Applications of derivatives involve using the concept of differentiation to solve real-world problems. The Class 12 Mathematics syllabus covers the following topics related to applications of derivatives:
Rate of Change: This topic covers the concept of rate of change and its interpretation as the derivative of a function. It includes problems related to finding the rate of change of physical quantities, such as velocity, acceleration, and temperature.
Increasing and Decreasing Functions: This topic covers the concept of increasing and decreasing functions, and their applications in finding maximum and minimum values of a function.
Tangents and Normals: This topic covers the concept of tangents and normals to a curve, their slopes, and equations. It includes problems related to finding the equation of a tangent or normal to a curve at a given point.
Approximations: This topic covers the concept of approximations using derivatives, such as linear approximations, differentials, and error estimates.
Maxima and Minima: This topic covers the concepts of maximum and minimum values of a function, and their applications in optimization problems, such as finding the maximum or minimum value of a cost or profit function.
Mean Value Theorem: This topic covers the Mean Value Theorem, which states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval where the derivative of the function is equal to the average rate of change of the function over the interval.
Applications of derivatives are important because they provide a powerful tool for solving real-world problems related to optimization, rates of change, and approximations. They are used in various fields such as physics, engineering, economics, and statistics. The concepts of increasing and decreasing functions, tangents and normals, and maxima and minima are used in solving problems related to optimization, related rates, and curve sketching. The Mean Value Theorem is used in proving results related to the continuity and differentiability of functions.