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You are looking for the sum of all numbers in a linear function. The sum is equivalent to the area under the curve which you normally use calculus and integrate

by kjay 13y ago

You are looking for the sum of all numbers in a linear function. The sum is equivalent to the area under the curve which you normally use calculus and integrate for. However, in the case of a linear function, it's just a triangle. So, you can take a short cut if you just "know" f(x)=x^2 / 2.

DoubleMalt 13y ago
[nitpick] actually it is x * (x - 1) / 2 in this case [/nitpick]
ImprovedSilence 13y ago
another way of looking at it is just find the average amount charged(the middle number in this case) -> 1500, and multiply by the # of times charged (3000), and there ya go, its really the same math at kjay's, but just approaching it a different way.
logicallee 13y ago
What does "know" mean? suspect? :)
- Double_Cast 13y ago
  imo, "can visualize". In my experience, those strong in math easily visualize math problems geometrically. This is opposed to, for example, having to grind through the quadratic formula. Try to mentally (or on paper) graph the freelancer's pay. The x-axis represents "hours" and the y-axis represents "pay per hour". The function we get is the linear function "f(x) = x". I.e. for the first hour, he's paid $1; for the 300th hour, he's paid $300. A very straightforward graph. If we cut off our x-axis at $300, the graph looks like a right triangle. If you want to find the total amount of money he's accumulated, all you have to do is find the area under the triangle (which is a special case of integration). For a triangle, geometry class taught us "Area = (1/2)(base)(height)". Calculus courses prefer that we integrate, which has broader application. Integration of "f(x) = x" gives us "f'(x) = (1/2)(x^2)". Our "integral" is equivalent to the "geometry-class equation" because for our particular triangle, the base is the same as the height. For comparison: Area = (1/2)(base)(height) f'(x) = (1/2)(x^2) DoubleMalt is saying kjay's visualized "shortcut" (intuitively faster approximation) is slightly off. This is because the triangle in our actual problem looks like a staircase, and not a smooth slope. But kjay's shortcut has a margin of error of only about "-0.033%" which is a good enough approximation by most people's standards. Also, DoubleMalt's equation should look like x * (x + 1) / 2 instead of x * (x - 1) / 2

by kjay 13y ago

DoubleMalt 13y ago

[nitpick] actually it is x * (x - 1) / 2 in this case [/nitpick]

ImprovedSilence 13y ago

another way of looking at it is just find the average amount charged(the middle number in this case) -> 1500, and multiply by the # of times charged (3000), and there ya go, its really the same math at kjay's, but just approaching it a different way.

logicallee 13y ago

What does "know" mean? suspect? :)

Double_Cast 13y ago
imo, "can visualize". In my experience, those strong in math easily visualize math problems geometrically. This is opposed to, for example, having to grind through the quadratic formula. Try to mentally (or on paper) graph the freelancer's pay. The x-axis represents "hours" and the y-axis represents "pay per hour". The function we get is the linear function "f(x) = x". I.e. for the first hour, he's paid $1; for the 300th hour, he's paid $300. A very straightforward graph. If we cut off our x-axis at $300, the graph looks like a right triangle. If you want to find the total amount of money he's accumulated, all you have to do is find the area under the triangle (which is a special case of integration). For a triangle, geometry class taught us "Area = (1/2)(base)(height)". Calculus courses prefer that we integrate, which has broader application. Integration of "f(x) = x" gives us "f'(x) = (1/2)(x^2)". Our "integral" is equivalent to the "geometry-class equation" because for our particular triangle, the base is the same as the height. For comparison: Area = (1/2)(base)(height) f'(x) = (1/2)(x^2) DoubleMalt is saying kjay's visualized "shortcut" (intuitively faster approximation) is slightly off. This is because the triangle in our actual problem looks like a staircase, and not a smooth slope. But kjay's shortcut has a margin of error of only about "-0.033%" which is a good enough approximation by most people's standards. Also, DoubleMalt's equation should look like x * (x + 1) / 2 instead of x * (x - 1) / 2