1. Find the value(s) of c guaranteed by Rolle’s Theorem for f(x)=x^2+3x on [0, 2]

A. c=-3/2
B. c=0, 3
C. Rolle’s Theorem doesn’t apply as f is not continuous on [0, 2]
D. Rolle’s Theorem doesn’t apply as f(0) does not equal f(2)
E. None of these
I got D. I plugged in 0 and 2 into the function to see if they equalled and they didn't.

2. Determine the open intervals where the graph of f(x)=-1/(x+1)^2 is concave up or concave down.
A. concave down (negative infinity, infinity)
B. concave down (negative infinity, -1); concave up (-1, infinity)
C. concave down (negative infinity, -1) and (-1, infinity)
D. concave up (negative infinity, -1) and (-1, infinity)
E. none of these
I got C. I found the first derivative. I found the second derivative I used the interval test to determine concavity.

3. Consider f(x)=x^2/(x^2+a), a>0. Determine the effect on the graph of f if a is varied.
A. Each y value is multiplied by a
B. As a increases, the vertical tangent lines move further from the origin
C. The graph of the curve is shifted sqrt(a) units to the left.
D. As a increases, the curve approaches its asymptote more slowly
E. None of these
I think I asked this problem earlier but I'm still confused. Is there a way to figure this out with out a graph?

1,2 correct.

3. divide numerator and denominator by x^2

f(x)=1/(1+a/x^2)

consider a=1, and a= 6000. What does that do to the graph?

I don't know if you will see this but Is number 3's answer D?

For the first question:

You correctly determined that Rolle's Theorem does not apply because the function is not continuous on the interval [0, 2]. Therefore, the correct answer is indeed D. Rolle’s Theorem doesn’t apply as f(0) does not equal f(2).

For the second question:
To determine the concavity of a function, you need to find the second derivative and then analyze the intervals where it is positive (concave up) or negative (concave down). You correctly found the first and second derivatives. Using the interval test to analyze concavity, the correct answer is indeed C. The function is concave down on the intervals (-∞, -1) and (-1, ∞).

For the third question:
To determine the effect on the graph of f if a is varied in the function f(x)=x^2/(x^2+a), you can analyze the behavior of the function as a changes. Let's consider the options one by one:

A. Each y value is multiplied by a: This is not correct because multiplying each y value by a would result in a different overall scaling of the function, which is not what happens when a is varied.

B. As a increases, the vertical tangent lines move further from the origin: This is also not correct. When a increases, the behavior of the tangent lines does not change in this manner.

C. The graph of the curve is shifted sqrt(a) units to the left: This option is not correct either. Shifting the curve to the left by sqrt(a) units does not accurately describe the effect of changing a.

D. As a increases, the curve approaches its asymptote more slowly: This is the correct answer. As a increases, the curve approaches its asymptote (the line y=1) more slowly. This can be observed by analyzing the behavior of the function as x approaches positive or negative infinity.

Therefore, the correct answer is D. As a increases, the curve approaches its asymptote more slowly.

It is possible to figure this out without a graph by analyzing the behavior of the function as described above.