Understand Key Concepts of Application of Derivatives for Class 12 Maths - Academic Year 2025-26
FAQs on Master Class 12 Maths Application of Derivatives with Free NCERT Book PDF
1. Which topics from Chapter 6, Application of Derivatives, are most important for the CBSE Class 12 Maths board exam 2025-26?
Based on previous board exam trends and the CBSE syllabus, the most important topics from Application of Derivatives are:
- Maxima and Minima: This is the highest-weightage topic, especially word problems leading to 5-mark questions.
- Tangents and Normals: Finding equations of tangents and normals, and conditions of orthogonality and parallelism are frequently asked.
- Increasing and Decreasing Functions: Expect questions asking to find the intervals where a function is strictly increasing or decreasing.
- Rate of Change of Quantities: This topic often appears in 2 or 3-mark questions involving geometric shapes.
2. What types of questions can be expected from the Maxima and Minima section in the board exam?
From Maxima and Minima, you can expect a variety of question types, including:
- Long Answer (LA) Questions (5 marks): These are typically complex word problems where you need to formulate a function and find its maximum or minimum value. Examples include finding the dimensions of a cylinder inscribed in a cone or a window of a certain shape with maximum area.
- Short Answer (SA) Questions (3 marks): These usually involve finding the local or absolute maxima and minima of a given function over a specific interval.
- Case-Based/MCQs (1 mark): These might test your understanding of the conditions for maxima/minima using the first or second derivative test.
3. How can I identify whether a word problem requires the concept of Maxima/Minima or the Rate of Change?
This is a crucial distinction for solving problems correctly. Look for keywords in the question:
- Rate of Change problems typically use phrases like “how fast is... changing”, “at what rate is... increasing/decreasing”, and involve variables with respect to time or another variable. For example, 'how fast is the volume of a balloon changing with respect to its radius'.
- Maxima and Minima problems ask you to find the “greatest”, “least”, “maximum”, “minimum”, or “optimal” value of a quantity. For example, 'find the maximum volume of a cone' or 'find the least amount of material required'. The goal is to optimise a function.
4. What are some expected question formats from the 'Tangents and Normals' topic?
For the topic of Tangents and Normals, important questions often involve:
- Finding the equation of a tangent or normal to a curve at a given point.
- Finding points on a curve where the tangent is parallel or perpendicular to a given line.
- Finding the equation of a tangent to a curve that passes through a specific external point.
- Proving that two curves touch each other or intersect at right angles (orthogonally). These are considered Higher Order Thinking Skills (HOTS) questions.
5. Why is the second derivative test often more important than the first derivative test for board exam questions on maxima and minima?
While both tests can find maxima and minima, the second derivative test is often more direct and conclusive for the types of functions given in CBSE exams. It provides a clear check: if f''(c) < 0, it's a local maximum, and if f''(c) > 0, it's a local minimum. The first derivative test requires you to check the sign of f'(x) on both sides of the critical point, which can be more time-consuming and prone to calculation errors, especially with complex polynomial or trigonometric functions.
6. What is a common mistake students make in questions about 'Increasing and Decreasing Functions' that costs them marks?
A very common and critical mistake is using closed intervals [a, b] instead of open intervals (a, b) when stating the final answer. Functions are said to be strictly increasing or decreasing over an open interval. Another frequent error is incorrectly determining the sign of the derivative f'(x) in different intervals, especially when dealing with complex factors. Always use a number line and test values carefully to avoid this.
7. How should I approach a complex 5-mark word problem on maxima and minima to ensure full marks?
To secure full marks in a maxima/minima word problem, follow these steps systematically:
- Step 1: Read the problem carefully to identify the quantity to be maximised or minimised and the given constraints.
- Step 2: Formulate the primary equation for the quantity to be optimised (e.g., Volume, Area, Cost) in terms of two or more variables.
- Step 3: Use the given constraints to create a secondary equation. Use this to express the primary equation in terms of a single variable. This is a critical step.
- Step 4: Find the first derivative of the function and set it to zero to find the critical points.
- Step 5: Apply the second derivative test to confirm whether the critical point corresponds to a maximum or minimum. Show this step clearly in your solution.
- Step 6: Conclude the answer by stating the required maximum or minimum value and the dimensions or values at which it occurs.
