English Assignment - Reflection from Videos (part 2)

A.      Derivative Notation

The derivative of f(x) is written as f'(x). We read it f prime of x. We can say it y' (y prime) because f(x) and y is interchangeable. Other we have notation dy/dx. It means the change in y's over the change in x's or we know it as slope.

We say that the derivative is a slope of tangent line at specific point. We remember that the slope is change in y values over the change in x values. For the derivative we need two points. The first is (x, f(x)), f(x) is represented y. We can say that (x, f(x)) is the starting point (x,y). The second point is ((x+h),f(x+h)) with h is change in x. From those points we can find the slope of the line between points by slope formula.

The formula is shown bellow.

Slope=(change in y's)/(change in x's)=(y2-y1)/(x2-x1)

Suppose that y1 is f(x), y2 is f(x+h), x1 is x, and x2 is x+h. So from the formula we get

(f(x+h)-f(x))/(x+h-x)= (f(x+h)-f(x))/h

h represent change in x values. If we put limit h approach to zero like this

f'(x)=lim┬(h→0)⁡〖(f(x+h)-f(x))/h〗

With the numerator is change in values of function during interval h. The denominator is amount interval of h. So the derivative is not only the limit of a slope but also limit change in values of f(x) over change in x or the instantaneous rate of change.

Example

        Suppose we have f(x)=4x^2-8x+3, find f'(2). Use the definition of the derivative!

We get x=2 because we want to look for f'(2).

f'(2)=lim┬(h→0)⁡〖(f(2+h)-f(2))/h〗

First, evaluate f(2+h).

f(2+h)=4(2+h)(2+h)-8(2+h)+3

f(2+h)=4(4+4h+h²)-16-8h+3

f(2+h)=16+16h+4h²-16-8h+3

f(2+h)=4h²+8h+3

Then, evaluate f(2)

f(2)=4(2)(2)-8(2)+3

f(2)=3

Then, we put them to the form.

f'(2)=lim┬(h→0)⁡〖(f(2+h)-f(2))/h〗

f^' (2)=lim┬(h→0)⁡〖(4h^2+8h+3-3)/h〗

f^' (2)=lim┬(h→0)⁡〖h(4h+8)/h〗

f^' (2)=lim┬(h→0)⁡〖4h+8〗

f'(2)=8

So we get that at point (2,3) the slope is 8.

Now we try to know what the general form for derivative is. Suppose the f(x) is 4x²-8x+3. Then we get:

f^' (x)=lim┬(h→0)⁡〖(f(x+h)-f(x))/h〗

f'(x)=lim┬(h→0)⁡〖((4x^2+8xh+4h^2-8h+3)-(4x^2-8x+3))/h〗

f^' (x)=lim┬(h→0)⁡〖(8xh+4h^2-8h)/h〗

f^' (x)=lim┬(h→0)⁡〖(h(4h+8x-8))/h〗

f^' (x)=lim┬(h→0)⁡〖4h+8x-8〗

f^' (x)=8x-8

So to find the slope of any point we only substitute the value for x to the form.

 

A.      Trigonometry Function

Trigonometry function is the ratios of different sides of a triangle with respect to an angle. So let say that you have the sides with trigonometry function you could determine the angles of triangle. With trigonometry function you only need to know values of the sides to find measure of an angle and figure out values of all parts of a triangle.

There are six basics trigonometry function.

1.       Sine

2.       Cosine

3.       Tangent

4.       Cosecant

5.       Secant

6.       Cotangent

Trigonometry functions sides of triangle and angle being measured.

opp: side opposite gamma

adj: side adjacent gamma

hyp: hypotenuse

So we get the general form of six basics trigonometry function.

1.       sin⁡γ=opp/hyp

2.       cos⁡〖γ=adj/hyp〗

3.       tan⁡〖γ=opp/adj〗

4.       csc⁡〖γ=hyp/opp〗

5.       sec⁡〖γ=hyp/adj〗

cot⁡〖γ=adj/opp〗

A.      Properties of Logarithm (JustMathTutoring.com)

In this video we discuss about logarithm.

1.       The Basic Form of Logarithm

log_b⁡x=y ↔b^y=x

This is the basic form of the logarithm. The number b is called base. We can read the form as logarithm x base b. So we know that b in the power of y is equal to x.

2.       Notation

log_10⁡x=log⁡x

                Base 10 is okay to not to be written in the logarithm form. We know that the base is 10.

log_e⁡x=ln⁡x

This form is called natural logarithm. e is natural number. It's a irrational number, just like phi (π).

Example

               
log_10⁡100=x

                From the form we know the base is 10. From the basic form of logarithm we may conclude that〖10〗^x=100. Then x=2.

3.       We have other form. The forms are shown bellow.

a.       log_b⁡〖(mn)〗=log_b⁡〖m+log_b⁡n 〗

b.      log_b⁡〖m/n〗=log_b⁡〖m-log_b⁡n 〗

c.       log_b⁡〖(x^n)〗=n.log_b⁡x

For multiplication inside the bracket, we have other form, it becomes logarithm addition. Here's an example.

log_3⁡〖(x^2 (y+1))/z^3 〗

Make it becomes a simple form.

log_3⁡〖(x^2 (y+1))/z^3 〗

log_3⁡〖x^2 (y+1)- log_3⁡〖z^3 〗  〗

log_3⁡〖x^2+log_3⁡〖(y+1)〗- log_3⁡〖z^3 〗  〗

〖2. log〗_3⁡〖x+log_3⁡〖(y+1)〗- 〖3.log_3〗⁡z  〗