A mix of limits, slope, and vectors

In summary, the conversation discusses finding the slope of a line that a moving particle approaches as t approaches infinity. The position and velocity vectors of the particle are given, and the position vector at t = 1 is also provided. The participants use L'Hôpital's rule and other mathematical techniques to find the slope, ultimately concluding that it is 2.
  • #1
jcook735
33
0
1. A moving particle has position (x(t),y(t)) at any time t. The position of the particle at t =1 is (2,6), and the velocity vector at any time t > 0 is given by (1 - (1/(t^2)), 2 + (1/(t^2))).

The particle approaches a line as t -> infinity. Find the slope of the line. Show the work that leads to your conclusion.



2. I found that the position vector is <t + (1/t), 2t - (1/t) + 5>



3.
I decided that the best thing to do was take the limit of y(t) over x(t), and applied L'Hôpital's rule until the equation turned into something tangible. I ended up with the limit as t approaches infinity to be 2. Is that right? Is that the slope of the line? Did I not go far enough or did I do it completely wrong?
 
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  • #2
That looks fine to me. I don't even think you really NEED l'Hopital. Just divide the numerator and denominator of y(t)/x(t) by t and you've got it.
 
  • #3
yay! thanks
 

1. What is the difference between a limit and a slope?

A limit is a mathematical concept that describes the behavior of a function as its input approaches a certain value. It is used to describe the behavior of functions at points where they are not defined. On the other hand, slope is a measure of how steep a line is and is used to describe the rate of change of a function at a specific point.

2. How are vectors used in calculus?

Vectors are used in calculus to represent both direction and magnitude of quantities such as velocity and acceleration. They are also used to represent forces and displacement in physics problems. In calculus, vectors are useful for understanding the behavior of curves and surfaces in multiple dimensions.

3. Can limits, slope, and vectors be applied to real-world problems?

Yes, these concepts are widely used in real-world applications such as physics, engineering, and economics. For example, limits are used in predicting the behavior of a population over time, slope is used in calculating the rate of change in stock prices, and vectors are used in designing structures and predicting the movement of objects in space.

4. How do limits and derivatives relate to each other?

Limits and derivatives are closely related concepts in calculus. Derivatives are used to find the instantaneous rate of change of a function at a specific point, while limits are used to analyze the behavior of a function as it approaches a certain point. In fact, the derivative of a function at a point is the limit of the slope of the function as the distance between two points approaches zero.

5. What is the geometric interpretation of a vector?

A vector can be thought of as an arrow in space, with a specific direction and length. The direction of the vector represents the direction of movement, while the length represents the magnitude or size of the movement. In calculus, vectors are often used to visualize and analyze the movement of objects in space.

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