Calculus: Sign Diagram

Question

Question: Find f'(x) of f(x) = (3x-2)/(x+3) and draw is sign diagram.

I was able to find f'(x) which was 11/(x+3)².

I am stuck on the part of drawing the sign diagram.

Answer for sign diagram:

(I'll try to explain it in words)

dotted line in the middle, -3

both sides (right and left) is +

Answer 1

Great job finding the derivative!

When creating a sign diagram for the first derivative, you want to first look for these key points:

a) Values of x where f'(x) = 0

b) Values of x where f'(x) is undefined

In this case, there are no values of x for which f'(x) = 0.... however, at x = -3, the derivative is undefined, so that's why that's a key point. However, if the function is undefined when x = -3, that means that x cannot ever equal -3, so that's why you have a dotted line there. (Sometimes people will label x = -3 with an open circle to indicate the same thing.)

Once you've found the key points (there may be more than one in other problems), you're going to evaluate f'(x) for random numbers to the left and right of the key points. In this case, we'll look for a random number (let's say x = -5), that's LESS than x = -3 and a random number (let's use x = 0) that is GREATER than x = -3.

f'(-5) = 11/(-5 + 3)^2 = 2.75

f'(0) = 11/(0 + 3)^2 = 11/9

Since both of these values are POSITIVE, that's why you have a "+" (plus sign) on both sides.

(As a quick reminder, what this means is that the SLOPE of the curve of the original function is positive throughout the function, except for at x = -3, where it is undefined. )