Grade 12
  1. Algebra  1.1. Matrices  1.1.1. Definition and Types of Matrices  1.1.2. Operations on Matrices  1.1.3. Transpose of a matrix  1.1.4. Determinants and their properties  1.1.5. Inverse of a Matrix  1.1.6. Solving Linear Equations Using Matrices  1.1.7. Applications of Matrices  1.2. Complex Numbers  1.2.1. Definition and representation  1.2.2. Algebra of complex numbers  1.2.3. Modulus and argument  1.2.4. Polar and Exponential Forms  1.2.5. De Moivre's Theorem  1.2.6. Roots of complex numbers  1.2.7. Applications in Geometry  1.3. Vectors and 3D Geometry  1.3.1. Vector Algebra  1.3.2. Scalar and Vector Products  1.3.3. Equation of a Line in Space  1.3.4. Equation of a Plane  1.3.5. Distance between lines and planes  1.3.6. Angle between Lines and Planes
  2. Calculus  2.1. Limits and Continuity  2.1.1. Concept of Limits  2.1.2. Continuity of Functions  2.1.3. Types of Discontinuities  2.1.4. Left-hand and Right-hand Limits  2.2. Differentiation  2.2.1. Derivative as a Rate of Change  2.2.2. Techniques of Differentiation  2.2.3. Higher Order Derivatives  2.2.4. Applications of Derivatives  2.2.5. Tangents and Normals  2.2.6. Increasing and Decreasing Functions  2.3. Integration  2.3.1. Indefinite Integrals  2.3.2. Definite Integrals  2.3.3. Integration Techniques  2.3.4. Area under a curve  2.3.5. Applications of Integrals in Geometry  2.4. Differential Equations  2.4.1. Formation of Differential Equations  2.4.2. Order and Degree of Differential Equations  2.4.3. Solutions of Differential Equations  2.4.4. Applications of Differential Equations  2.4.5. Solving First-Order Linear Differential Equations
  3. Probability and Statistics  3.1. Probability  3.1.1. Conditional Probability  3.1.2. Bayes' Theorem  3.1.3. Random Variables  3.1.4. Probability Distributions  3.1.5. Binomial and Normal Distributions  3.2. Statistics  3.2.1. Mean and Standard Deviation  3.2.2. Variance and Covariance  3.2.3. Correlation and Regression  3.2.4. Hypothesis Testing
  4. Linear Programming  4.1. Graphical Method  4.1.1. Feasible and Infeasible Regions  4.1.2. Optimal solutions  4.1.3. Sensitivity Analysis in Graphical Method  4.2. Simplex Method  4.2.1. Formulation of Problems  4.2.2. Solving linear programming problems  4.2.3. Dual Simplex Method
  5. Relations and Functions  5.1. Types of Relations  5.1.1. Reflexive Relations  5.1.2. Symmetric Relations  5.1.3. Transitive Relations  5.1.4. Equivalence Relations  5.2. Types of Functions  5.2.1. Onto Functions  5.2.2. One-to-One Functions  5.2.3. Inverse Functions  5.2.4. Composite Functions  5.3. Inverse Trigonometric Functions  5.3.1. Principal Values in Inverse Trigonometric Functions  5.3.2. Properties and Graphs of Inverse Trigonometric Functions  5.3.3. Applications in Calculus
  6. Advanced Trigonometry  6.1. Trigonometric Equations  6.1.1. General solutions of equations  6.1.2. Transformation formulae  6.2. Hyperbolic Functions  6.2.1. Definitions and properties  6.2.2. Relationships between hyperbolic and trigonometric functions
  7. Mathematical Reasoning  7.1. Logical Reasoning  7.1.1. Statements and Logical Connectives  7.1.2. Tautologies and Contradictions  7.2. Proofs  7.2.1. Direct proof  7.2.2. Indirect Proof  7.2.3. Proof by contradiction  7.2.4. Proof by Mathematical Induction
  8. Applications of Mathematics  8.1. Applications of Calculus  8.1.1. Maxima and Minima Problems  8.1.2. Economics and Biology Applications  8.1.3. Rate of Change in Physics  8.2. Applications of Probability  8.2.1. Risk Analysis  8.2.2. Decision Making  8.2.3. Game Theory

Grade 12 ↓


Calculus


Introduction to calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of small quantities. It is mainly divided into two parts: differential calculus and integral calculus. Differential calculus deals with rates of change (derivatives), while integral calculus deals with the accumulation of quantities (integrals and area under curves).

