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# Midterm I Review

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### Midterm I Review

1. 1. Review for Midterm I Math 1a October 21, 2007 Announcements Midterm I 10/24, Hall 7-9pm, Hall A and D Old exams and solutions on website problem sessions every night, extra MQC hours
2. 2. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
3. 3. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
4. 4. The concept of Limit Learning Objectives state the informal deﬁnition of a limit (two- and one-sided) observe limits on a graph guess limits by algebraic manipulation guess limits by numerical information
5. 5. Heuristic Deﬁnition of a Limit Deﬁnition We write lim f (x) = L x→a and say “the limit of f (x), as x approaches a, equals L” if we can make the values of f (x) arbitrarily close to L (as close to L as we like) by taking x to be suﬃciently close to a (on either side of a) but not equal to a.
6. 6. The error-tolerance game L a
7. 7. The error-tolerance game L a
8. 8. The error-tolerance game L a
9. 9. The error-tolerance game This tolerance is too big L a
10. 10. The error-tolerance game L a
11. 11. The error-tolerance game Still too big L a
12. 12. The error-tolerance game L a
13. 13. The error-tolerance game This looks good L a
14. 14. The error-tolerance game So does this L a
15. 15. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
16. 16. Computation of Limits Learning Objectives know basic limits like limx→a x = a and limx→a c = c use the limit laws to compute elementary limits use algebra to simplify limits use the Squeeze Theorem to show a limit
17. 17. Limit Laws Suppose that c is a constant and the limits lim f (x) and lim g (x) x→a x→a exist. Then 1. lim [f (x) + g (x)] = lim f (x) + lim g (x) x→a x→a x→a 2. lim [f (x) − g (x)] = lim f (x) − lim g (x) x→a x→a x→a 3. lim [cf (x)] = c lim f (x) x→a x→a 4. lim [f (x)g (x)] = lim f (x) · lim g (x) x→a x→a x→a
18. 18. Limit Laws, continued lim f (x) f (x) = x→a 5. lim , if lim g (x) = 0. x→a g (x) lim g (x) x→a x→a n n 6. lim [f (x)] = lim f (x) (follows from 3 repeatedly) x→a x→a 7. lim c = c x→a 8. lim x = a x→a 9. lim x n = an (follows from 6 and 8) x→a √ √ 10. lim n x = n a x→a n 11. lim f (x) = lim f (x) (If n is even, we must additionally n x→a x→a assume that lim f (x) > 0) x→a
19. 19. Direct Substitution Property Theorem (The Direct Substitution Property) If f is a polynomial or a rational function and a is in the domain of f , then lim f (x) = f (a) x→a
20. 20. Theorem (The Squeeze/Sandwich/Pinching Theorem) If f (x) ≤ g (x) ≤ h(x) when x is near a (as usual, except possibly at a), and lim f (x) = lim h(x) = L, x→a x→a then lim g (x) = L. x→a
21. 21. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
22. 22. Limits involving inﬁnity Learning Objectives know vertical asymptotes and limits at the discontinuities of ”famous” functions intuit limits at inﬁnity by eyeballing the expression show limits at inﬁnity by algebraic manipulation
23. 23. Deﬁnition Let f be a function deﬁned on some interval (a, ∞). Then lim f (x) = L x→∞ means that the values of f (x) can be made as close to L as we like, by taking x suﬃciently large. Deﬁnition The line y = L is a called a horizontal asymptote of the curve y = f (x) if either lim f (x) = L or lim f (x) = L. x→∞ x→−∞ y = L is a horizontal line!
24. 24. Theorem Let n be a positive integer. Then 1 limx→∞ =0 xn limx→−∞ x1n =0
25. 25. Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C0 D∞
26. 26. Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C0 D∞
27. 27. Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) =3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2 Upshot When ﬁnding limits of algebraic expressions at inﬁnitely, look at the highest degree terms.
28. 28. Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) =3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2 Upshot When ﬁnding limits of algebraic expressions at inﬁnitely, look at the highest degree terms.
29. 29. Inﬁnite Limits Deﬁnition The notation lim f (x) = ∞ x→a means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x suﬃciently close to a but not equal to a. Deﬁnition The notation lim f (x) = −∞ x→a means that the values of f (x) can be made arbitrarily large negative (as large as we please) by taking x suﬃciently close to a but not equal to a. Of course we have deﬁnitions for left- and right-hand inﬁnite limits.
30. 30. Vertical Asymptotes Deﬁnition The line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following is true: limx→a f (x) = ∞ limx→a f (x) = −∞ limx→a+ f (x) = ∞ limx→a+ f (x) = −∞ limx→a− f (x) = ∞ limx→a− f (x) = −∞
31. 31. Finding limits at trouble spots Example Let t2 + 2 f (t) = t 2 − 3t + 2 Find limt→a− f (t) and limt→a+ f (t) for each a at which f is not continuous.
32. 32. Finding limits at trouble spots Example Let t2 + 2 f (t) = t 2 − 3t + 2 Find limt→a− f (t) and limt→a+ f (t) for each a at which f is not continuous. Solution The denominator factors as (t − 1)(t − 2). We can record the signs of the factors on the number line.
