Featured 1 week ago

BTW, the graph of this equation is very beautiful

Here's the graph:

Featured 1 week ago

The answer is

This is an improper integral

Start by calculating the definite integral

Let

Therefore,

Perform the integration by parts

Therefore,

Therefore,

The integral converges

Featured 1 week ago

Making

Featured 5 days ago

Transform the function in this way:

We can use now the well known trigonometric limit:

and using the trigonometric identity:

we have:

While the third function is continuous so:

and we can conclude that:

graph{(tanx-sinx)/x^3 [-1.25, 1.25, -0.025, 1]}

Featured 5 days ago

See the proof below

Let

and

It is given that

Then

The dot product is

Therefore,

Featured 3 days ago

The function is concave down on

Concavity is determined by the second derivative.

#f''(x) = -8#

This is negative on all

Hopefully this helps!

Featured 3 days ago

Featured 3 days ago

You were on the correct track with your setup on number 8, but when you took the derivative of the

The proper setup is

It is difficult to read the image included, but I believe the setup involves

A quick sanity check on the sign of the result can help catch errors as they are in progress. As the ships continue moving, ship A will get closer to the lighthouse, while ship B will move farther away.

This action will serve to make the angle

Your original attempt ended with a negative answer, which would indicate the angle is *decreasing* - a result that contradicts the nature of the situation. It might not help you find the issue, but it can at least give you a hint that you're on the wrong track somewhere.

Featured 2 days ago

In the Ideal Gas Law formula, there are 3 variables and 2 constants.

This is a "classic" related rates problem. We need to take derivatives of every variable (

We have all of the values we need provided to us to substitute into this related rates expression:

A quick "sanity check" verifies that the sign of our answer is what we'd expect. Since P is inversely related to V and directly related to T, if the volume is decreasing, and the temperature is increasing, we'd definitely expect the pressure P to be increasing (positive). Since

Featured 2 days ago

After setting

So,

=

=

=

=

=