Silly Mistakes 1

Question: 
If f (x) = 3x² + 2x +1, find the first derivative of f at x = 0.

Answers:
1st version:
f '(x) = 6x + 2 = 6(0) + 2 = 2 

2nd version:
f '(x) = 6x + 2 
f '(0) = 6(0) + 2 = 2

The truth  f '(x) ≠ 2 but f '(0) = 2.

Interprete Minitab Output for Multiple Comparison.

When the null hypothesis is rejected (the result showed a significance difference), we want to know where the difference among the means is. Which group of mean causes this difference? We can do further analysis by carry out multiple comparisons. The most commonly used methods are LSD and Tukey's tests.

Example: Concentration of Mercury.

Since H0 is rejected, there is a difference in mean concentration of mercury in the three regions. We can do further analysis to know which region mean causes the difference.

Conclusion: Only one and third comparisons exhibited significant difference in mercury concentration. Region 2 has the highest concentration of mercury. 

Interprete Minitab Output for One-Way ANOVA

Hypothesis for this example:

H0 : μ1 =  μ2 =  μ3

H1 : At least one mean is different from the others.

Interpretation:

It can be said that the concentration of Mercury in the three regions is different. This indicates that there are different sources of pollution that affect the three regions. It could be due to different activities operating in each region.

Check out the multiple comparison here.

Type I and Type II Errors

There are four possible outcomes in the hypothesis testing situation. The four possible outcomes are shown in the figure above. There are two possibilities for a correct decision and two possibilities for a wrong decision.

Correct decision happens when

• H0 is false and it is rejected (This is our target)
• H0 is true and it is not rejected

Incorrect decision happens when

• H0 is false and it is not rejected
• H0 is true and it is rejected (We must reduce this error)

Who Says Maths is Boring?

Who says Maths is boring?
Lets dance!
Music please... ♫♫♪♪♫♫♪

Kalkulator: Mod Degree dan Radian

Memperkenalkan...Mr. Karl!
Ta daa...

Apabila pengiraan melibatkan fungsi trigo (sin, cos dan tan), Mr. Karl perlu dipastikan berada dalam mod yang betul sama ada mod degree atau radian. 

Pelajar sering melakukan kesilapan mudah apabila mod kalkulator tidak betul. Sangat rugi! Kesilapan mudah seperti ini perlu dielakkan. Anda tak boleh salahkan Mr. Karl dalam hal ini!

Contoh 1: 
Mod degree : sin (π) = 0.0548
Mod radian : sin (π) = 0
Jawapan yang betul ialah 0 (dikira dalam mod radian)

Contoh 2: 
Mod degree : sin (90) = 1
Mod radian : sin (π/2) = 1
Kedua-dua jawapan adalah betul.

Kenapa macam ni? Sebabnya

Contoh 1: π radian = 180°. 
                Jadi untuk mod degree : sin (180) = 0
Contoh 2: 90° = π/2 radian

Contoh kesalahan pelajar:

Komen: 
Mr. Karl perlu berada dalam mod radian untuk mengira sin (1) dan cos (1) kerana 1 ini bermaksud 1 radian bukan 1°. Langkah -langkah penyelesaian soalan ini adalah betul tetapi jawapan terakhir pelajar adalah salah kerana pengiraan nilai sin (1) dan cos (1) yang tidak betul.

Tips:
Untuk berada dalam keadaan yang selamat, pastikan Mr. Karl sentiasa berada dalam mod radian.

Completing the Square

By completing the square

Examples of Integration by Trigonometric Method (tangent secant)

Integration by Trigonometric Method (tangent secant) has been discussed in the previous entry. Now let us discuss the examples:

1.) ∫ tan6 x sec4 x dx 

Solution:

Since power of sec is even, we use method (1).

2.) ∫ tan5 x sec7 x dx 

Solution:

Since power of tan is odd, we use method (2).

3.) ∫ tan5 x sec4 x dx 

Solution:

Since power of sec is even and power of tan is odd, we use method (3).

Here's more:

Integration by Trigonometric Method (sine cosine)

Examples of Integration by Trigonometric Method (sine cosine) 

Integration by Trigonometric Method (tangent secant)

Integration by Trigonometric Method (tangent secant)

Integration by Trigonometric Method (sine cosine) has been discussed in the previous entry. In this entry, we will discuss about trigonometric method when the integrand, f (x)  is a product of  tan x and sec x.

How to evaluate  ∫ tanm x secn x dx   ?

1.) Power of sec is even : n = even no.(and m = even no.)

• Split off a factor of sec2 x
• Use identity, substitute sec2 x = 1 + tan2 x
• Let u = tan x

eeeeeeExamples:  ∫ tan6 x sec4 x dx

2.) Power of tan is odd : n = odd no. (and m = odd no.)

• Split off a factor of sec x tan x
• Use identity, substitute tan2 x = sec2 x -1
• Let u = sec x

eeeeeeExamples:  ∫ tan5 x sec9 x dx

3.) If power of sec is even and power of tan is odd

• Use either method (1) or (2) but it is easier to convert the term with the smallest power.

eeeeeeExamples:  ∫ tan3 x sec4 x dx

Click here for examples.

Here's  more:

Integration by Trigonometric Method (sine cosine)

Examples of Integration by Trigonometric Method (sine cosine)

Examples of Integration by Trigonometric Method (sine cosine)

Integration by Trigonometric Method (sine cosine) has been discussed in the previous entry. Now let us discuss the examples:

1.) ∫ sin5 x cos2 x dx 

Solution:

Since power of sin is odd, we use method (1).

2.) ∫ sin6 x cos3 x dx 

Solution:

Since power of cos is odd, we use method (2).

3.) ∫ sin2 x cos2 x dx 

Solution:

Since all powers are even, we use method (4).

Here's more: 

Integration by Trigonometric Method (sine cosine)

Integration by Trigonometric Method (tangent secant)

Examples of Integration by Trigonometric Method (tangent secant)

Integration by Trigonometric Method (sine cosine)

Use trigonometric method when the integrand, f (x)  is a product of sin x and cos x or tan x and sec x. In this entry, we will discuss about sine cosine. Tangent secant will be discuss in the next entry.

How to evaluate  ∫ sinm x cosn x dx ? 

1.) Power of sin is odd : m = odd no. (and n = even no.)

• Split off a factor of sin x
• Use identity, substitute sin2 x = 1 - cos2 x
• Let u = cos x

eeeeeeExamples:  ∫ sin5 x cos2 x dx,  ∫ sin5 x dx,  
                             ∫ sin3 x cos2 x dx 

2.) Power of cos is odd : n = odd no. (and m = even no.)

• Split off a factor of sin x
• Use identity, substitute sin2 x = 1 - cos2 x
• Let u = cos x

eeeeeeExamples: ∫ sin6 x cos3 x dx, ∫ cos3 x dx,  
                            ∫ sin2 x cos5 x dx

3.) Both powers are odd

• Use either method (1) or (2) but it is easier to convert the term with the smallest power.

4.) Both powers are even

• Use half-angle identities

eeeeeeExamples: ∫ sin2 x cos2 x dx, ∫ sin4 x dx,  ∫ sin2 x dx

Click here for examples.