Calculus (IB)

1. Introduction to Calculus

Calculus is the branch of mathematics that studies continuous change. It is divided into two major parts:

• Differential Calculus: Concerns the study of rates of change and slopes of curves.

• Integral Calculus: Involves accumulation of quantities and areas under and between curves.

Calculus is fundamental in various fields such as physics, engineering, economics, and biology.

2. Limits and Continuity

2.1 Definition of a Limit

The limit of a function as approaches is written as: This means that as gets arbitrarily close to , approaches .

2.2 Techniques for Evaluating Limits

• Direct Substitution: Plugging in directly.

• Factoring: Factor and simplify before substitution.

• Rationalization: Multiply by the conjugate if necessary.

• L'Hôpital's Rule: Used for indeterminate forms (e.g., ).

2.3 Continuity

A function is continuous at if:

1. exists.

2. is defined.

3. .

Discontinuities can be classified as:

• Removable Discontinuities: Can be fixed by redefining .

• Jump Discontinuities: Sudden jumps in function values.

• Infinite Discontinuities: Asymptotic behavior.

3. Differentiation

3.1 Definition of the Derivative

The derivative of is defined as: It represents the instantaneous rate of change of .

3.2 Differentiation Rules

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

3.3 Higher-Order Derivatives

• Second derivative: represents acceleration in physics.

• Higher derivatives: , , etc.

3.4 Applications of Derivatives

• Tangent and Normal Lines: Equation of the tangent at is:

• Increasing and Decreasing Functions:

  • is increasing.

  • is decreasing.

• Concavity and Inflection Points:

  • is concave up.

  • is concave down.

• Optimization: Finding local maxima and minima.

4. Integration

4.1 Definition of the Integral

The integral is the reverse process of differentiation. The indefinite integral is: where is the constant of integration.

The definite integral is defined as: and represents the signed area under the curve from to .

4.2 Integration Rules

• Power Rule: (for ).

• Substitution Rule: .

• Integration by Parts: .

• Partial Fractions: Used for rational functions.

4.3 Applications of Integration

• Area Under a Curve: gives the area.

• Volume of Revolution:

  • Disk Method: .

  • Shell Method: .

• Average Value of a Function:

• Kinematics: Finding displacement and total distance traveled.

5. Differential Equations

5.1 Definition

A differential equation relates a function with its derivatives. It can be:

• Ordinary Differential Equations (ODEs): Involving one independent variable.

• Partial Differential Equations (PDEs): Involving multiple independent variables.

5.2 Solving Differential Equations

• Separable Equations: can be rewritten as:

• First-Order Linear Equations: In the form , solved using integrating factors.

5.3 Applications of Differential Equations

• Exponential Growth and Decay: leads to .

• Newton’s Law of Cooling: .

• Logistic Growth: models population growth.

6. Conclusion

Calculus is an essential mathematical tool used to model and analyze change in various disciplines. Mastery of differentiation and integration, along with their applications, is crucial for solving real-world problems.