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**LESSON 1 Mathematics for Physics**

Mr. Komsilp Kotmool (Aj Tae) Department of Physics, MWIT Web site : .ac.th/~tae_mwit

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**Warm Up the farmer needs 3 hens to produce 12 eggs in 6 days.**

A single chicken farmer has figured out that a hen and a half can lay an egg and a half in a day and a half. How many hens does the farmer need to produce one dozen eggs in six days? A single chicken farmer also has some cows for a total of 30 animals, and all animals in farm have 74 legs in all. How many chickens does the farmer have? the farmer needs 3 hens to produce 12 eggs in 6 days.

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**CONTENTS Why do we use Mathematics in Physics?**

What is Mathematics in this course? Functions Limits and Continuity of function Fundamental Derivative Fundamental Integration Approximation methods

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**Why do we use Mathematics in Physics (and other subjects)?**

ข้อมูลเชิงวิทยาศาสตร์ (ฟิสิกส์) Qualitative Physics – เชิงคุณภาพ เป็นรูบแบบการศึกษาที่บ่งบอกถึงความรู้สึก มีหรือไม่มี เช่น บางส่วนของวิชา Quantum Physics Quantitative Physics – เชิงปริมาณ เป็นรูปแบบการบ่งชี้เชิงข้อมูที่แม่นตรง มาก-น้อยแค่ไหน สามารถนำไปประยุกต์ใช้ในกิจกรรมของมนุษย์ได้ย่างมีประสิทธิภาพ (ส่วนใหญ่ของฟิสิกส์)

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**What is Mathematics in this course?**

REVIEW: basic problem Displacement S = ? = v x t NOT AT ALL condition: v must be constant !! But in reality, v is not constant !!! General definition Better or CALCULUS or

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**Functions What is FUNCTION? REVIEW : some basic equations**

y = Ax+B linear y = Ax2+Bx+C parabola y = sin(x) trigonometric y = circular y is the function of x for the 1-3 equation, but the 4th is not!! Rewrite: y is the function of x to be y(x) [y of x]

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**Functions What is FUNCTION? REVIEW again: For physical view**

The motion of a car is expressed with S = 5t-5t2 equation. Displacement (S) depends on time (t) We can say that S is the function of t, and can be rewritten: S(t) = 5t-5t2 Note: We usually see f(x) and g(x) in many text books.

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**Functions Mathematical definitions**

1). If variable y depends on a variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x. 2). A function f is a rule that associates a unique output with each input. If a input denoted by x, then the output is denoted by f(x) (read “f of x”)

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Exercise Consider variable y whether it is the function or not and find the value of y at x=2 1). y – x2 = 5 2). 4y2 + 9x2 = 36 3). y = 0 4). ycsc(x) = 1 5). y2 +2y –x =0

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**Limits and Continuity of function**

What is limits? Why do we use limits? When do we use limits? ?????

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**Limits and Continuity of function**

Newton’s Law of Universal Gravitation We can not calculate force at r → ∞. How do we solve this problem?? But we can find the close value!!!!!! at r → ∞.

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**Limits and Continuity of function**

Electric field at arbitrary from solid sphere E = ? At r = a

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**Limits and Continuity of function**

For mathematical equation f(x=0) = ? We can not find f(x) at x = 0 But we can find its close value of f(x) at x → 0 Therefore, f(x) → 2 at x → 0 “the limit of f(x) as x approaches 0 is 2”

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**Limits and Continuity of function**

How about at x → 0? -1 ; x<0 1 ; x>0 Limit from the left Limit from the right Not exist !! For discontinuity

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**Limits and Continuity of function**

THEOREMS. Let a and k be real numbers, and suppose that and 1 2 3 4 5 and

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**Limits and Continuity of function**

A function f(x) is said to be continuous at x = c provided the following condition are satisfied : 1). f(c) is defined 2). exists 3).

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**History of Calculus Sir Isaac Newton (1642-1727)**

Gottfried Wilhelm Leibniz ( )

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**Fundamental Derivative**

In reality, the world phenomena involve changing quantities: the speed of objects (rocket, vehicles, balls, etc.) the number of bacteria in a culture the shock intensity of an earthquake the voltage of an electrical signal It is very easy for the ideal situations {many exercises in your text books} For example: constant of velocity S = v x t rate of change of S is constant v = ΔS/Δt Consider: can we have this situation in the real world?

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**Fundamental Derivative**

There are many conditions in nature affecting to complicate phenomena and equations!!!

