2301107 Calculus I 1. Limits and Continuity Outline 1.1. Limits 1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws 1.1.4 Mathematical definition of a it 1.1.5 Infinite it 1.1. Continuity Chapter 1:Lmits and Continuity 2
1.1.1. Motivation:Tangent Tangent derived from the Latin word, tangens, which means touching. Tangent line in a circle, intersect a circle only once Is it possible to identify this line, mathematically? Chapter 1:Lmits and Continuity 3 numerically? Given the parabola y = x2, find the tangent line to the parabola at a point P(1, 1). slope of PQ= x2 1 x 1 0 Q(x, x 2 ) P(1, 1) Required two points to compute the slope. x PQ 2 3 1.5 2.5 1.1 2.1 1.01 2.01 1.001 2.001 Chapter 1:Lmits and Continuity 4
situation? Velocity problem Suppose a ball has been drop from ENG4, 400 m. above the ground, find velocity after 5 sec. s(t) = 4.9 t 2 Average = distance traveled/time elapsed Time interval Average velocity (m/s) 5 < t < 6 53.9 5 < t < 5.1 49.49 5 < t < 5.05 49.245 5 < t < 5.01 49.049 5 < t < 5.001 49.0049 The (instantaneous) velocity after 5 sec. is 49 m/s. Chapter 1:Lmits and Continuity 5 numerical computation. Consider y = x 2 x + 2 for values of x near 2. 0 4 2 (2, 4) x f(x) x f(x) 1 2 3 8 1.5 2.75 2.5 5.75 1.9 3.71 2.1 4.31 1.99 3.97 2.01 4.03 1.999 3.997 2.001 4.003 when x is close to 2 (on either side of 2), f(x) is close to 4. Note, f(2) is not significant here! Chapter 1:Lmits and Continuity 6
Mathematical notation of it The it of f(x), as x approaches a, equal L is written as f x =L x 1 1. Guess the value of using a calculator. x 1 x 2 1 x f(x) x f(x) Limit is. We do not compute value at a. Chapter 1:Lmits and Continuity 7 t 2 9 3 2. Estimate the value of by calculator. t 0 x f(x) x f(x) t 2 Limit is. Chapter 1:Lmits and Continuity 8
sin x 3. Guess the value of using a calculator. x 0 x x f(x) x f(x) Limit is. Chapter 1:Lmits and Continuity 9 4. Investigate x 0 sin x. x f(x) x f(x) x f(x) Chapter 1:Lmits and Continuity 10
5. Find x 1. x 1 x f(x) x f(x) x f(x) Chapter 1:Lmits and Continuity 11 6. Investigate x 0 sin x. x f(x) x f(x) x f(x) Chapter 1:Lmits and Continuity 12
0 if t 0 7. Investigate H t ={ 1 if t 0}. x f(x) Chapter 1:Lmits and Continuity 13 One-Sided Limits Definition: We write f x =L as the left-hand it of f(x) as x approaches a [it of f(x) as x approaches a from the left] is equal to L if we can make f(x) arbitrarily close to L by taking x (less than a) to be sufficiently close to a. f x =L The right-hand it of f(x) as x approaches a [it of f(x) as x approaches a from the right] is equal to L if we can make f(x) arbitrarily close to L by taking x (greater than a) to be sufficiently close to a. Chapter 1:Lmits and Continuity 14
Graph and its it Rule: f x =L if and only if f x =L and f x =L. Determine the it of g. g x x 3 g g x 4 x 3 3 g x x 7 2 g x 1 x 7 0 3 7 x 0 g x Chapter 1:Lmits and Continuity 15 Infinite its Determine if the it exists x 0 1 x 2. x 0 1 x 2= y= 1 x 2 0 x Chapter 1:Lmits and Continuity 16
Infinite its Definition: Let f be a function defined on both sides of a, except possibly at a itself. Then f x = means f(x) can be made arbitrarily large as x sufficiently close to a, but not equal to a. Chapter 1:Lmits and Continuity 17 Infinite its Determine if the it exists x 0 1 x 2. 0 x y= 1 x 2 Chapter 1:Lmits and Continuity 18
Infinite its Definition: Let f be a function defined on both sides of a, except possibly at a itself. Then f x = means f(x) can be made arbitrarily small (large negative) as x sufficiently close to a, but not equal to a. Chapter 1:Lmits and Continuity 19 Vertical asymptote Definition: The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true: f x = f x = f x = f x = f x = f x = Chapter 1:Lmits and Continuity 20
8. Find x 3 2 x x 3, x 3 2 x x 3. Chapter 1:Lmits and Continuity 21 9. Find the vertical asymptotes of f(x) = tan x. Chapter 1:Lmits and Continuity 22
1.1.3 Limit laws Suppose that c is a constant and its of the following exist f x, g x then 1. 2. 3. 4. 5. f x g x = f x g x = c f x =c f x f x g x = f x f x g x = g x f x f x f x if g x g x g x g x 0 Chapter 1:Lmits and Continuity 23 1.1.3 Limit laws (cont.) Moreover, 6. 7. 8. 9. 10. 11. [ f x ] n = c=c x=a x n =a n where n Z n x= a n where n Z n f x = n f x n where n Z f x where n f x exists Chapter 1:Lmits and Continuity 24
