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 ↓


Applications of Differentiation


Differentiation is a fundamental concept in calculus, which deals with how functions change or, more mathematically, how the rate of change of a function can be expressed. This powerful tool helps us explore various phenomena in many fields such as physics, engineering, economics, and others. The main utility of differentiation lies in its ability to provide insight into the behavior of a function, especially when analyzing the motion of objects, optimization problems, and understanding complex systems.

1. Basics of differentiation

Before diving into the applications, it is essential to understand the basics of differentiation. The derivative of a function is defined as the rate at which the value of the function changes with respect to a change in its input value. Mathematically, if y = f(x), the derivative is expressed as f'(x) or dy/dx.

Example: 
If f(x) = x^2, then f'(x) = 2x.

2. Finding the tangent and normal

One of the first applications of differentiation is to find the equation of the tangent and normal to a curve. The tangent to a curve at a point is a straight line that touches the curve at that point but does not intersect it. The normal line is perpendicular to the tangent.

Example:
For y = x^2, at x = 1, the slope of the tangent is given by f'(1) = 2*1 = 2.
Therefore, the equation of the tangent is y - 1 = 2(x - 1).
The slope of the normal is -1/2. So its equation is y - 1 = -1/2(x - 1).

Visually, if you draw a parabola y = x^2 and plot these lines, you will see the tangent and normal at the point (1, 1).

3. Rate of change

One of the main applications of differentiation is its ability to provide the rate of change of a quantity. For example, if you know the position of a car over time, the derivative of the position with respect to time gives the speed of the car.

Example:
If the position s(t) = 5t^2 (where s is in meters and t is in seconds), then the speed v(t) = s'(t) = 10t.

4. Optimization problems

Differentiation helps to find the maximum and minimum values of a function, this process is known as optimization. It is used in many real-life scenarios, for example, minimizing costs, maximizing profits, or even finding the best possible design.

To find such extreme values, we look for critical points where the derivative of the function becomes zero or is undefined. These points often lead to maximum or minimum values.

Example:
For the function f(x) = -x^2 + 4x,
The derivative is f'(x) = -2x + 4.
To find the critical point, setting f'(x) = 0 gives x = 2.
Examining the second derivative f''(x) = -2, we find that it is a maximum.
Thus, f(2) = -2^2 + 4*2 = 4 is the maximum value.

5. Motion along a line

Differentiation kinematics is applied to the study of motion. When an object moves in a straight line, its position at any instant can be expressed as a function of time. Differentiation of the position function gives the velocity function.

Example:
Let s(t) = t^3 - 6t^2 + 9t.
So velocity v(t) = s'(t) = 3t^2 - 12t + 9.
Acceleration is the derivative of velocity: a(t) = v'(t) = 6t - 12.

This mathematical model helps to understand how fast the object is moving and how its speed changes over time. If you visualize, you can plot position, velocity, and acceleration as functions of time.

6. Solving problems involving related rates

Relative rate problems involve finding the rate of change of one quantity relative to another. These problems require differentiating a relationship involving several variables related to time.

Example:
A balloon is inflated such that its volume V is increasing at a constant rate of 100 cubic centimeters per second. Given that the radius r of the sphere is related to its volume by V = (4/3)πr^3, find the rate of change of the radius when the radius is 10 cm.

Differentiate between the two sides with respect to time t.
dV/dt = 4πr^2 (dr/dt)

Substituting dV/dt = 100 and r = 10 gives:
100 = 4π(10)^2 (dr/dt)
Solve for dr/dt, resulting in dr/dt ≈ 0.0796 cm/s.

7. Curvature and concavity

The second derivative of a function gives information about the curvature and concavity of the function. If the second derivative is positive, the function is concave up (like a cup), and if negative, the function is concave down (like a hat).

Example:
Consider the function f(x) = x^3 - 3x.
First derivative: f'(x) = 3x^2 - 3
Second derivative: f''(x) = 6x

For 0 < x, f''(x) > 0 and the function is concave.
For x < 0, f''(x) < 0 and the function is concave down.

8. Conclusion

Differentiation is a versatile and powerful tool that has many practical applications in real life. From understanding rates of change and finding tangents to solving engineering problems and modeling natural phenomena, differentiation is an essential component of mathematical analysis. Consistently, its applications allow us to better understand and describe the world around us.

Although we have explored many examples, the scope of differentiation is much broader. Its importance in mathematics is profound and widely appreciated in every field that relies on the precise investigation of transformations.


Grade 11 → 4.3


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Applications of Differentiation

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