$ to $n = 10$.

### Solution
None



# Infinite series and convergence

Infinite series are mathematical objects that represent the sum of an infinite number of terms. Convergence is a property of infinite series that indicates whether the sum of the series is a finite value.

There are several types of convergence for infinite series, including absolute convergence, conditional convergence, and divergence.

Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges.

## Exercise
Instructions:

Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges.

### Solution
None



# The Riemann integral and the concept of area under a curve

The Riemann integral is a mathematical concept that represents the area under a curve. It is defined as the limit of the sum of the areas of rectangles that approximate the curve.

Find the Riemann integral of the function $f(x) = x^2$ from $x = 0$ to $x = 2$.

## Exercise
Instructions:

Find the Riemann integral of the function $f(x) = x^2$ from $x = 0$ to $x = 2$.

### Solution
None

The concept of area under a curve is essential in various fields, including physics, engineering, and economics. It is used to calculate the work done by a force, the area of a region, and the growth rate of a function.

Use the concept of area under a curve to find the work done by a force of $f(x) = x^2$ from $x = 0$ to $x = 2$.

## Exercise
Instructions:

Find the work done by a force of $f(x) = x^2$ from $x = 0$ to $x = 2$.

### Solution
None



# Applications of calculus in real-world scenarios

Calculus has numerous applications in real-world scenarios, including physics, engineering, and economics.

In physics, calculus is used to model the behavior of objects in motion, to find the maximum and minimum values of a function, and to analyze the growth rate of a function.

Use calculus to find the maximum and minimum values of the function $f(x) = x^2$.

## Exercise
Instructions:

Find the maximum and minimum values of the function $f(x) = x^2$.

### Solution
None

In engineering, calculus is used to design and analyze structures, to model the behavior of objects in motion, and to find the maximum and minimum values of a function.

Use calculus to find the maximum and minimum values of the function $f(x) = x^3$.

## Exercise
Instructions:

Find the maximum and minimum values of the function $f(x) = x^3$.

### Solution
None

In economics, calculus is used to model the growth rate of a function, to find the equilibrium points of a function, and to analyze the behavior of a function in response to external factors.

Use calculus to find the equilibrium points of the function $f(x) = x^2$.

## Exercise
Instructions:

Find the equilibrium points of the function $f(x) = x^2$.

### Solution
None

