Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 ↓


Calculus


Calculus is a branch of mathematics that helps us understand changes. It is a tool that allows us to investigate how things change and how those changes affect other things. Calculus is used in a wide variety of disciplines, including physics, engineering, economics, statistics, and even in understanding space and time.

Understanding the tasks

Before diving into calculus, it is important to understand the concept of a function. A function is a relation between a set of inputs and a set of possible outputs where each input relates to exactly one output. A function is often expressed as f(x) where x is the input variable.

For example, consider the function f(x) = 2x + 3 This function tells us that for any input value x, the output value will be twice the input value plus three.

Limitations

Limits help us understand the behavior of a function at specific points. The limit of a function describes the value that the function output approaches as the input approaches a certain point. Limits are fundamental in calculus.

For example, let's find the limit of the function f(x) = (x^2 - 1)/(x - 1) as x approaches 1.

    Let f(x) = (x^2 - 1)/(x - 1). Simplify x^2 - 1 to (x - 1)(x + 1). Then f(x) = ((x - 1)(x + 1))/(x - 1). This simplifies to f(x) = x + 1 when x ≠ 1. As x approaches 1, f(x) approaches 2. So, the limit of f(x) as x -> 1 is 2.

Derivatives

The derivative is a fundamental tool in calculus that helps us understand how a function is changing at any given time. The derivative of a function at a point measures its rate of change or the slope of the tangent line at that point.

Consider the function f(x) = x^2. To find the derivative, we look at the change in f(x) as x changes by a small amount, ∆x.

Mathematically, the derivative of f at x is the limit:

    f'(x) = lim (∆x -> 0) [f(x + ∆x) - f(x)] / ∆x

For f(x) = x^2, the derivative is:

    f'(x) = lim (∆x -> 0) [(x + ∆x)^2 - x^2] / ∆x = lim (∆x -> 0) [x^2 + 2x∆x + (∆x)^2 - x^2] / ∆x = lim (∆x -> 0) [2x∆x + (∆x)^2] / ∆x = lim (∆x -> 0) [2x + ∆x] = 2x

Application of derivatives

Derivatives have many applications in real life. They are used to find the maximum and minimum values of a function, optimize certain functions, and determine the concavity and inflection points of a graph.

If we want to find the maximum area of a rectangle with perimeter 20 units, then derivatives can be used.

    Let the length be x and the width be y. Given the perimeter 2x + 2y = 20, solve for y: y = 10 - x. The area A = xy = x(10 - x) = 10x - x^2. To maximize the area, find the derivative A'(x) = 10 - 2x. Setting A'(x) = 0, 10 - 2x = 0, gives us x = 5. So the width y = 10 - 5 = 5. The maximum area is 25 square units.

Integrals

Integrals are another essential concept in calculus. They are the opposite process of derivatives and are used to find areas under curves, among other things. If derivatives measure the rate of change, integrals allow us to add up small changes to determine the total accumulation.

The integral of a function f(x) is represented as:

    ∫ f(x) dx

For the function f(x) = x^2, the integral is calculated as:

    ∫ x^2 dx = (1/3)x^3 + C

Here, C is the constant of integration because the derivative of a constant is zero, and it can be any number.

Definite integral

Definite integrals are used to calculate a number, where indefinite integrals give a family of functions. A definite integral calculates a quantity over a specific interval. The notation for a definite integral of f(x) from a to b is:

    ∫ a b f(x) dx

For example, to find the area under the curve f(x) = x^2 from x = 1 to x = 3, you would calculate:

    ∫ 1 3 x^2 dx = [(1/3)x^3] from 1 to 3 = (1/3)[3^3 - 1^3] = (1/3)[27 - 1] = (1/3)[26] = 26/3

Real-world applications of calculus

Calculus is used in many real-world situations. It is used by engineers to determine the most efficient design and operation of machinery. Economists use it to find the maximum profit under certain constraints. Biologists use it to model population dynamics.

Now, let's consider an example where calculus is used in a simple physical problem:

Suppose a ball is thrown straight up into the air. Its position after t seconds is given by the function s(t) = 60t - 5t^2. The velocity of the ball is the derivative of its position.

    v(t) = s'(t) = d(60t - 5t^2)/dt = 60 - 10t

When t = 2 sec the velocity will be:

    v(2) = 60 - 10(2) = 60 - 20 = 40 m/s

Conclusion

Calculus is an amazing tool that has truly changed the way we understand the world around us. From its fundamental concepts of limits, derivatives, and integrals, calculus is not just a subject, but a gateway to solving real-world problems. It allows us to model, predict, and understand phenomena across a variety of disciplines. Through practice and application, calculus becomes not just a branch of mathematics, but a lens through which we can view the dynamic nature of our environment.


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