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LESSON 1 Mathematics for Physics Mr. Komsilp Kotmool (Aj Tae) Department of Physics, MWIT Web site :

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Warm Up 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)? ( ) ( ) Qualitative Physics – Quantum Physics - ( ) Quantitative Physics – - ( )

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What is Mathematics in this course? REVIEW: basic problem NOT AT ALL 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

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Functions What is FUNCTION? REVIEW : some basic equations y = Ax+Blinear y = Ax 2 +Bx+Cparabola y = sin(x)trigonometric y = circular y is the function of x for the 1-3 equation, but the 4 th 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-5t 2 equation. Displacement (S) depends on time (t) We can say that S is the function of t, and can be rewritten: S(t) = 5t-5t 2 Note: We usually see f(x) and g(x) in many text books.

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Functions Mathematical definitions y is a function of x. 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 – x 2 = 5 2). 4y 2 + 9x 2 = 36 3). y + = 0 4). ycsc(x) = 1 5). y 2 +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 Newtons 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

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Limits and Continuity of function 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 ( ) 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 constantv = 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)t 2 Velocity does not be constant with time !!! How do we define velocity from t 1 to t 2 ? Average velocity ?

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Fundamental Derivative Slopes and rate of change S(t) t t1t1 t2t2 S(t 1 ) S(t 2 ) v av can not be exactly represented this motion! Take t 2 close to t 1 S(t) t t1t1 t2t2 S(t 1 ) S(t 2 ) more exactly !!! Close to tangent line Slope at t = t 1 is tangent line

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Fundamental Derivative Take t 2 close to t 1 that t 2 -t 1 = Δt 0 v int 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 1. y(x) = x 2. y(x) = x 2 3. 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

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Fundamental Derivative What is the derivative of these complicate functions? Case 1) y(x) = sin(x 2 ) 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 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 inxx x x

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Fundamental Integration Integration 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: say v(t) is the derivative of S(t) !! 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 : Reverse problem : If we know v(t) of a object Can/How do we find S(t) of its ? S(t) is the antiderivative of v(t) !! 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 v0v0 t v0v0 v t t S = v 0 t S = (1/2)(v 0 +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 t 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 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. Method of Exhaustion disadvantage: do not proper with asymmetric shape

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Fundamental Integration v(t) t v t ΔtΔtΔtΔt t*t* v(t * ) For smoother area Therefore, we get Read S(t) equal integral of v(t) dee t

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

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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). 1. y(x) = 3x 3 + 4x y(x) = 0 3. y(x) = sin(x) 4. y(x) = xcos(x 2 ) 5. 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 All of these are call the indefinite integral

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Fundamental Integration The definite integral Recall: the indefinite integral 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? 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. 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) = 4x 3 + 2x + 5 in the interval [2,5] of x. Sol

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Fundamental Integration Exercises: Find the value of these definite integral

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Additional applications of Integration In Physics Center of mass H L 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

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Additional applications of Integration In Physics Work done by variable force ? Work done by a constant force F S

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References Anton H., Bivens I., and Davis S. Calculus. 7 th Ed. John Willey&Son, Inc

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