M2: Algebra and geometry Part 2

When the rate of change considering the segment length is considered, one of the direct application is mechanical deduction. Differentiation build up a strong relationship between displacement, velocity and acceleration so differential equation would be useful here.

Locus of a certain point in coordinate geometry is a big topic in core/M2 mathematics, and it has significant applications, remarkably some pivot systems.

Example 3a. (Locus on linear system) Let A, B and C be 3 points on positive x-axis, positive x-axis and negative x-axis respectively. AB = 5, BC = 7. Let |OA| = x, |OC| = y, find dy/dx.

What I want to illustrate here is that, the differentiation operator has a very nice behavior that allows us to do questions in many ways. Denote |OB| = h.

Solution 1. Set up relation between x,y through h.
Obviously there are pairs of right angled triangle, hence and , then (since y is positive), then .

Solution 2. Implicit differentiation.
First we set up the explicit relation and . By differentiating w.r.t. t, , and do the same thing to to get .

Now
Therefore . Note that we can multiply both sides by dt/dx, and eliminate the terms dt by chain rule.

Solution 3. Differentation through angle.
Let angle BAC be . Then , , . Now ,  .

The same question can be solved by different approaches, and that's the art of problem solving.

M2: Algebra and geometry Part 1

In HKCEE A-maths or HKDSE M2 paper, the topic "rate of change" as the application of differentiation always appear in the paper and sometimes it has been modelled into a geometric question like the velocity of a certain position of a pivot. What I want to illustrate is that geometric question can be modelled into a algebra question in many ways like vectors, coordinate geometry, and this time we can deal with calculus.

With the assistance of trigonometry we built up a strong relationship between segment lengths and angles, allowing us to relate different quantity, then the rate of change of a certain quantity can be transformed easily. We will demonstrate four questions, one unrelated to calculus, one A-math, one M2 and one Tokyo U questions, showing that differentiation can deal with meaningful conclusion instead of simply finding some derivatives.

Recall: Sine law and Cosine law.

Theorem. (Sine Law)

Theorem. (Cosine Law)

Question 1. (Incircle) Let a,b,c be three sides of right-angled triangle ABC where c is the hyp. side. Show the incircle radius

Define be the area of the triangle ABC.

Now notice that



Since it is right angled triangle, , and therefore



Comparing with the x-y-z approach of incircle (Fact: AF = AG, CF = CE, BG = BE due to tangent properties or congruent triangle.) this is completed in a more algebrical way. (Using the x-y-z approach requires a cartesian plane which the right angled properties is shown in a geometrical simulation, but in this approach it is shown in a algebrical way).