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1 Math 6 Notes 1.1 A PREVIEW OF CALCULUS There are main problems in calculus: 1. Finding a tangent line to a curve though a point on the curve.. Finding the area under a curve on some interval. SEE and DISCUSS the pictures on pages 4-43 in your tet. Key picture: As Q approaches P, the secant line approaches the tangent line. As Δ gets smaller, Q gets closer to P and the slope of the tangent line can be found. We would then use the point-slope formula to actually obtain the equation of the tangent line. m sec ( Δ ) ( Δ ) ( Δ ) f c + f( c) c + c f c + f( c) Δ To find the area under a curve, we use representative rectangles. Eample: page 46 Eploration Find the area under y on the interval ( ) 0,1.
2 Math 6 Notes Section 1. INTRODUCTION TO LIMITS The it as approaches c (from either side) is written: ( ). A c f L it is defined as the value approached from the left or right, regardless of the value of the function at that point. There are 3 ways in which No Limit Eists : 1. f( ) approaches different values from each side.. f( ) approaches ± on either or both sides. 3. f( ) oscillates between two fied values. Formal Definition of a Limit Let f be a function defined on an open interval containing c (ecept possibly at c) and let L be a real number. ( ) means that for each c f L ε>0 there eists a δ>0 such that if 0 < c < δ, then f( ) L < ε. Picture: Pictures: problems - page ( + 4) 5 a) Find the it. Methods of Finding Limits: 1. Look at the graph.. Use an -y chart if the graph is hard to create or read. 3. Use the formal method ( ε and δ ). 4. Use other numerical techniques. b) Find δ>0 such that f( ) L < 001. whenever 0 < c < δ. Page 55 Eamine and discuss graphs for eercises 13, 14, 17, 0 and 4. Notes:
3 ( + ) a) Find the it. b) Find δ>0 such that f( ) L < 001. whenever 0 < c < δ. 44. ( ) 4 In problems 37-48, find the it L. Then use the ε δ definition to prove that the it is L. 4. ( 1)
4 Notes Section 1.3 FINDING LIMITS ANALYTICALLY Some its can be found by direct substitution. This happens when the it as approaches c actually equals f( c). For constant functions,. c b b For the identity function,. c c For power functions,. c n c n In fact, for all polynomial, trigonometric, and p ( ) rational functions f( ) q o q( ), ( ), you can plug in c in order to find the it. Properties: If b and c are real numbers and n is a positive integer, and if ( ) and, c f L ( ) c g K then 1. [ bf ( ) ] bl c scalar multiplication. [ ( ) ( ) ] c f ± g L ± K sum or difference 3. [ ( ) ( ) ] c f g LK product f( ) L 4. c g ( ) K [ ( ) ] c f n L quotient g ( ) 0 n 5. power For composite functions, 1. If ( ) and, then c g L ( ) ( ) L f f L. rationalize the numerator. Two Special Trig Limits: 1.. sin cos 0 0 The Squeeze Theorem: If you have a hard time finding a it, you can sometimes find the it by squeezing it between two functions with known its. If h ( ) f( ) g ( ) for all in an open interval containing c, ecept possibly at c itself, and if ( ), then eists c h L c g ( ) ( ) c f and equals L too. Problems - pages Find ( 1) 14. Find Find ( cos 3 ) π ( ( )) ( ). c f g f L. If two functions, f( ) g( ) for all ( ) c f ( ) c g c, then 50. Find 4 If you try to evaluate a it and get 0, called an 0 indeterminate form, you can either try to 1. factor and cancel, or
5 54. Find tan sin 0 sin3 60. Find ( Δ ) + Δ 0 Δ 88. Use the Squeeze Theorem to find ( ). c f c a b a f( ) b + a ( cos )
6 Notes Section 1.4 CONTINUITY AND ONE-SIDED LIMITS If a function is continuous at c, there is no interruption in the graph of f at c. A function f is continuous at c if the following three conditions are met: 1. f( c) is defined.. ( ) eists c f 3. ( ) ( ). c f f c A function is continuous on an open interval ( ab, ) if it is continuous at each point in that interval. A function that is continuous on ( +, ) is said to be continuous everywhere. There are types of discontinuities: 1. removable Properties of Continuity: If b is a real number and f and g are continuous functions at c, then the following functions are also