Friday, October 30, 2009

Water Trough Practical Application

A 10ft long trough whose sides are shaped like isosceles triangles with a height of 1ft and a base of 3ft, fills up with water at a rate of 12 cubic feet per minute. How fast is the water rising when the water is 6 inches deep?

I created an excel spread sheet that illustrates whats going on here very nicely. But I'll leave one hint right here. The key is to write an equation that relates the depth of the water to the volume of the water. Follow the link to download the xls file.

WaterTroughProblem.xls

x^2y^2 + xsiny = 4, Implicit Differentiation

x^2y^2 + xsiny = 4
This problem requires you to use implicit differentiation in conjunction with the product rule and the chain rule. The thing that might scare you about this problem is the xsiny part but don't worry

Thursday, October 29, 2009

x^2 + xy - y^2, Implicit Differentiation

This problem requires implicit differentiation because the equation cannot be solved for y. The fun part about this problem is that you get to use the product rule on the xy term in the middle of the equation.

Click on the image below to see a lager version of my solution.

Piecewise Function, y = |x^2-9| Plot & Find Derivative

Problem:
Find the derivative of y =|x^2-9|. Plot the function and the derivative.

Solution:
Lets say you had the graph of just x^2-9 (without the absolute value, represented by the orange line in the graph below) that would look like the graph of x^2 shifted down 9 units.

But, there is an absolute value sign so the part of the graph that goes below zero becomes positive. That’s why you have that blue bubble in the middle of the graph. This bubble is just a reflection of the orange line about the x axis. So the blue hump in the middle of the graph is y = -x^2 + 9. In order to reflect a function about the x axis just multiply the right hand side of the equation by -1.

If you redefine the function in |x^2-9| in a piece wise fashion, you can find the derivative on each part.
y = { x^2 - 9 when x <= -3
9-x^2 when -3< x < -3 (the reflection)
x^2 -9 when x >= 3

y' = {2x when x <=-3
-2x when -3>x<3
2x when x >= 3

Now every where you have a negative slope the value of the derivative is negative and everywhere you have a positive slope the value of the derivative is positive.

Notice that where the red line goes vertical you actually have jump discontinuity. So at x = -3 and x = 3 the left and right hand limits of the function are not equal and therefore the derivative of the function does not exist at these points.


if you'd like to experiment with the excel file I made for this problem here is a link PiecewiseProblem_1.xls

The Difference Quotient

Here is a link to an excel spread sheet that I put together to explain the difference quotient:
DifferenceQuotient.xls

In the spread sheet I use the graph of y = sin(x) to illustrate the application of the difference quotient. Below is a screen shot of the interactive graph which is contained in the spread sheet.