Math post

呢題曾經post過，

今次用LibreOffice Writer執番靚仔：

呢題中二程度

Proofread version:

Enrichment question with unproofread solution:

跟住呢題同HKDSE M2被廢除嘅微分部分有關：

dx/dy = 1/(dy/dx)

Questions:

(a) Find d ln(x) /dx

(b) Find d x^(1/3) / dx

Solution:

(a) Recall d e^x /dx = e^x

Now d lnx /dx = d lnx/d e^(ln x)

= 1/ [e^(ln x)] = 1/x

(b) Recall d x^3 /dx = 3 x^2

Now d x^(1/3) /dx = d x^(1/3)/d ((x^3)^(1/3))

= 1/ [3 (x^(1/3))^2] = 1/ (3 x^(2/3)) = x^(-2/3) /3

Question: Using any 5 methods, show that:

sin x + cos x ≤ √2

Solution:

Method 1: Linear programming

Sketch C: x^2 + y^2 = 1 and L: y=kx

and denote the radius locus as r(x, y), |r|=1

Then max(x, y) = r(cos π/4, sin π/4), k=1

i.e. max(sin x + cos x) = 2 × √2/2 = √2

Method 2: Differentiation w.r.t. x

Method 3: Use auxiliary angle formula

Method 4: Use sin x = cos (x+π/2)

Method 5: Take a square and get sin(2x):

(sin x + cos x)^2 = 2 + 2 sin x cos x

Note: sin x < 1 and cos x < 1

(sin x + cos x)^2 = 1 + 2 sin x cos x ≤ 2

Note: sin x ≤ 1 and cos x ≤ 1

It is not difficult to show that sin x cos x ≤ 1/2.

By linear programming again,

sin x and cos x have 2 common edges,

i.e. x=π/4 or x= π + π/4 for x ≤ 2π

such that sin x = cos x = 1/√2 or -1/√2.

Both cases yield max( sin x cos x) =2/√2 = √2

Use your intuition to learn the proof,

such that you don't need HKDSE M2.

Use mathjax

Errata:

max( sin x + cos x) =2/√2 = √2

max( sin x × cos x) =(1/√2)^2 = 1/2

Note that we cannot apply AM>GM inequality in this question, becausr trigonometric functions are always oscillating and thus can't form either arithmetic progression or geometric progression.

If you don't believe so, you will compute some nonsense like sin x + cos x ≥√2 .

because*

Prove the following identity:

Solution:

Recall what we have learnt from junior high:

Then using the nature of sine function:

We have the operation:

Q.E.D.

Prove that sin(2x) = 2sin(x)cos(x) intuitively.

Solution: