AOS3 Topic 1: Basics of Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It forms the foundation for much of modern science and engineering. Here are the fundamental concepts:

1. Limits: The Foundation of Calculus

Definition: A limit is the value that a function (or sequence) approaches as the input (or index) approaches some value.

Notation: \( \lim_{{x \to c}} f(x) = L \)

Concept: Understanding limits is crucial for grasping how functions behave near specific points, even if they don't actually reach those points.

2. Derivatives: The Rate of Change

Definition: The derivative measures how a function changes as its input changes. It is the instantaneous rate of change, or the slope of the function at any point.

Notation: \( f'(x) \) or \( \frac{dy}{dx} \)

Basic Formula: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

Concept: Derivatives are used to find tangents to curves, optimize functions, and model dynamic systems.

Continuity

A function \( f(x) \) is said to be continuous at a particular point \( x = a \), if the following three conditions are satisfied:

  • \( f(a) \) is defined
  • \( \lim_{{x \to a}} f(x) \) exists
  • \( \lim_{{x \to a}} f(x) = f(a) \)

Continuity and Differentiability

A function is always continuous if it is differentiable at any point, whereas the vice-versa condition is not always true.

Integral Calculus Basics

Integral calculus is the study of integrals and their properties. It is mostly useful for the following two purposes:

  • To calculate \( f \) from \( f' \) (i.e., from its derivative). If a function \( f \) is differentiable in the interval of consideration, then \( f' \) is defined in that interval.
  • To calculate the area under a curve.

Integration

Integration is the reciprocal of differentiation. As differentiation can be understood as dividing a part into many small parts, integration can be said as a collection of small parts in order to form a whole. It is generally used for calculating areas.

Definite Integral

A definite integral has a specific boundary within which the function needs to be calculated. The lower limit and upper limit of the independent variable of a function are specified; its integration is described using definite integrals. A definite integral is denoted as:

\[ \int_{a}^{b} f(x) \, dx \]

Indefinite Integral

An indefinite integral does not have a specific boundary, i.e., no upper and lower limit is defined. Thus the integration value is always accompanied by a constant value (C). It is denoted as:

\[ \int f(x) \, dx = F(x) + C \]

3. Integrals: The Accumulation of Quantity

Definition: The integral of a function represents the accumulation of quantities, such as area under a curve.

Notation: \( \int f(x) \, dx \)

Basic Formula: If \( F(x) \) is the antiderivative of \( f(x) \), then \( \int f(x) \, dx = F(x) + C \).

Concept: Integrals are used to calculate areas, volumes, and other quantities that accumulate continuously.

4. Fundamental Theorem of Calculus

Statement: This theorem links the concept of differentiation and integration. It has two parts:

First Part: If \( F \) is an antiderivative of \( f \) on an interval \([a, b]\), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).

Second Part: If \( f \) is continuous on \([a, b]\), then the function \( F(x) = \int_{a}^{x} f(t) \, dt \) is continuous on \([a, b]\), differentiable on \((a, b)\), and \( F'(x) = f(x) \).

Applications of Calculus

Calculus is a mathematical model that helps us analyze a system to find an optimal solution and predict the future. In real life, concepts of calculus play a major role in various fields, whether it is related to solving the area of complicated shapes, ensuring the safety of vehicles, evaluating survey data for business planning, managing credit card payment records, or understanding how changing conditions of a system affect us.

Calculus is the language of physicians, economists, biologists, architects, medical experts, statisticians, and it is often used by them. For example, architects and engineers use concepts of calculus to determine the size and shape of curves to design bridges, roads, and tunnels.

Using calculus, some concepts are beautifully modeled, such as birth and death rates, radioactive decay, reaction rates, heat and light, motion, electricity, and more.

Example 1

Limits

Find the limit of \( \frac{x^2 - 1}{x - 1} \) as \( x \) approaches 1.

Solution:

\[ \lim_{{x \to 1}} \frac{x^2 - 1}{x - 1} = \lim_{{x \to 1}} \frac{(x - 1)(x + 1)}{x - 1} = \lim_{{x \to 1}} (x + 1) = 2 \]

Example 2

Finding a Limit

Evaluate the limit:

\[ \lim_{{x \to 3}} \frac{x^2 - 9}{x - 3} \]

Solution:

At first glance, direct substitution gives a zero in the denominator. So, we need to simplify the expression:

\[ \frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3} \]

Now, cancel out the common factor \( (x - 3) \):

\[ \frac{(x - 3)(x + 3)}{x - 3} = x + 3 \]

Now, substitute \( x = 3 \):

\[ \lim_{{x \to 3}} (x + 3) = 3 + 3 = 6 \]

Thus, the limit is 6.

Example 3

Finding a Derivative

Find the derivative of the function \( f(x) = 3x^2 + 5x - 7 \).

Solution:

Using the basic derivative rules:

\[ f'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(5x) - \frac{d}{dx}(7) \]

The derivative of \( 3x^2 \) is \( 6x \), the derivative of \( 5x \) is \( 5 \), and the derivative of a constant is zero:

\[ f'(x) = 6x + 5 \]

So, the derivative is \( f'(x) = 6x + 5 \).

Example 4

Continuity

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

Solution:

For the function to be continuous at \( x = 0 \), three conditions must be satisfied:

  • \( f(0) \) must be defined, but it is not (division by zero).
  • Thus, the function is not continuous at \( x = 0 \).

Example 5

Definite Integral

Evaluate the definite integral:

\[\int_{0}^{2} (3x^2) \, dx\]

Solution:

First, find the indefinite integral:

\[ \int 3x^2 \, dx = x^3 + C \]

Now, apply the limits from 0 to 2:

\[ \int_{0}^{2} 3x^2 \, dx = \left[ x^3 \right]_{0}^{2} = (2^3) - (0^3) = 8 - 0 = 8 \]

Thus, the value of the definite integral is 8.

Example 6

Applying the Fundamental Theorem of Calculus

Let \( f(x) = 3x^2 \). Use the Fundamental Theorem of Calculus to evaluate the integral:

\[ \int_{1}^{3} 6x \, dx \]

Solution:

First, find the antiderivative:

\[ F(x) = 3x^2 \]

Now, apply the limits:

\[ \int_{1}^{3} 6x \, dx = F(3) - F(1) = 3(3)^2 - 3(1)^2 = 27 - 3 = 24 \]

Thus, the value of the definite integral is 24.

Exercise &&1&& (&&1&& Question)

What is the value of the limit?

\[\lim_{{x \to 0}} \frac{\sin(x)}{x}\]

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Exercise &&2&& (&&1&& Question)

Find the derivative of the function \( f(x) = 5x^3 \).

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Exercise &&3&& (&&1&& Question)

Evaluate the definite integral:

\[\int_{0}^{2} 4x \, dx\]

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Exercise &&4&& (&&1&& Question)

What is the antiderivative of \( f(x) = 3x^2 \)?

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Exercise &&5&& (&&1&& Question)

At which point is the derivative of the function \( f(x) = |x| \) not defined?

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Exercise &&6&& (&&1&& Question)

What is the period of the function \( f(x) = \sin(x) \)?

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