lecture2

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Unformatted text preview: Click to edit Master subtitle style Page ‹#› Lecture #2: Mathematics Tutorial Mathematics Tutorial 1 1 Lecture #2: Mathematics Tutorial Half of all Americans do not understand math, and the other two- thirds don't care.-- Garrison Keillor Baseball is 90% mental, the other half is physical. Page ‹#› Lecture #2: Mathematics Tutorial Lecture #2: Mathematics Tutorial 2 2 Mathematics Tutorial Page ‹#› Lecture #2: Mathematics Tutorial Mathematics Tutorial 3 3 Lecture #2: Mathematics Tutorial Page ‹#› Lecture #2: Mathematics Tutorial Why all the math? Perspective from the great master, Alfred Marshall: In a letter to his protégée, A.C. Pigou, he [Marshall] laid out the following system: “(1) Use mathematics as shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life Lecture #2: Mathematics Tutorial 4 4 Page ‹#› Lecture #2: Mathematics Tutorial Why all the math? Perspectives from the economics blogosphere: Nobel laureate Paul Krugman notes that “Math in economics can be extremely useful”, and that math can serve an essential analytic function by helping to clarify one’s thoughts. Greg Mankiw notes that “Math is good training for the mind. It makes you a more rigorous thinker.” Jason DeBacker observes that math helps to quantify tradeoffs, and that using math “… puts in plain sight the assumptions that lie Lecture #2: Mathematics Tutorial 5 5 Page ‹#› Lecture #2: Mathematics Tutorial Why all the math? Lecture #2: Mathematics Tutorial 6 6 Page ‹#› Lecture #2: Mathematics Tutorial (Just about) all the math you will need Next, we turn our attention to a study of the mathematical principles upon which this course is based. In this lecture. . . the number e and natural logarithms differentiating and Taylor 7 7 Lecture #2: Mathematics Tutorial Page ‹#› Lecture #2: Mathematics Tutorial The Number e a number, 2.71828182845905 Suppose we have a function of x written as ex ; we may also write this function as exp ( x ). The function ex is just the number 2.7183… raised to the power of x ; e 2 is just 2.7183…2 = 7.389056…, e 1 is 2.7183. . . and e 0 = 1. 8 8 Lecture #2: Mathematics Tutorial Page ‹#› Lecture #2: Mathematics Tutorial Lecture #2: Mathematics Tutorial 9 9 The Number e The function ex is the solution to the following infinite series: Graphically, ex looks like this: 2 3 1 1 1 ... . 2 6 ! i x i x e x x x i ∞ = = + + + + = ∑-1-0.5 0.5 1 1.5 2 x 1 2 3 4 5 6 7 e x Page ‹#› Lecture #2: Mathematics Tutorial Lecture #2: Mathematics Tutorial 10 10 The Function ex The function ex has the special property that the slope, or gradient of the function is also ex ....
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