Calculus - Lesson 5 | What are Functions? | Don't Memorise

TL;DR

Functions are mathematical relations between variables, and in calculus, they help determine rates of change.

Transcript

Here’s a plant, and what you see here is its shadow. Can you list the things that the length of the shadow is dependent on? One, it is dependent on the position of the source of light. Anything else you can think of? If the height of the plant grows, then the shadow’s length will also change right? So the length of the shadow is dependent on the po... Read More

Key Insights

• ❓ Functions are mathematical relations between variables, where each value of the independent variable corresponds to one value of the dependent variable.
• ☠️ Calculus uses functions to determine rates of change, such as speed or the rate of area growth.
• ☠️ By finding the limit of average speeds or rates as time intervals approach zero, calculus allows us to find instantaneous speeds or rates.
• 🐎 Functions can be used to find the speed of objects at any instant, including objects falling freely towards the ground.
• 👻 The relationship between independent and dependent variables in functions allows us to understand how one variable influences the other.
• ❓ Functions can be represented by equations, with the independent variable on one side and the dependent variable on the other.
• 🌍 Understanding functions and calculus helps us analyze physical phenomena, solve real-world problems, and make predictions.

Questions & Answers

Q: What are functions?

Functions are mathematical relations between variables, where each value of the independent variable corresponds to one value of the dependent variable.

Q: How do functions work in calculus?

In calculus, functions are used to determine rates of change. By finding the limit of average speeds or rates as time intervals approach zero, we can determine instantaneous speeds or rates.

Q: How can functions help find the speed of an object at any instant?

By knowing the relationship between time and distance traveled, we can find the average speeds between smaller and smaller time intervals, eventually leading to the exact speed at a specific instant.

Q: Can functions be used to find the speed of any object falling towards the ground?

Yes, the relationship of speed being equal to two times the time elapsed is followed by any object falling freely towards the ground, allowing us to find the speed at any instant.

Q: Why are independent and dependent variables important in functions?

Independent variables, such as time or the length of a square's side, determine the values of dependent variables, such as distance traveled or area. Functions allow us to understand the relationship between these variables.

Summary & Key Takeaways

• Functions are relations between variables, where each value of the independent variable corresponds to one value of the dependent variable.

• In calculus, functions are used to determine rates of change, such as speed or the rate of area growth.

• Calculus allows us to find the instantaneous speed of an object by taking the limit of average speeds as time intervals approach zero.