33. 33. − + 0 (t − 1) 1
34. 34. − + 0 (t − 1) 1 − + 0 (t − 2) 2
35. 35. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2)
36. 36. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) f (t) 1 2
37. 37. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) + f (t) 1 2
38. 38. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ + f (t) 1 2
39. 39. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − + f (t) 1 2
40. 40. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − ∞ + f (t) 1 2
41. 41. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − ∞ + + f (t) 1 2
42. 42. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
43. 43. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
44. 44. Continuity Learning Objectives intuitive notion of continuity deﬁnition of continuity at a point and on an interval ways a function can fail to be continuous at a point
45. 45. Deﬁnition of Continuity Deﬁnition Let f be a function deﬁned near a. We say that f is continuous at a if lim f (x) = f (a). x→a
46. 46. Free Theorems Theorem (a) Any polynomial is continuous everywhere; that is, it is continuous on R = (−∞, ∞). (b) Any rational function is continuous wherever it is deﬁned; that is, it is continuous on its domain.
47. 47. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
48. 48. The Limit Laws give Continuity Laws Theorem If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: 1. f + g 2. f − g 3. cf 4. fg f 5. (if g (a) = 0) g
49. 49. Transcendental functions are continuous, too Theorem The following functions are continuous wherever they are deﬁned: 1. sin, cos, tan, cot sec, csc 2. x → ax , loga , ln 3. sin−1 , tan−1 , sec−1
50. 50. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
51. 51. The Intermediate Value Theorem Learning Objectives state IVT use IVT to show that a function takes a certain value use IVT to show that a certain equation has a solution reason with IVT
52. 52. A Big Time Theorem Theorem (The Intermediate Value 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) = f (b). Then there exists a number c in (a, b) such that f (c) = N.
53. 53. Illustrating the IVT f (x) x
54. 54. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] f (x) x
55. 55. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] f (x) f (b) f (a) x a b
56. 56. Illustrating the IVT 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) = f (b). f (x) f (b) N f (a) x a b
57. 57. Illustrating the IVT 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) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a c b
58. 58. Illustrating the IVT 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) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a b
59. 59. Illustrating the IVT 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) = f (b). Then there exists a number c in (a, b) such that f (c) = N. f (x) f (b) N f (a) x a c1 c2 c3 b
60. 60. Using the IVT Example Prove that the square root of two exists. Proof. Let f (x) = x 2 , a continuous function on [1, 2]. Note f (1) = 1 and f (2) = 4. Since 2 is between 1 and 4, there exists a point c in (1, 2) such that f (c) = c 2 = 2.
61. 61. True or False At one point in your life your height in inches equaled your weight in pounds.
62. 62. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
63. 63. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
64. 64. Concept Learning Objectives state the deﬁnition of the derivative Given the formula for a function, ﬁnd its derivative at a point “from scratch,” i.e., using the deﬁnition Given numerical data for a function, estimate its derivative at a point. given the formula for a function and a point on the graph of the function, ﬁnd the (slope of, equation for) the tangent line
65. 65. The deﬁnition Deﬁnition Let f be a function and a a point in the domain of f . If the limit f (a + h) − f (a) f (a) = lim h h→0 exists, the function is said to be diﬀerentiable at a and f (a) is the derivative of f at a.
66. 66. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
67. 67. The Derivative as a function Learning Objectives given a function, ﬁnd the derivative of that function from scratch and give the domain of f’ given a function, ﬁnd its second derivative given the graph of a function, sketch the graph of its derivative
68. 68. Derivatives Theorem If f is diﬀerentiable at a, then f is continuous at a.
69. 69. How can a function fail to be continuous?
70. 70. The second derivative If f is a function, so is f , and we can seek its derivative. f = (f ) It measures the rate of change of the rate of change!
71. 71. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
72. 72. Implications of the derivative Learning objectives Given the graph of the derivative of a function... determine where the function is increasing and decreasing determine where the function is concave up and concave down sketch the graph of the original function ﬁnd and interpret inﬂection points
73. 73. Fact If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is decreasing on (a, b), then f (x) ≤ 0 for all x in (a, b). Fact If f (x) > 0 for all x in (a, b), then f is increasing on (a, b). If f (x) < 0 for all x in (a, b), then f is decreasing on (a, b).
74. 74. Deﬁnition A function is called concave up on an interval if f is increasing on that interval. A function is called concave down on an interval if f is decreasing on that interval.
75. 75. Fact If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b) If f is concave down on (a, b), then f (x) ≤ 0 for all x in (a, b). Fact If f (x) > 0 for all x in (a, b), then f is concave up on (a, b). If f (x) < 0 for all x in (a, b), then f is concave down on (a, b).
76. 76. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving inﬁnity Concept Continuity Intepretations Concept Implications Examples Computation
77. 77. Computing Derivatives Learning Objectives the power rule the constant multiple rule the sum rule the diﬀerence rule derivative of x → e x is e x (by deﬁnition of e)
78. 78. Theorem (The Power Rule) Let r be a real number. Then dr x = rx r −1 dx
79. 79. Rules for Diﬀerentiation Theorem Let f and g be diﬀerentiable functions at a, and c a constant. Then (f + g ) (a) = f (a) + g (a) (cf ) (a) = cf (a)
80. 80. Rules for Diﬀerentiation Theorem Let f and g be diﬀerentiable functions at a, and c a constant. Then (f + g ) (a) = f (a) + g (a) (cf ) (a) = cf (a) It follows that we can diﬀerentiate all polynomials.
81. 81. Derivatives of exponential functions Fact dx = ex dx e