MAE101 CAL v1 chapter 2 limits

ĐẠI HỌC FPT CẦN THƠ Chapter LIMITS (Page 123-194, Calculus Volume 1) Dr Tran Quoc Duy Email: duytq4@fpt.edu.vn Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONTENTS 2.1 A Preview of calculus 2.2 The limit of a Function 2.3 The Limit Laws 2.4 Continuity 2.5 The Precise Definition of a Limit Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ LIMITS 2.1 A Preview of Calculus Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE TANGENT PROBLEM Example x2 - We know that the slope of the secant line PQ is mPQ = x -1 How to find an equation of the tangent line to the parabola y = x2 at the point P(1,1)? Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE VELOCITY PROBLEM Example Investigate the example of a falling ball ▪ Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground ▪ Find the velocity of the ball after seconds Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE VELOCITY PROBLEM If the distance fallen after t seconds is denoted by s(t) and measured in meters, then Galileo’s law is expressed by the following equation s(t) = 4.9t2 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE VELOCITY PROBLEM change in position average velocity = time elapsed = s ( 5.1) − s ( ) = 49.49 m/s 0.1 Thus, the (instantaneous) velocity after 5s is: v = 49 m/s Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE AREA PROBLEM Example We begin by attempting to solve the area problem: Find the area of the region S that lies under the curve y = f(x) from a to b Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ LIMITS 2.2 The Limit of a Function Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE LIMIT OF A FUNCTION In general, we write lim f (x ) = L x ®a if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a but not equal to a Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONTINUITY Definition A function f is continuous on an interval if it is continuous at every number in the interval – If f is defined only on one side of an endpoint of the interval, we understand ‘continuous at the endpoint’ to mean ‘continuous from the right’ or ‘continuous from the left.’ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONTINUITY Theorem If f and g are continuous at a; and c is a constant, then the following functions are also continuous at a: f + g f - g cf fg f if g (a)  g Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONTINUITY Theorem The following types of functions are continuous at every number in their domains: – Polynomials – Rational functions – Root functions – Trigonometric functions Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Theorem CONTINUITY If f is continuous at b and lim g ( x) = b then x ®a lim f ( g ( x)) = f (b) x ®a In other words, ( lim f ( g ( x)) = f lim g ( x) x →a x →a ) – If x is close to a, then g(x) is close to b; and, since f is continuous at b, if g(x) is close to b, then f(g(x)) is close to f(b) Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONTINUITY Theorem If g is continuous at a and f is continuous at g(a), then the composite function ( f g ) ( x) = f ( g ( x)) is continuous at a This theorem is often expressed informally by saying “a continuous function of a continuous function is a continuous function.” Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTERMEDIATE VALUE THEOREM Theorem Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f (a) ¹ f (b) Then, there exists a number c in (a, b) such that f(c) = N Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ INTERMEDIATE VALUE THEOREM Show that there is a root of the equation x - x + 3x - = Example between and f ( x ) = x x + 3x - – Let – We are looking for a solution of the given equation that is, a number c between and such that f(c) = – Therefore, we take a = 1, b = 2, and N = in the theorem – We have f (1) = - + - = -1 < and f (2) = 32 - 24 + - = 12 > Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ QUIZ QUESTIONS 1) If f(1)>0 and f(3)a Then lim f ( x) = L x → means that   0, M  if x  M then f ( x) − L   Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ DEFINITION The line y=L is called the horizontal asymptote of f(x) if we have one of the following: lim f ( x ) = L x → Dr Tran Quoc Duy lim f ( x ) = L x →− Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example Find the asymptotes of the function x3 − f ( x) = x + x2 − Solution 1− x3 x3 − lim = lim =1 x → x + x − x → 1+ − x x y=1 is horizontal asymptote ( x − 1) ( x + x + 1) x3 − = x + x − ( x − 1)( x + x + 2) x3 + lim = x →1 x + x − Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example Compute a lim sin x → x b lim( x + − x) lim sin x Does not exist x → c d x → lim( x − x ) x → Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ QUIZ QUESTIONS 1) Find lim cos x x → a 2) Find a b infinity c d Does not exist lim cos x x → x b infinity Dr Tran Quoc Duy c d Dose not exist Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ QUIZ QUESTIONS 3) If Then lim f ( x) = , lim g ( x) =  x →0 x →0 lim[ f ( x) − g ( x)] = x →0 a True b False 4) A function can have two different horizontal asymptotes a True Dr Tran Quoc Duy b False Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ...ĐẠI HỌC FPT CẦN THƠ CONTENTS 2. 1 A Preview of calculus 2. 2 The limit of a Function 2. 3 The Limit Laws 2. 4 Continuity 2. 5 The Precise Definition of a Limit Dr Tran Quoc Duy... Engineering ĐẠI HỌC FPT CẦN THƠ LIMITS 2. 1 A Preview of Calculus Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE TANGENT PROBLEM Example x2 - We know that the slope of the... ONE-SIDED LIMITS Similarly, “the right-hand limit of f(x) as x approaches a is equal to L” and we write lim+ f (x ) = L x®a lim- g (x ) x 2 lim g (x ) lim- g (x ) x 2 lim+ g (x ) x 2 x ®5 lim+