Differential calculus

Let's start with differential calculus. Imagine you're driving a car and you want to know how fast you're going at a particular time. This is where the concept of differential comes into play; it helps us find the instantaneous rate of change.

In simple terms, differentiation can be understood as finding the slope of a curve. If you have a function f(x), then the derivative f'(x) tells you the slope of the tangent at any point x on that curve.

Basics of finding the derivative

Let's take a simple function f(x) = x^2. To find its derivative:

f'(x) = 2x

This derivative tells us that at any point x, the slope of the function is 2x. When x = 2, the slope (or rate of change) of the function is 4 Let's look at this graphically:

X y = x^2 tangent line

Rules of differentiation

There are several rules to make the functions easily differentiable:

  • Power Rule: d/dx [x^n] = n*x^(n-1)
  • Sum Rule: d/dx [u + v] = du/dx + dv/dx
  • Multiplication Rule: If u(x) and v(x) are functions, then d/dx [u*v] = u'v + uv'
  • Quotient Rule: d/dx [u/v] = (u'v - uv')/v^2
  • Chain Rule: If a function is composed of other functions, f(g(x)), then the derivative is f'(g(x)) * g'(x)

Integral calculus

Now, let's discuss integral calculus. If differential calculus means taking things apart to see how they change, integral calculus means bringing things together to understand the whole.

Integrals are used to calculate areas under curves, volumes, central points, etc. For any function f(x), the integral is represented as:

∫ f(x) dx

Definite and indefinite integrals

Integrals can be classified into indefinite and definite integrals. An indefinite integral has no specific limit:

∫ x^2 dx = x^3/3 + c

Here, C is the constant of integration.

A definite integral calculates the area under a curve between two points a and b :

∫[a, b] x^2 dx = [x^3/3] from a to b = (b^3/3) - (a^3/3)
A B area under the curve

Fundamental theorem of calculus

The fundamental theorem of calculus connects differentiation and integration and states two important points:

  1. The derivative of the integral of a function is the original function itself.
  2. If you integrate the derivative from a to b, you'll get the value of the original function at b minus its value at a.

This is mathematically represented as:

d/dx (∫[a, x] f(t) dt) = f(x)
∫[a, b] f'(x) dx = f(b) - f(a)

Application of calculus: Real-world example

Calculus is widely used in various fields such as physics, engineering, economics, statistics, and even in the analysis of biological phenomena.

Example: Calculus in physics

In physics, calculus is used to study motion. Suppose a particle is moving in a straight line. Its position at any time t is given by s(t) = t^3 - 3t^2 + 2t. Let's find the velocity and acceleration of the particle:

  • Velocity: Velocity v(t) is the first derivative of the position function. 
    v(t) = ds/dt = 3t^2 - 6t + 2
  • Acceleration: Acceleration a(t) is the derivative of the velocity function. 
    a(t) = dv/dt = 6t - 6
Post Tea

Looking at the above position-time graph gives an idea of the path of the particle over time. Its velocity graph will look steeper at certain intervals, where the slope of the position graph is higher, indicating higher speed.

Conclusion

Calculus, though complex, is a beautiful chapter in mathematics that enhances our understanding of changes and areas under curves. From engineering feats to natural phenomena, calculus has applications almost everywhere. While derivatives help to understand instantaneous changes, integrals help to summarize everything to show the bigger picture.

The key to mastering Calculus lies in practicing its various questions. Look at each problem as a small puzzle, and with time and patience, the solutions will come naturally. It is not just about calculations, but also about appreciating the mathematical world in its finest.


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