Farmer Steve plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?

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Solution:

The first order of business in optimizing functions is to design a function which represents what you want to optimize. We want to minimize fencing, so our formula will be F. Once we come up with the formula (based on the diagram), we use what we know about the area of the the field to eliminate a variable. We then take the derivative and set equal to zero (we are finding a max or min!) and proceed to the answer as follows:

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Calculus II (BC): Euler's Method

        Consider the differential equation , y(0) = 0.

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Solution:

Remember that Euler's method is just a way to approximate solutions to differential equations (to 'step through the slope field'). It uses slopes of tangent lines and given points to approximate the direction the function will take. As y' is the slope of the tangent line, we need it to come up with the tangent line equation (for the tangent line to the solution graph at x = 0). We can easily see that y'(0) = 1 + x - y = 1 + 0 - 0 = 1, since we are given the first point on the graph of the solution is (0,0). So the tangent line has a slope of one at the point (0,0). We follow this slope over our step of 0.4 to get the next approximated point on the graph of (0.4, 0.4)--if the slope is one, this makes sense--go 'right' the same amount as you go 'up.'

Calculating the next point (using the same method) gives us (0.8,0.8); that stinkin' x-y part of the problem is making our coordinate pair cancel each other out (since they're equal). Thus, it's no leap of the imagination to see that the remainder of the points will be (1.2,1.2), (1.6,1.6) and (2,2). Ergo, we approximate that the function will contain the point (2,2).

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