Limit of polynomial and rational functions If f is a polynomial or a rational function and a is in the domain of f, then f x = f a For example, f(x) = 3x 4 5x 2 + x 3, then 3 x 4 5 x 2 x 3=3 a 4 5 a 2 a 3 Chapter 1:Lmits and Continuity 25 10. Find x 1 x 2 1 x 1. Chapter 1:Lmits and Continuity 26
x 1 if x 1 11. Find g x where x 1 g x ={ if x=1}. Chapter 1:Lmits and Continuity 27 12. Show that x =0. x 0 Chapter 1:Lmits and Continuity 28
x 13. Prove that does not exist. x 0 x Chapter 1:Lmits and Continuity 29 14. Determine if f whether it x ={ x 4 if x 4 8 2 x if x 4} f x exists. x 4 Chapter 1:Lmits and Continuity 30
Theorem of its Theorem: If f(x) < g(x) when x is near a and the its of f and g both exist as x approaches a, then f x g x The Squeeze Theorem: If f(x) < g(x) < h(x) when x is near a and then f x = g x =L h x =L Chapter 1:Lmits and Continuity 31 15. Show that x 0 x 2 sin 1 x =0. Chapter 1:Lmits and Continuity 32
16. Given f x = 3, g x =0, find the its that exists. If not, explain. 1. f x h x 2. h x =8 f x 2 3. 3 h x Chapter 1:Lmits and Continuity 33 17. Given f x = 3, find the its that exists. If not, explain. 1. f x h x 2. 1 f x g x =0, 3. h x =8 f x g x Chapter 1:Lmits and Continuity 34
s 18. Evaluate the its, if they exist. x 2 x 6 t 2 9 1. 2. x 2 x 2 x 3 2t 2 7 t 3 Chapter 1:Lmits and Continuity 35 s 19. Evaluate the its, if they exist. x 2 4 x x 3 1 1. 2. x 4 x 2 3 x 4 x 1 x 2 1 Chapter 1:Lmits and Continuity 36
s 20. Evaluate the its, if they exist. 1 h 4 1 9 t 1. 2. h 0 h t 9 3 t Chapter 1:Lmits and Continuity 37 1.1.4 Mathematical definite of it Definition: Let f be a function defined on some open interval that contains a, except possibly at a. The it of f(x) as x approaches a is L, written f x =L if for every number ε > 0 there is a number δ > 0, f(x) L < ε whenever 0 < x a < δ Chapter 1:Lmits and Continuity 38
1.1.4 Mathematical definite of it For every number ε > 0 there is a number δ > 0, f(x) L < ε whenever 0 < x a < δ ε ε L a f x =L Limit of f(x) as x approaches a 0 δ δ Chapter 1:Lmits and Continuity 39 Alternative interpretation Alternatively, If 0 < x a < δ then f(x) L < ε. x a f(x) f(a) ( δ δ a x ) ( ε ε L f(x) ) Chapter 1:Lmits and Continuity 40
Definite of left & right-hand it Definition: f x =L if for every number ε > 0 there is a number δ > 0, f(x) L < ε whenever a δ < x < a Definition: f x =L if for every number ε > 0 there is a number δ > 0, f(x) L < ε whenever a < x < a + δ Chapter 1:Lmits and Continuity 41 1.1.5 Infinite it definition Definition: means for every positive number M there is a number δ > 0, f(x) > M whenever 0 < x a < δ Definition: f x = f x = means for every negative number N there is a number δ > 0, f(x) < N whenever 0 < x a < δ Chapter 1:Lmits and Continuity 42
1.2 Continuity Definition: A function f is continuous at a number a if Note f x = f a f(a) is defined. (that is, a is in the domain of f) Limit of f as x approaches a exists The value of the it is the same as function value. If f is not continuous at a, we say that f is discontinuous at a. Chapter 1:Lmits and Continuity 43 Graph and continuity Determine from the graph g, which points of g are discontinuous and why? g 4 2 0 2 Chapter 1:Lmits and Continuity 44
21. Where are the following functions discontinuous? 1. f x = x2 x 2 2. f x = x x 2 where x = the largest integer that is less than or equal to x. Chapter 1:Lmits and Continuity 45 3. f x ={x 2 x 2 if x 2 if x 0 x 2 4. f x ={1 x 1 if x=2} 2 1 if x=0} Chapter 1:Lmits and Continuity 46
One-sided continuity Definition: A function f is continuous from the right at a number a if f x = f a Definition: A function f is continuous from the left at a number a if f x = f a Definition: A function f is continuous on an interval if it is continuous at every number in the interval. Chapter 1:Lmits and Continuity 47 22. Show that the function f(x) = 1 1 x 2 is continuous on the interval [-1, 1]. Chapter 1:Lmits and Continuity 48
