The volume of a prism is given by its base area multiplied by its height.
The bases are two right triangles. If we project everything on the xy plane, we can see that the vertices of the base triangle is
So, leg AB is 3 units long, and leg AC is 4 units long. This means that the area of the triangle is
The height of the prism connects, for example, points and
, so it's 5 units long. So, the volume of the prism is
The absolute value function is defined as
If x is strictly positive (x > 0), then |x| = x, and d|x|/dx = dx/dx = 1.
If x is strictly negative (x < 0), then |x| = -x, and d|x|/dx = d(-x)/dx = -1.
But if x = 0, the derivative doesn't exist!
In order for the derivative of a function f(x) to exist at x = c, the limit
must exist. This limit does not exist for f(x) = |x| and c = 0 because the value of the limit depends on which way x approaches 0.
If x approaches 0 from below (so x < 0), we have
whereas if x approaches 0 from above (so x > 0), we have
But 1 ≠ -1, so the limit and hence derivative doesn't exist at x = 0.
Putting everything together, you can define the derivative of |x| as
For this, the first thing to do is to assume that the function of temperature with respect to r is written in one of the following ways:
Way 1:
Way 2:
To find the instant variation we must find the derivative of the temperature with respect to the distance r.
We have then:
For function 1:
Rewriting
For function 2:
Rewriting
Answer:
The instantaneous variation of the temperature with respect to r is given by:
Assuming function 1:
Assuming Function 2: