Calculus + Continuous Functions?

Question

This was posted by my Calc. Professor:

The gravitational force exerted by Earth on a unit mass at a distance r from the center of the planet is

F(r) { GMr/R^3 if r { GM/r^2 if r >_ (Greater than or equal to) R

where M is the mass of the Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?

The rational functions don't seem to present any problems, the denominators are exponential so the won't equal 0. It seems that F is a continuous function of r.

Can anyone help me with this problem?

Answer 1

The function that your teacher gave you is a piecewise defined function. It has two different equations depending on the input value. To make sure this function is continuous, you need to examine the point at which you change from one function to the other. You need to examine the limit as r approaches this point from the left and from the right. If the limits are equal then it is continuous.

Another point to worry about is r = 0. The function isn't defined at this point. You can assume that the domain of the function is (0, infinity) because the distance from the center of the earth (r) is never negative.

Answer 2

Hmm. lacking derivatives at some specific element, specific. lacking any derivatives in any respect? i do no longer think of so. y = (absolute fee of x) is a V-formed graph that should lack a by-product at x=0, because of the fact the slope could be -a million coming from the left and +a million coming from the perfect.