Mathematics Learning Path
High School (11–12) Mathematics
Explore structured mathematics courses, solved problems, simulations and Python activities.
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Area Under a Curve
How much paint covers a wall with a curved top? Cut the region into thin strips, add the areas of the strips, and let the strips become thinner and thinner. The sum becomes a definite integral. You find the area under a curve, the area measured along the y-axis and the area of a region where the curve crosses the axis. The line, the parabola, the circle and the ellipse are the standard cases. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level, IB and AP.
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Area Between Curves
A road runs between a hillside and a stream, and a surveyor wants the area of the land between them. Find where two curves meet, decide which one lies above, and integrate the difference. Then split a region where the upper curve changes, and choose vertical or horizontal strips to keep the working short. Parabolas, circles, ellipses and lines are the shapes you meet. It covers the second part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level, IB and AP.
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Volumes, Averages and Other Uses of the Integral
Turn a curve around an axis and it sweeps out a solid, like a bowl or a vase made on a lathe. Find its volume with thin discs and washers. Then find the average value of a function, the distance travelled from a velocity graph and the total of other rates, and end with the length of a curve. The Indian syllabus stops at area. These topics belong to A-Level Further Maths, IB Higher Level and AP Calculus, and they use the same idea of adding thin pieces.
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Integration as the Reverse of Differentiation
You know that the derivative of is . Now go the other way: which function has the derivative ? Find antiderivatives, understand the constant of integration, learn the standard integrals and the rules, and then use substitution, with the help of trigonometric identities, to reach integrals that look hard. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Methods of Integration
Some integrals do not yield to substitution alone. Learn the integrals of the special forms, including those with a quadratic under a root, split a fraction into partial fractions, and use integration by parts, which comes from the product rule. Choose the right method by looking at the shape of the integrand. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Definite Integrals
How much distance does a bus cover if you know its speed at every moment? The answer is the area under the speed graph. Meet the area function, see why its derivative is the original function, and use the Fundamental Theorem of Calculus to evaluate definite integrals. Then use substitution and the properties of definite integrals, and see the integral as the limit of a sum. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Rates of Change and Monotonic Functions
The derivative tells how fast one quantity changes when another changes. Use it to follow a moving particle and a changing cost. Solve related rates with a ladder, a balloon, a filling cone and a shadow. Then find where a function increases or decreases from the sign of its derivative. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Tangents, Normals and Approximations
The derivative is the gradient (also called the slope) of a curve at a point. Write the equations of the tangent and the normal, find where a tangent is horizontal, parallel or perpendicular to a given line, and measure the angle between two curves. Then use differentials to estimate small changes, roots and percentage errors. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Maxima, Minima and Optimisation
Where does a graph turn? Find the critical points of a function and decide whether each is a maximum, a minimum or neither, with the first and the second derivative tests. Find the largest and smallest values on a closed interval. Then solve optimisation problems: the biggest box, the cheapest tin, the shortest fence. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Continuity of Functions
A graph with no break is continuous, and a graph with a gap is not. Turn that picture into a test with left-hand and right-hand limits, name the three kinds of break, and check piecewise functions. Then use the Intermediate Value Theorem to show that an equation has a root. It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topic of A-Level, IB and AP.
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Differentiability and the Rules of Differentiation
Where does a graph have a corner? Define the derivative with limits, see why a differentiable function must be continuous, and then learn the rules: the chain rule, inverse trigonometric, exponential and logarithmic functions, and implicit, logarithmic and parametric differentiation. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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Higher Derivatives and the Mean Value Theorems
The first derivative is the gradient of a graph, and the second derivative tells how that gradient changes. Find second and higher derivatives, including those of implicit and parametric functions. Then meet Rolle's theorem and the Mean Value Theorem, and see what they say about speed, roots and inequalities. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.
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