Derivatives
Chapter Name : Limits And Derivatives |
Sub Topic Code : 104_11_13_05_03 |
Topic Name : Derivatives |
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Sub Topic Name : Derivative Of Polynomial And Trigonometric Functions |
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Introduction
• In day to day life there are many applications in which quantities associated with some process are linked with some other quantity. • Change in one quantity will result in change in related quantity also. • Polynomials and trigonometric functions have many day to day applications and certain rules are defined for their differentiation.
Pre-Requisites:
• Knowledge of algebraic functions and derivatives. • Knowledge of polynomials and trigonometric functions.
Activity:
You will observe as you increase volume of balloon the other two quantities also change.
Real Life Question:
A water tank is being filled with water through pipes. The volume of water in tank at any given time is given by V(t) = 2+5t-10t². Find the rate at which volume is increasing when t=1.
Key Words / FlashCards
| Key Words | Definitions (pref. in our own words) |
|---|---|
| Derivative | Rate of change of one physical quantity with other. |
| Polynomial | An expression which have constants, exponents, variables and finite terms. |
| Trigonometric Function | A function of an angle expressed as ratio of two sides of a right angled triangle including that angle. Trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent. |
Learning aids / Gadgets
| Gadgets | How it can be used |
|---|---|
| Balloon | Fill balloon with air at a constant rate. If the ratio of length of balloon to its radius is fixed. At what rates do the length and radius of balloon change? |
Real life uses :
• Derivative of trigonometric functions is used in engineering, oceanography, astronomy. • Derivative of polynomials are used in business, stock market etc.
Places to visit :
Fill bottle with water from water filter and find the rate at which volume of bottle is increasing.
Practical examples around us
| Examples | Explainations |
|---|---|
| Oceanography(describing Waves) | We all know ocean waves. Sine function is used to measure size of wave which is a trigonometric function. Energy in wave depends on wind on water. So energy of wave is derivative of water. Rules of Derivative of trigonometric functions are applied to sine function of wave. One of the giant ocean waves which we know is Tsunami. |
What you learn in Theory:
Theoretically derivatives have some rules with the help of these rules problems on derivatives can be solved.
What you learn in Practice:
For example, as volume of cylinder depends on height and radius of cylinder. If we change one quantity the other two quantities also change. Product rule of derivative can be applied to know how the change in one quantity will affect other quantity as well.