# Thread: Integral rate of flow

1. ## Integral rate of flow

The differential equation dy/dx = 2x^3 + 2 cos(x/2) - (1.2) ^x gives the rate of flow of water in/out of a reservoir in gallons per minute.

a) Find the general solution to the differential equation and its particular solution through the point (1, 2). Interpret the meaning for the particular solution.

b) For this particular solution list the points of max, Min, and points of Inflection, domain, Range, Intervals for Continuity, and Differentiability.

For A, this is what I got. [(5/6) ^-x]/ [ln (5/6)] + 4sin(x/2) + (x^4)/2 + C
2 = [(5/6) ^-1]/ [ln (5/6)] + 4sin (1/2) + (1^4)/2 + C; 2 = -164.544 + 1.918 + .5 + C; C = 162.126
[(5/6) ^-x]/ [ln (5/6)] + 4sin(x/2) + (x^4)/2 + 162.13

I think I am doing this correctly but when I look at b, it becomes clear that the function and the integral will always be positive based on my work. That being the case, there would be no min max or inflection point. Also, I am not sure if it wants me to find the questions in b for the original function or the integral.

2. Originally Posted by ffuh2
The differential equation dy/dx = 2x^3 + 2 cos(x/2) - (1.2) ^x gives the rate of flow of water in/out of a reservoir in gallons per minute.

a) Find the general solution to the differential equation and its particular solution through the point (1, 2). Interpret the meaning for the particular solution.

b) For this particular solution list the points of max, Min, and points of Inflection, domain, Range, Intervals for Continuity, and Differentiability.

For A, this is what I got. [(5/6) ^-x]/ [ln (5/6)] + 4sin(x/2) + (x^4)/2 + C
2 = [(5/6) ^-1]/ [ln (5/6)] + 4sin (1/2) + (1^4)/2 + C; 2 = -164.544 + 1.918 + .5 + C; C = 162.126
[(5/6) ^-x]/ [ln (5/6)] + 4sin(x/2) + (x^4)/2 + 162.13

I think I am doing this correctly but when I look at b, it becomes clear that the function and the integral will always be positive based on my work. That being the case, there would be no min max or inflection point.
Why do you say that? The function f(x)= sin(x)+ 2 is always positive but has an infinite number of maximums, minimums and inflection points.

Also, I am not sure if it wants me to find the questions in b for the original function or the integral.
The problem says "for this particular solution list ...". It is clearly talking about the integral.

3. Based on the question, I think he was referring to a global maximum or minimum as opposed to a local max, min, and inflection. If there is an infinite number then, I would need to find if the local max, mins, & inflections are periodic otherwise which I am not sure how to do here. Otherwise, I would just state that there are an infinite number of them.

Does my methodology look ok for finding the integral?

4. How would I go about finding the periodic max, mins, & inflections?

5. Originally Posted by ffuh2
How would I go about finding the periodic max, mins, & inflections?
Same way you do for any function. For max and min, find where the derivative of the function is 0, and check if the derivative changes sign there, and for inflections, find where the second derivative is 0 and check if the second derivative changes sign there.

For this problem that is particularly easy because you are given the derivative to start with.