Calculus of vector function

In summary, the position vector of a particle at time t is given by r(t)= 2sin(2t)i + cos(2t)j + 2tk where t >=0. The velocity and speed can be calculated as v(t) = 4cos(2t)i - 2sin(2t)j + 2k and speed = √(12cos^2(2t)+8) respectively. To find the maximum and minimum speed, you can use the first derivative test and set the derivative equal to zero to find the critical points. At these points, the derivative will be undefined, indicating a maximum or minimum value.
  • #1
hellboy324
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Homework Statement


The position vector of a particle at time t is given by r(t)= 2sin(2t)i + cos(2t)j + 2tk where t >=0.

Homework Equations


v(t) = 4cos(2t)i - 2sin(2t)j + 2k
speed = | v(t) | =√(16cos^2(2t)+4sin^2(2t)+4) = √(12cos^2(2t)+8)

The Attempt at a Solution


I found the velocity and speed, but I have no idea what to do to find out the maximum and minimum speed for this question, someone can teach me?
 
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  • #2
How do you usually find the extrema of a function?
 
  • #3
Things are easier in that the question is asking about scalar speed. What does it mean for the derivative when speed hits a peak/trough?
 

1. What is the definition of a vector function?

A vector function is a mathematical function that takes one or more variables and returns a vector as its output. It is commonly written as f(t) = (x(t), y(t), z(t)), where x(t), y(t), z(t) are scalar functions of the independent variable t.

2. What is the significance of calculus in vector functions?

Calculus is used to analyze and manipulate vector functions in order to understand their behavior and properties. It helps in finding derivatives, integrals, and limits of vector functions, which are essential for solving real-world problems in physics, engineering, and other fields.

3. How is the derivative of a vector function defined?

The derivative of a vector function f(t) = (x(t), y(t), z(t)) is defined as f'(t) = (x'(t), y'(t), z'(t)), where x'(t), y'(t), z'(t) are the derivatives of the scalar functions x(t), y(t), z(t) with respect to t. This represents the instantaneous rate of change or slope of the vector function at a given point.

4. How is the integral of a vector function calculated?

The integral of a vector function f(t) = (x(t), y(t), z(t)) is calculated by integrating each component of the vector function separately. This means that the integral of f(t) is equal to (∫x(t)dt, ∫y(t)dt, ∫z(t)dt). The integral of a vector function represents the area under the curve of the function in a given interval.

5. What is the relationship between vector functions and parametric equations?

Vector functions and parametric equations are closely related as both involve a set of equations that describe a curve or a surface in space. The main difference is that parametric equations use scalar variables and vector functions use vector variables. In some cases, a vector function can be converted into a set of parametric equations and vice versa.

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