Grade 11 Math Notes

 Grade 11

Math I

Capter 1

Remainder Theorem and Factor Theorem

1.1 Dividing Polynomials

1.2 Synthetic Division 

Long and Synthetic Division

1.3 Remainder Theorem Pg. 

Remainder and Factors

1.4 Factor Theorem Pg. 7

1.4 Factor Theorem Pg. 7 

Theorem 1.3

Notes - No. Of Factors

Theorem 1.4 &1.5

Notes & Example

Possible Rational Roots 

Binomial (Direct Solve) and 

Polynomial (1St Rational Root step by step still to be binomial)

Notes & Example 

Constant Term, 

Leading Coefficient, 

How to do possible rational roots, and 

rational roots in lowest term

Part 1

Part 2

Part 3 

Lowest Term

Finding Factors of Polynomials by rational roots to get binomial for final solution

Thinking Process for No.6 from Ex 1.2

Guides for findinf two unknown coefficients

Chapter 2 

The Binomial Theorem Pg.15

Binomial Expansion

Coefficient Pacterm

Example 1

2.2 The Binomial Theorem Pg.17 

Power Level and Solution

Combinatorial Coefficient

Theorem 2.1 Pg.18

Collorary 2.2 Pg.18

General Term of (x+y)n

Chapter (11) 

Methods of Differentiations

👉Consider real-valued functions whose domains arbitrary subset of R. 

👉Limit of a function and derivatives 

👉sun rule, different rule, product rule, quotient rule, and chain rule (To calculate derivative more effeciently) 

👉Method of Differentiation for implicit functions 

👉Undifined (Limit) (My opinion - Asymtope) and Value (Continue)

👉If any is divided by infinity, result is always "0". eg. 6/ꟹ, 9/ꟹ

👉Rational Function, Divide x as power of x in denorminator to both of denorminator and upper part and continue solve as usual. If denorminator is "0", it is undefined.

👉Square Root function, negative is undefined.

👉Simple Plynomial Function, Multiply the highest powered x to the whole polynomial and solve continue as usual.

Easy Notes for Chapter 11

Methods of Differentiation

From the Myanmar Grade 11 Mathematics Textbook

Chapter 11 mainly teaches:

1. Limits of Functions
2. Derivatives
3. Differentiation Rules
4. Implicit Differentiation

11.1 Limit of Functions

Simple Idea

A limit means:

“What value does a function approach when 

$x$

Example:

If

$$f(x)=x+2$$

when 

$x\to 3$

,

$$f(x)\to 5$$

because:

$$3+2=5\mathrm{<br>}\mathrm{<br>}\mathrm{<br>}$$

 

Basic Notation

$$\underset{x\to a}{\lim }f(x)<span\; style="text-align:\; justify;"><br></span><span\; style="text-align:\; justify;">means:</span>$$

“the limit of f(x) as $x$ approaches a.

Example 1

Find:

$$\underset{x\to 2}{\lim }(x^2+3)$$

Substitute $x=\mathrm{<span\; class="base"><span\; class="strut"></span><span\; class="mord">2</span></span>:}$

$$2^2+3=4+3=7$$

Answer: 7

Important Idea

A limit helps us find the instantaneous rate of change, which leads to derivatives.

11.2 Derivatives

Meaning of Derivative

Derivative shows:

• how fast something changes
• slope of a curve
• gradient at a point

Symbol:

$$\frac{dy}{d\mathrm{x}}$$

or

$$f^{\mathrm{\prime }}(x)$$

---

Derivative Formula

$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$

This is the basic definition of differentiation.

---

Example 2

Find derivative of:

$$f(x)=x^2$$

Using rule:

$$\frac{d}{dx}(x^2)=2x$$

Answer: 2x

11.3 Differentiation Rules

These rules are very important.

---

1. Constant Rule

If $c$ is a constant:

$\frac{d}{dx}(c)=0$

Example:

$$\frac{d}{dx}(7)=0$$

---

2. Power Rule

$\frac{d}{dx}(x^n)=nx^{n-1}$

Examples:

$$\frac{d}{dx}(x^3)=3x^2$$$$\frac{d}{dx}(x^5)=5x^4$$

---

3. Constant Multiple Rule

$$\frac{d}{dx}[cf(x)]=c\frac{d}{dx}[f(x)]$$

Example:

$$\frac{d}{dx}(5x^2)=5(2x)=10x$$----------------------------------------------------------------------------------------------------------------------

4. Sum Rule

$$\frac{d}{dx}[f(x)+g(x)]=f^{\mathrm{\prime }}(x)+g^{\mathrm{\prime }}(x)$$

Example:

$$\frac{d}{dx}(x^2+3x)=2x+3$$

---

5. Difference Rule

$$\frac{d}{dx}[f(x)-g(x)]=f^{\mathrm{\prime }}(x)-g^{\mathrm{\prime }}(x)$$

Example:

$$\frac{d}{dx}(x^2-4x)=2x-4$$

---

6. Product Rule

Used when two functions multiply.

$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$

Example:

Find derivative of:

$$y=x^2(x+1)$$

Let

$$u=x^2,v=x+1$$

Then:

$$\frac{du}{dx}=2x$$
$$\frac{dv}{dx}=1$$

Apply rule:

$$\frac{dy}{dx} = x^2(1)+(x+1)(2x)$$
$$=x^2+2x^2+2x$$
$$=3x^2+2x$$

Answer:

$$\boxed{{\displaystyle 3x^2+2\mathrm{x}}}$$

---

7. Quotient Rule

Used for division.

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

Example:

$$y=\frac{x^2}{x+1}$$

Use quotient rule.

----------------------------------------------------------------------------------------

11.4 Implicit Differentiation

Sometimes $y$ is mixed with $x$.

Example:

$$x^2+y^2=25$$

Differentiate both sides.

$$\frac{d}{dx}(x^2)+\frac{d}{dx}(y^2)=0$$
$$2x+2y\frac{dy}{dx}=0$$

Solve:

$$2y\frac{dy}{dx}=-2x$$
$$\frac{dy}{dx}=-\frac{x}{y}$$

Answer:

$$\boxed{{\displaystyle \frac{dy}{dx}=-\frac{x}{\mathrm{y<span\; class="vlist"><span\; class="boxpad"><span\; class="mord"><span\; class="mord"><span\; class="mord"><span\; class="mfrac"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist-s"></span></span><span\; class="vlist-r"><span\; class="vlist"></span></span></span></span><span\; class="mclose\; nulldelimiter"></span></span></span></span></span><span\; class="pstrut"></span><span\; class="stretchy\; fbox"></span></span><span\; class="vlist-s"></span>}}}}$$

---

Short Tricks to Remember

Rule	Shortcut
Constant	Answer is 0
$x^n$	Bring power down
Addition	Differentiate separately
Multiplication	Product Rule
Division	Quotient Rule
Mixed $x,y$	Implicit Differentiation

---

Important Formulas Summary

Power Rule

$\frac{d}{dx}(x^n)=nx^{n-1}$

Product Rule

$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$

Quotient Rule

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

---

Exam Tips

• Learn formulas by heart.
• Practice many differentiation problems.
• Simplify answers carefully.
• Watch signs (+ and −).
• Write steps clearly in exams.

$$\boxed{{\displaystyle \mathrm{<span\; class="vlist-s"></span>}}}$$

$$\boxed{{\displaystyle \mathrm{<span\; class="vlist-s"></span>}}}$$