Write an expression involving an improper integral

In summary, we have an object moving along a curve in the xy-plane with given position and velocity functions. We must find the slope of the tangent line at a given point, and evaluate the limit of this slope as t approaches infinity. We must also find the horizontal asymptote of the curve, represented by an improper integral. To find the y values at infinity, we must integrate the given velocity function and find the constant of integration using the given point.
  • #1
nns91
301
1

Homework Statement


AN object moving along a curve in the xy-plane is at position (x(t),y(t)) at time t, where

dx/dt=Arcsin(1-2*e^(-t)) and dy/dt= 4t/(1+t^3)

for t>or= 0. At time t=2, the object is at the point (6,-3).

a. Let m(t) denote the slope of the line tangent to the curve at the point (x(t),y(t)). Write an expression for m(t) in terms of t and use it to evaluate lim m(t) as t approaches infinity.

b. The graph of the the curve has a horizontal asymptote y=c. Write an expression involving an improper integral that represents this value c

Homework Equations



None

The Attempt at a Solution



a. So I got m(t)= 4t/ (1+t^3)*Arcsin(1-2*e^(-t)). Then will my limit be 0 ?

b. Will the expression be: c= integral ( (dy/dt) / (dx/dt)) from 0 to infinity ??
 
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  • #2
M(t) is (dy/dt)/(dx/dt), right? Not (dy/dt)*(dx/dt). And the limit will be 0.

To see the "end behavior" of the y values, find limit of y(t) as t -> inf.

Hint: To get y(t) you need to integrate dy/dt with respect to t, or at least write it as a function involving an integral (don't forget to find the constant of integration using the given point.)
 
  • #3
Thanks. Yeah, that's what I mean, I forgot the parentheses.

For part d, will it be y= integral (dy/dt dt) from 0 to infinity ??
 

1. What is an improper integral?

An improper integral is a type of integral that involves infinite limits of integration or a function that is not defined at certain points within the interval of integration. This means that the integral cannot be evaluated using the standard methods of integration and requires a different approach.

2. How is an improper integral different from a regular integral?

An improper integral differs from a regular integral in that it involves infinite limits of integration or a function that is not defined at certain points within the interval of integration. This means that the integral cannot be evaluated using the standard methods of integration and requires a different approach.

3. What is the process for solving an improper integral?

The process for solving an improper integral involves breaking it down into smaller integrals, evaluating each one separately, and then taking the limit as the boundaries of integration approach infinity or as the undefined points approach a defined value.

4. Can you provide an example of an expression involving an improper integral?

One example of an expression involving an improper integral is ∫0 x2 dx. This integral involves an infinite limit of integration and would require the use of a different method, such as the limit comparison test or the p-series test, to evaluate it.

5. Why are improper integrals important in mathematics?

Improper integrals are important in mathematics because they allow us to extend the concept of integration to functions that are not defined or behave strangely within a given interval. They also have many applications in physics, engineering, and other fields, making them a fundamental tool in problem-solving and modeling real-world scenarios.

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