Applied Mathematics · Class 11 Commerce
Ch 8Calculus — Class 11 Applied Mathematics, concept-first.
This concept map shows how the chapter fits together. It starts from functions, their domain and range and types (with their graphs). The idea of instantaneous change leads to limits and continuity, then to differentiation and finally to the applications of derivatives such as rates of change and tangents.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Introduction
You've been solving problems your whole life. When you see a new type of question, you first figure out what it's asking, then decide which tools to use, then carry out the steps.
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Concept Map
This concept map shows how the chapter fits together. It starts from functions, their domain and range and types (with their graphs).
Introduction
Calculus is the mathematics of change and motion. In earlier classes, you studied algebra and geometry, which mostly dealt with fixed quantities and static shapes.
Functions
3 QA function is a rule that links each input to exactly one output — think of it as a machine that takes a number from a set called the domain and gives back a unique number from another set called the…
+−Worked Examplesi1 question
Graphical Representation of Functions
Graphs turn abstract functions into something you can see. Instead of just working with symbols like , you get a curve that shows how the output changes with the input — a picture of the relationship…
Domain and Range of a Function
Every function is a rule that pairs each input with exactly one output. The domain is the set of all possible inputs for which the rule is defined, while the range is the set of all outputs the functi…
+−Exercise 8.3i2 questions
Polynomial Functions
Polynomials are the gentlest functions we meet in calculus — smooth, predictable, and built from nothing more than powers of with constant coefficients.
Rational Function
A rational function is simply a ratio of two polynomials, like . The key idea is that its behaviour changes dramatically near the points where the denominator becomes zero — the function "blows up" th…
Exponential Function
The exponential function is not just another formula to memorize — it is the mathematical way of describing things that grow (or decay) at a rate proportional to themselves.
Logarithmic Functions
Logarithmic functions are the natural inverse of exponential functions, and they unlock a new way to describe growth that slows down over time.
Greatest Integer Function
The greatest integer function, often written as or , is a special kind of step function that rounds any real number down to the nearest integer less than or equal to it. For example, , , and .
Modulus Function
The modulus function, written as , is a simple but powerful idea: it gives the distance of a number from zero on the number line, so it always returns a non-negative value.
Signum Function
The signum function is a compact way to capture the sign of a real number — whether it is positive, negative, or zero. It is defined piecewise: for , for , and .
Limit of a Function
4 QWe now move from the idea of a sequence — a list of numbers approaching a target — to the behaviour of a function as its input gets arbitrarily close to some point.
+−Worked Examplesi2 questions
Continuity of a Function
We now turn to the idea of continuity — a concept that captures, in precise mathematical language, the intuitive notion of a function whose graph can be drawn without lifting the pen.
+−Exercise 8.6i2 questions
Instantaneous Rate of Change and Derivative
3 QWe have seen how the average rate of change tells us the slope over an interval, but what if we want the speed of a car at a precise instant, or the slope of a curve at a single point? That is the ide…
+−Worked Examplesi1 question
Derivatives of Algebraic Functions Using Chain Rule
The chain rule is the tool you reach for when a function is built inside another — when you have a function of a function, like or .
Tangent Line and Equation of Tangent
The idea of a tangent is something you already know from geometry — a line that just touches a circle at one point.