Second derivatives (vector-valued functions) | Advanced derivatives | AP Calculus BC | Khan Academy

TL;DR

This video explains vector valued functions, their notations, and how to find their first and second derivatives.

Transcript

• [Voiceover] So I have a vector valued function h here, and when I say vector value, it means you give me a t, it's a function of t, and so you give me a t, I'm not just going to give you a number, I'm going to give you a vector. And as we'll see, you're going to get a two dimensional vector. You could view this as the x component of the vector an... Read More

Key Insights

• ↩️ Vector valued functions return vectors and can represent various two-dimensional quantities.
• 💁 Notations for vector valued functions can differ, but they all convey the same information.
• 🫡 Finding the first derivative involves taking the derivative of each component with respect to the parameter.

Questions & Answers

Q: What does it mean for a function to be vector valued?

A vector valued function returns a vector when given a parameter. It can represent various two-dimensional quantities, like position, velocity, or acceleration.

Q: How are vector valued functions represented differently?

Vector valued functions can be represented using engineering notation, where the horizontal and vertical components are multiplied by unit vectors. They can also be represented with an arrow on top, although this is not always necessary.

Q: How do you find the first derivative of a vector valued function?

To find the first derivative, take the derivative of each component of the vector with respect to the parameter. Apply the power rule to each term.

Q: What does the second derivative of a vector valued function represent?

The second derivative of a vector valued function represents acceleration. It is found by taking the derivative of each component of the first derivative.

Summary & Key Takeaways

• Vector valued functions return a vector when given a parameter, usually representing position, velocity, or acceleration.

• Different notations, like using unit vectors or arrows, can be used to denote vector valued functions.

• To find the first derivative, take the derivative of each component with respect to the parameter. The second derivative is found using the same process.