continuous: 1. bf scalar multiplication. f ± g sum or difference 3. fg product 4. f g, g 0 quotient If g is continuous at c and f is continuous at gc ( ), then f g( ) f g( ) is continuous at c. ( ) Intermediate Value Theorem: If f is continuous on the closed interval [ab, ] and k is any number between f( a) and f( b), then there eists at least one number c in [ ab, ] such that f( c ) k. (This is an eistence theorem.) problems - pages Find the it, if it eists: + 4. non-removable The Eistence of a Limit Let f be a function and let c and L be real numbers. The it of f( ) as approaches c is L if and only if ( ) c f L and + ( ). c f L 1. Find the it, if it eists: This definition opens the door to one-sided its and allows us to define continuity on a closed interval. A function f is continuous on a closed interval [ ab], if it is continuous on the open interval ( ab, ) and + ( ) ( ) a f f a and ( ) ( ). b f f b We say the function is continuous from the right at a and continuous from the left at b. picture:
7 , 18. Find the it, if it eists: 11, 1 > Find a so that g ( ) is continuous on the entire real line. 4sin, < 0 g ( ) a, > Find the value(s) at which f is not continuous. Which discontinuities are removable? 1 f( ) Discuss the continuity of h ( ) fg ( ( )) 1 f( ) g() Find the value(s) at which f is not continuous. Which discontinuities are removable? 1 f( ) Find the value(s) at which f is not continuous. Which discontinuities are removable? 3 f( ) Find the interval(s) on which the function is continuous. f( ) + 3
8 Notes Section 1.5 INFINITE LIMITS Graphing Rational Functions 1. There is an -intercept everywhere the numerator equals zero.. There is a vertical asymptote whenever the denominator is zero. 3. If the degree of the numerator is less than the degree of the denominator, then the -ais is a horizontal asymptote. 4. If the degree of the numerator equals the degree of the denominator, then the ratio of the leading coefficients is the horizontal asymptote. 5. If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote. 6. Use an -y chart to fill in etra points. Graph: y Graph: y Graph: y 4 Graph: y 1 ( ) Graph: y + 8 4
9 Let f be a function that is defined at every real number in some open interval containing c (ecept possibly at c itself). E. h ( ) ( ) c f means that for each M > 0, there eists a δ>0such that f( ) > M whenever 0 < c < δ. ( ) c f means that for each N < 0, there eists a δ>0such that f( ) < N whenever 0 < c < δ. Properties: If ( ) and, then c f ( ) c g L e. Find ( ) ( ) c f ± g 1. [ ] ( ) ( ) c f g g ( ) c f( ) 0. [ ] 3. if L > 0 if L < 0 4. Find 0 1 problems - page 88 In eercises 33-48, find the vertical asymptotes (if any) of the function, 34. g ( ) A patrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of ½ revolutions per second. The rate at which the light beam moves along the wall is r 50πsec ϑ ft sec. 36. f( ) + 16 a. Find the rate r when ϑ is π 6. e. f( ) sec( π )
10 b. Find the rate r when ϑ is π 3. Page 90 True or False? Determine whether the statement is true of false. If it is false, eplain why or give an eample that shows it is false. 67. If p() is a polynomial, then the graph of the p ( ) function given by f( ) has a vertical 1 asymptote at 1. c. Find the rate r when as ϑ π. 68. The graph of a rational function has a least one vertical asymptote. 58. Boyle s Law. For a quantity of gas at a constant temperature, the pressure P is inversely proportional to the volume V. Find + the it of P as V The graphs of polynomial functions have no vertical asymptotes. 70. If f has a vertical asymptote at 0, then f is undefined at 0.
11 Notes Section 1R problems - page 91 CHAPTER ONE REVIEW 4. cos( π + Δ) + 1 Δ 0 Δ 16. Find Find s s s 1 E. + 0 sec 36. (modified) Determine the intervals on which 3 f( ) is continuous. f( ) 1 E
12 4. Determine the intervals on which the function is + 1 continuous. f( ) page 81 - Intermediate Value Theorem 84. Verify that the Intermediate Value Theorem applies to the interval [ 03], with f( c ) 0. f( ) 6 + 8
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