Calculus

Calculus is a branch of mathematics that deals with the study of rates of change. It is a fundamental tool in many fields of science and engineering, and is essential for understanding the behavior of functions and solving problems involving motion, growth, and change. In this module, we will introduce two basic concepts of calculus that are essential for understanding the subject: differentiation and integration.

Differentiation

Differentiation is the process of finding the rate at which a function changes with respect to one of its variables. The derivative of a function is a measure of how the function changes as its input changes. It is defined as the limit of the average rate of change of the function over a small interval as the interval approaches zero.

The derivative of a function $f(x)$ with respect to the variable $x$ is denoted by $f'(x)$ (prime notation), $\frac{df}{dx}$ (Leibniz notation), or $\dot{f}(x)$ (Newton's notation). It represents the rate of change of the function with respect to $x$. The derivative of a function can be interpreted geometrically as the slope of the tangent line to the graph of the function at a given point.

The derivative of a function can be calculated using the rules of differentiation, which are based on the properties of elementary functions such as polynomials, exponentials, logarithms, and trigonometric functions.

Rules of Differentiation

The following are some of the basic rules of differentiation that we will be using in this course:

1. Power Rule: If $f(x) = x^n$, then $f'(x) = nx^{n-1}$.
2. Sum Rule: If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$.
3. Product Rule: If $f(x) = g(x)h(x)$, then $f'(x) = g'(x)h(x) + g(x)h'(x)$.
4. Quotient Rule: If $f(x) = \frac{g(x)}{h(x)}$, then $f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}$.
5. Chain Rule: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x))h'(x)$.
6. Trigonometric Functions: $\frac{d}{dx} \sin(x) = \cos(x)$, $\frac{d}{dx} \cos(x) = -\sin(x)$, $\frac{d}{dx} \tan(x) = 1 + \tan^2(x)$.

Integration

Integration is the process of finding the area under a curve or the accumulation of a quantity over an interval. The integral of a function is a measure of the total quantity represented by the function over a given interval. It is defined as the limit of the sum of the function values times the width of the intervals as the number of intervals approaches infinity.

The integral of a function $f(x)$ with respect to the variable $x$ is denoted by $\int f(x) dx$. It represents the area under the curve of the function over the interval $[a, b]$. The integral of a function can be interpreted geometrically as the area under the curve of the function between two points.

Antiderivative

The antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$. In other words, the antiderivative of a function is the reverse process of differentiation. The antiderivative of a function is also known as the indefinite integral of the function.

Rules of Integration

The following are some of the basic rules of integration that we will be using in this course:

1. Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.
2. Sum Rule: $\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx$.
3. Constant Multiple Rule: $\int k f(x) dx = k \int f(x) dx$, where $k$ is a constant.
4. Substitution Rule: If $u = g(x)$, then $\int f(g(x)) g'(x) dx = \int f(u) du$.
5. Trigonometric Functions: $\int \sin(x) dx = -\cos(x) + C$, $\int \cos(x) dx = \sin(x) + C$, $\int \tan(x) dx = -\log|\cos(x)| + C$.

Summary

• Calculus is a powerful mathematical tool that is used to study the behavior of functions and solve problems involving motion, growth, and change.
• Differentiation is the process of finding the rate at which a function changes with respect to one of its variables.
• Integration is the process of finding the area under a curve or the accumulation of a quantity over an interval.

Problem Set

Problem 1

Find the derivative of the following functions:

1. $f(x) = x^2 + 3x - 5$
2. $g(x) = \sin(x) + \cos(x)$
3. $h(x) = \frac{1}{x^3} + \tan(x)$
4. $k(x) = \cos^2(x^2)$

Problem 2

Find the integral of the following functions:

1. $\int (x^2 + 3x - 5) dx$
2. $\int (\sin(x) + \cos(x)) dx$
3. $\int (\frac{1}{x^3} + \cos(x)) dx$