Theorem for 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 1. f + g 2. f g 3. cf 4. fg 5. if g a 0 g Theorem: Any polynomial and rational function is continuous wherever it is defined. The following types of functions are continuous at every number in their domains: Polynomials, Rational functions, Root functions, Trigonometric functions. Chapter 1:Lmits and Continuity 49 Continuity of a composite function Theorem: If f is continuous at b and In other words, f g x = f b f g x = f g x g x =b then Theorem: If g is continuous at a and f is continuous at g(a), then the composite function f g given by (f g)(x) = f(g(x)) is continuous at a. Chapter 1:Lmits and Continuity 50
23. Where are the following functions continuous? (a) h(x) = sin(x 2 ) (b) F x = 1 x 2 7 4 Chapter 1:Lmits and Continuity 51 Intermediate Value Theorem Intermediate Value Theorem: Suppose f is continuous on [a, b] and let N be any number between f(a) and f(b), where f(a) f(b). Then there exists a number c in (a, b) such that f(c) = N. Chapter 1:Lmits and Continuity 52
24. Show that there is a root of the equation between 1 and 2. 4x 3 6x 2 + 3x 2 = 0 Chapter 1:Lmits and Continuity 53 25. If f(x) = x 3 x 2 + x, show that there is a number c such that f(c) = 10. Chapter 1:Lmits and Continuity 54
26. Explain why the function is discontinuous at the given number a. Sketch the graph of the function. 1. f x = 1 x 1 2 a=1 2. f x ={ 1 if x 1 x a=1 2 if x=1} 3. f x ={x 2 x 12 x 3 5 if x 3 a= 3 if x= 3} Chapter 1:Lmits and Continuity 55 27. If f and g are continuous functions with f(3) = 5 and 2 f x g x =4 determine g(3). x 3 Chapter 1:Lmits and Continuity 56
Tangent line using it Definition: The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope f x f a m= x a provided that this it exists. P(a, f(a)) Q Q Q Q 0 Chapter 1:Lmits and Continuity 57 28. Find an equation of the tangent line to the parabola y = x 2 at the point P(1, 1). Chapter 1:Lmits and Continuity 58
Tangent line Alternatively, m= h 0 f a h f a h. Chapter 1:Lmits and Continuity 59 29. Find an equation of the tangent line to the hyperbola y = 3/x at the point (3, 1). Chapter 1:Lmits and Continuity 60
30. Find the slopes of the tangent lines to the graph of the function f(x) = x at the points (1, 1), (4, 2) and (9, 3). Chapter 1:Lmits and Continuity 61 Velocities The function f that describes the motion is called the position function of the object, s = f(t). The change in position from t = a to t = a+h is f(a + h) f(a) The average velocity over this time interval is f a h f a average velocity= h We define the (instantaneous) velocity v(a) at time t = a to be the it of the average velocities: f a h f a v a = h 0 h Chapter 1:Lmits and Continuity 62
Rates of change Suppose y depends on x, y = f(x). The change in x from x 1 to x 2 is called the increment of x is x = x 2 - x 1 and the corresponding change in y is y = f(x 2 ) f(x 1 ) The difference quotient y x = f x 2 f x 1 x 2 x 1 is called the average rate of change of y with respect to x over the interval [x 1, x 2 ]. Chapter 1:Lmits and Continuity 63 Rates of change Instantaneous rate of change y x 0 x = f x 2 f x 1 x 2 x 1 x 2 x 1 Chapter 1:Lmits and Continuity 64
31. Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m. above the ground. What is the velocity of the ball after 5 seconds? How fast is the ball traveling when it hits the ground? Chapter 1:Lmits and Continuity 65 32. The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s = 4t 3 +6t+2, where t is measured in seconds. Find the velocity of the particle at times t = a, t = 1, t = 2 and t = 3. Chapter 1:Lmits and Continuity 66
33. Find an equation of the tangent line to the curve at the given point. 1. y=1 2 x x 3, 1, 2 2. y= 2 x 1, 4,3 3. y= x 1, 3, 2 x 2 4. y= 2 x x 1, 0,0 2 Chapter 1:Lmits and Continuity 67 34. Find the slope of the tangent to the parabola y = 1 + x + x 2 at the point where x = a. Find the slopes of the tangent lines at the points whose x-coordinate are (i) -1, (ii) -½, (iii) 1. Chapter 1:Lmits and Continuity 68
35. If a ball is thrown into the air with a velocity of 40 ft/s, its height (in feet) after t seconds is given by y = 40t 16t 2. Find the velocity when t = 2. Chapter 1:Lmits and Continuity 69 36. The displacement (in meters) of a particle moving in a straight line is given by s = t 2 8t + 18, where t is measured in seconds. 1. Find the average velocity over each time interval: (i) [3, 4], (ii) [3.5, 4], (iii) [4, 5] Chapter 1:Lmits and Continuity 70
2. Find the instantaneous velocity when t = 4. Chapter 1:Lmits and Continuity 71