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**Fundamental Derivative**

Consider: equation of motion (free fall) S(t) = ut – ½(g)t2 Velocity does not be constant with time !!! How do we define velocity from t1 to t2 ? Average velocity ?

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**Fundamental Derivative**

Slopes and rate of change Take t2 close to t1 S(t) S(t) t t1 t2 S(t1) S(t2) Close to tangent line t1 t2 S(t1) S(t2) t vav can not be exactly represented this motion! more exactly !!! Slope at t = t1 is tangent line

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**Fundamental Derivative**

Take t2 close to t1 that t2-t1 = Δt → 0 vint → v(t) instantaneous velocity Mathematical notation (read “dee S by dee t”)

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**Fundamental Derivative**

Exercises: Find the derivative of y(x) and its value at x=2 y(x) = x y(x) = x2 y(x) = sin(x)

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**Fundamental Derivative**

FORMULARS: If f(x) and g(x) are the function of x and c is any real number 1. 2. 3. 4. 5. 6.

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**Fundamental Derivative**

What is the derivative of these complicate functions? Case 1) y(x) = sin(x2) Case 2) y(x) = xcos(x) Case 3) y(x) = tan(x) Chain rule:

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**Fundamental Derivative**

FORMULARS : If u is a function of x {u(x)}, and c is any real number Frequency use in Physics 6. 7. 1. 2. 3. 4. 5. g(x) ≠ 0

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**Additional applications of Derivative**

Maximum and Minimum problems Ex : An open box is composed of a 16-inch by 30-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares to be obtained a box with the largest volume? 16 in 30 in x

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**Fundamental Integration**

What is the integration (calculus)? Why do we use the integration (calculus)? When do we use the integration (calculus)?

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**Fundamental Integration**

Recall: If we know S(t) of a object we can find v(t) of its and say v(t) is the derivative of S(t) !! Reverse problem : If we know v(t) of a object Can/How do we find S(t) of its ? We can say that S(t) is the antiderivative of v(t) !! What is the antiderivative?

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**Fundamental Integration**

Consider graphs of constant and linear velocity with time (v(t)-t) v(t) t v0 v S = v0t S = (1/2)(v0+v)t S(t) was represented by area under function of v(t) !!!

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**Fundamental Integration**

How about the complicate function of v(t)? v(t) t v What about S(t) of this curve? S(t) also was represented by area under of this curve! Next problem…. How do we find this area?

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**Fundamental Integration**

Method of Exhaustion The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater and greater number of sides. disadvantage: do not proper with asymmetric shape

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**Fundamental Integration**

v(t) t v For smoother area t* v(t*) Therefore, we get Δt Read “S(t) equal integral of v(t) dee t”

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**Fundamental Integration**

Mathematical Method Note : f(x) is the derivative of F(x), but in the other hand, F(x) is a antiderivative of f(x). For Polynomial Function F(x) = x2 f(x) = 2x Constants depend on conditions !! F(x) = x2 + 5 f(x) = 2x F(x) = x2 + 10 f(x) = 2x

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**Fundamental Integration**

Recall: Set n=n-1 Therefore, we get For the flexible form When C is the arbitrary constant.

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**Fundamental Integration**

Exercises: Find the antiderivative of the following y(x). y(x) = 3x3 + 4x2 + 5 y(x) = 0 y(x) = sin(x) y(x) = xcos(x2) y(x) = tan(x) Integration by Substitution

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**Fundamental Integration**

FORMULARS: If f(x) and g(x) are the function of x and c is any real number 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. All of these are call the indefinite integral

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**Fundamental Integration**

The definite integral Recall: We get the general solution of S(t) from the indefinite integral. Problem: How do we get displacement (S) in the interval [a,b] of time? v(t) t a b a is the lower limit b is the upper limit this is the definite integral !

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**Fundamental Integration**

Definition: (a) If a is in the domain of f(x), we define (b) If f(x) is integrable on [a,b], then we define Theorem. If f(x) is integrable on a close interval containing the three nuber a, b, and c, then

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**Fundamental Integration**

Example: Find the antiderivative of y(x) = 4x3 + 2x + 5 in the interval [2,5] of x. Sol

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**Fundamental Integration**

Exercises: Find the value of these definite integral. 1. 2. 3. 4. 5.

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**Additional applications of Integration**

In Physics Center of mass The triangle plate has mass M and constant density ρ that is shown in the figure. Find its center of mass that can find from these equations H L

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**Additional applications of Integration**

In Physics Work done by a constant force Work done by variable force ? F S

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**References Anton H., Bivens I., and Davis S. Calculus. 7th Ed.**

John Willey&Son, Inc

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Calculus,10/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

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