Calculus Help

[pogejr]

Got two problems I have no idea what to do.

1. Given the graphs below, four functions are pictured, f,f',f'', and g, but I can't seem to tell which is which.

a. Determine which color graph is f, its first derivative, and its second derivative.

b. Which is greater: f''(3) or f'''(2) Explain.

http://i164.photobucket.com/albums/u...r/image010.jpg


2. Water is flowing into a spherical tank at a constant rate. Let V(t) represent the volume of water in the tank and H(t) represent the height of water in the tank at time t.

a) What is the meaning of V'(t)? Is the value of V'(t) positive, negative, or zero? Explain

b) What is the meaning of H'(t)? Is the value of H'(t) positive, negative, or zero? Explain.

c) Is the value of V''(t) positive, negative, or zero? Explain

d) Is the value of H''(t) positive, negative or zero when the tank is
i) one quarter full? ii) half-full? iii) three-quarter full?

Explain

[Eliseos]

For the first part, look at the slopes and see how they relate to one another. Start with assuming one is f and see what functions match the behavior of a derivative for that function. Look at inflection points and the slopes specifically.

For the second part, you're going to have to think of what a derivative represents for a function. You have a function representing water flow into a tank, and another that represents the height of the water in that tank. Think of how position, velocity, and acceleration are related, and then apply the same relationships to V(t) and H(t).

[pogejr]

I have the first derivatives down for the second problem, but I'm not sure what the second derivative means. For the first problem, I'm confused on the relationship between Point of inflections, min/max and zeros, as in which corresponds with which derivative.

[Slapper]

[Eliseos]

When you have zero slope, the derivative of that function will be zero. You can use this to determine where the corresponding graph crosses zero. If there is a point on a function that has zero slope, but does not have a corresponding function that has a zero crossing, you can eliminate that one. Then chain together the other ones using the zero crossings.

As for the tank problem, let's use the position/velocity/acceleration example. Say position has units of meters. Taking the derivative of position gives us velocity, say in meters per second. It tells how fast (or slow) our position is changing over time. Taking the derivative of velocity (and consequently the second derivative of position) gives us our acceleration, how fast (or slow) our velocity is changing. It is in meters per second per second. Now, let's say we are using a car to travel somewhere. Using our starting position as a reference, we go a block away, make a u-turn, and come back. Say we have a really awesome car, and by the time we reach the point we are making a u-turn, we are at the speed limit for the street. During the first half of the trip, our velocity is changing, which means we are accelerating. Our velocity is therefore positive (we're counting positive distance travelling away from the origin, negative distance coming back to where we started), going from the start to the u-turn. Our acceleration is also positive since our velocity changes in the positive direction (our velocity had to change, since we went from 0 to the speed limit). Immediately at the u-turn, our velocity is zero since we aren't moving anywhere (well, we're moving radially but just ignore that to keep this simple). Also to keep this simple and unrealistic, our speed remains the same out of the u-turn as it was coming into it. Therefore at this point our velocity is positive, but our acceleration is 0 since we didn't change our velocity at all. Driving back to the origin, we hit the brakes so that at exactly at the origin our speed is zero at that point. Our velocity is negative, since we are traveling back to the origin, and our acceleration is also negative since we are slowing down.

For your specific problem, think of these ideas and how the flow of fluid relates to the position/velocity/acceleration example. Also for the last part, pay specific attention to the fact that this is a spherical tank.

[pogejr]

Thank you for helping me!

Here is what I got.

1. Given the graphs below, four functions are pictured, f,f',f'', and g, but I can't seem to tell which is which.
a. f(x) is the green line, f'(x) is the red line, and f''(x) is the blue line.
b. f''(3) is greater because if th derivative of f'' is taken to get f''', a smaller exponent will be yielded.

2. Water is flowing into a spherical tank at a constant rate. Let V(t) represent the volume of water in the tank and H(t) represent the height of water in the tank at time t.
a. V'(t) would represent the rate at which the volume of water changes in accordance with the time (dv/dt). This would be positive because the volume can only increase when water is being filled into the sphere.
b. H'(t) would represent the rate at which the height of the water changes in accordance with time (dh/dt). This would also be positive because as more water enters the sphere, the height can only increase.
c. The value of V''(t) would be zero because, assuming we graph the volume, the graph of the volume would be a sloped line, the derivative of that would be a horizontal line, and the derivative of that would be zero.
d.Is the value of H''(t) positive, negative or zero when the tank is
i. When the tank is one quarter full, H''(t) is positive because H'(t) is increasing at that point.
ii. When the tank is half full, H''(t) is zero because H'(t) is zero at that point.
iii. When the tank is three-quarters full, H''(t) is negative because H'(t) is decreasing at that point.

[Eliseos]

b. f''(3) is greater because if th derivative of f'' is taken to get f''', a smaller exponent will be yielded.

This reasoning isn't correct, but your answer is. A function can have a very large slope in a very short amount of time, and (assuming the initial slope is positive) the derivative at that point would be greater than the original function. Plus this doesn't work for functions like sign, cos, exponentials, composite functions, etc. Look at the slope at the point of f''(2) and the value of f''(3). There is something there that guarantees f''(3) will be greater than f'''(2). The rest of your answers look correct.

[pogejr]

Oh! Since f''(2) is close to the max, then the derivative, f'''(2) would be closer to zero than f''(3).

[Eliseos]

f''(2) has a negative slope, so f'''(2) is guaranteed to be negative, but f''(3) has a positive value since it's above the x-axis so it's guaranteed to be positive.