Vector-Valued Functions, Differentiation, Integration, Projectile Motion Problems - Calculus

A vector-valued function is a function where the output is a vector, typically defined by one or more component functions. It maps a single real number (or a vector) to a vector in a multi-dimensional space. These functions are often used to represent a curve in space by defining the position of a point on the curve at any given value of the parameter, commonly denoted by '\(t\)'.

💡How they work
• Input and output: A vector-valued function takes an input, which is usually a scalar real number (like '\(t\)'), and outputs a vector.
• Representing a curve: The function can be expressed in terms of its component functions, such as \(r(t)=\langle x(t),y(t),z(t)\rangle \) for a three-dimensional curve.
• Tracing a path: As the input parameter '\(t\)' changes, the output vector changes, effectively tracing out a path or curve in space. The set of all terminal points of these vectors (with initial points at the origin) forms the curve.

💡Key characteristics
• Domain and range: The domain of the function can be a single real number or a vector, and its range is always a vector space.
• Parameterization: A vector-valued function provides a parameterization for a curve, where the parameter '\(t\)' can be thought of as time.
• Applications: They are used to describe paths in space, such as the trajectory of a projectile, and form the basis for studying motion in three dimensions.

💡Worksheets are provided in PDF format to further improve your understanding:
• Questions Worksheet: https://drive.google.com/file/d/1tolpJ9mIz-hrSc3BJyC0piUzPQ8cKD3b/view?usp=drive_link
• Answers: https://drive.google.com/file/d/1tiM6iM6LTV0bkAhLvC8ij3bjvbloV5Jj/view?usp=drive_link

💡Chapters:
00:00 Parametric arc length integral
01:56 Worked example

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