Midterm I Review
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  • 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. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 3. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. 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. The error-tolerance game L a
  • 7. The error-tolerance game L a
  • 8. The error-tolerance game L a
  • 9. The error-tolerance game This tolerance is too big L a
  • 10. The error-tolerance game L a
  • 11. The error-tolerance game Still too big L a
  • 12. The error-tolerance game L a
  • 13. The error-tolerance game This looks good L a
  • 14. The error-tolerance game So does this L a
  • 15. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. 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. 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. 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. 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. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. 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. Theorem Let n be a positive integer. Then 1 limx→∞ =0 xn limx→−∞ x1n =0
  • 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. 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. 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. 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. 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. 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. 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. 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. − + 0 (t − 1) 1
  • 34. − + 0 (t − 1) 1 − + 0 (t − 2) 2
  • 35. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2)
  • 36. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) f (t) 1 2
  • 37. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) + f (t) 1 2
  • 38. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ + f (t) 1 2
  • 39. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − + f (t) 1 2
  • 40. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − ∞ + f (t) 1 2
  • 41. − + 0 (t − 1) 1 − + 0 (t − 2) 2 + (t 2 + 2) ±∞ − ∞ + + f (t) 1 2
  • 42. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 43. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. 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. 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. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. 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. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. 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. Illustrating the IVT f (x) x
  • 54. Illustrating the IVT Suppose that f is continuous on the closed interval [a, b] f (x) x
  • 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. 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. 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. 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. 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. 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. True or False At one point in your life your height in inches equaled your weight in pounds.
  • 62. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 63. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. 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. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. Derivatives Theorem If f is differentiable at a, then f is continuous at a.
  • 69. How can a function fail to be continuous?
  • 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. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. 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. 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. 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. Outline The Intermediate Value Limits Theorem Concept Computation Derivatives Limits involving infinity Concept Continuity Intepretations Concept Implications Examples Computation
  • 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. Theorem (The Power Rule) Let r be a real number. Then dr x = rx r −1 dx
  • 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. 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. Derivatives of exponential functions Fact dx = ex dx e