Questions concerning Calculus! Please help!?

Question

1. I am confused about max/min & 1st/2nd derivative. I thought the first derivative gives you max/min; but does the 2nd derivative just verify it or something?

2. I need help with this related rates problem (like how to set it up/START it):
V=1/3(pie)r^2h
*If you pour liquid in a container at 6 cubic feet per minute; the container has a height of 7 feet and the radius is 4 feet. What is the rate the liquid is rising in the container when the liquid is 3 feet deep?

3. How does a tangent line 'approximation' compare to finding the equation of a line tangent to a curve?

4. How are the mean value theorem and the intermediate value theorem related...in terms of derivatives?

5. If you have a limit problem, how do you know when to use the squeeze/sandwich theorem??? In general, WHEN do you use the aforementioned theorems?

6. How do you find the length of an interval (in regards to the bisection method)?

Thanks!

Answer 1

That's way too many questions for one post, but I'll help you out with #1.

When the first derivative is zero, you have a critical point. Most critical points are maxima or minima. When the second derivative is negative, the derivative is decreasing, and hence is positive to the left of the critical point and negative to the right of the critical point. Hence the function is increasing to the left of the critical point and decreasing to the right of the critical point. Hence the function has a maximum.

Turn all that around if the second derivative is positive.

If the second derivative is zero, it is often the case that you have neither a minimum nor a maximum, but rather a point of inflection. The classic example of this is (0,0) for the function y = x^3.