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[數學,M1&M2] Math is Fun: Essential Skills for Freshmen



Math is Fun: Essential Skills for Freshmen

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Outline 2 methods to tackle this queston.

Source: Kaoyan Mathematics (Syllabus A) past paper

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Solution

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Annex: Skills on Integration

When using the method of substitution,
write out the differentials (as on the right) to speed up the integration.

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Approximation by Taylor Series

Using the small O notations, we have:

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To determine what small O notation we shall use,
refer to the below examples:

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[ 本帖最後由 Aeroblast 於 2019-7-12 03:30 AM 編輯 ]

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Finding limits by orthodox definition

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Below are essential techniques:

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Higher Derivatives

Note that polynomials have finite derivatives. Since exponential functions, logarithmic functions and trigonometric functions are transcendental, they have infinite derivatives, so we have:

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Mean Value Theorem

Note the last two forms(Lagrange and Cauchy respectively),
which often appear in HKAL Pure Math past papers.

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Integration Bee

The fundamental:

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Your turn:
http://www.mit.edu/~pax/integrationbee.html




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Improper Integration by transformation

Here we will use improper integral to deal with some kind of integration:

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Note that the reflection of e^x on a line y=x is ln(x)
i.e. the line y=x is the common asymptote of both e^x and ln(x).

Then the integral below:

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Is equivalent to:

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An important limit in DSE level Math

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We often see (e^x-1)/x.
When x approaches 0, the limit become 1.

Proof 1:
http://www.ies-math.com/math/java/calc/lim_e/lim_e.html

Proof 2:
引用:
We have e^x = 1 + x/1! + x^2/2! + ... by Taylor Expansion
Then e^x - 1 = x/1! + x^2/2!
Denote y = e^x -1
Then y/x = 1/1! + x/2! + x^2/3! + ....
Put x= 0, we have y/x = 1
QED.

[ 本帖最後由 Aeroblast 於 2019-7-12 06:22 AM 編輯 ]

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Hypothesis Testing

Key concepts:

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Chi Square Test is manifested above.

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Parameter Estimation

This is an important device in econometrics:

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Recommended Math Reference for Engineering

Note that advanced economics does include engineering math topics:

引用:
Line Integral
Partial Derivatives
Differential Equation
System of Linear Equations
Characteristic Polynomials
Generating Functions


So below are some books available for purchase online:

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Recommended Math Reference for Computing

In computer sciences there are many topics related to discrete math:

引用:
Greedy Algorithm
Generating Functions
Recursion
Boolean Algebra
Logic Gate
Number Theory



So below are some references from online bookstores:

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[隱藏]
回覆 10#  Aeroblast 的帖子
proof 1
是不是有 typing error?

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