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4.5 continuous functions and differentiable functions

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4.5 continuous functions and differentiable functions

  1. 1. Continuous and Differentiable Functions
  2. 2. Continuous and Differentiable FunctionsIn this section we highlight some important facts aboutcontinuous functions and differentiable functions.
  3. 3. Continuous and Differentiable FunctionsIn this section we highlight some important facts aboutcontinuous functions and differentiable functions.Elementary Functions
  4. 4. Continuous and Differentiable FunctionsIn this section we highlight some important facts aboutcontinuous functions and differentiable functions.Elementary FunctionsRecall that a formula that may be constructed from the“basic formulas” by applying the +, – , *, / and thecomposition operations in finitely many steps is calledan elementary formula.
  5. 5. Continuous and Differentiable FunctionsIn this section we highlight some important facts aboutcontinuous functions and differentiable functions.Elementary FunctionsRecall that a formula that may be constructed from the“basic formulas” by applying the +, – , *, / and thecomposition operations in finitely many steps is calledan elementary formula. Most commonly used formulasare elementary formulas.
  6. 6. Continuous and Differentiable FunctionsIn this section we highlight some important facts aboutcontinuous functions and differentiable functions.Elementary FunctionsRecall that a formula that may be constructed from the“basic formulas” by applying the +, – , *, / and thecomposition operations in finitely many steps is calledan elementary formula. Most commonly used formulasare elementary formulas. These formulas are also theones whose derivatives can be computed easily
  7. 7. Continuous and Differentiable FunctionsIn this section we highlight some important facts aboutcontinuous functions and differentiable functions.Elementary FunctionsRecall that a formula that may be constructed from the“basic formulas” by applying the +, – , *, / and thecomposition operations in finitely many steps is calledan elementary formula. Most commonly used formulasare elementary formulas. These formulas are also theones whose derivatives can be computed easily.But under this definition, one may constructed anelementary function whose discontinuity is quitemessy. http://mathoverflow.net/questions/17901/exis tence-of-antiderivatives-of-nasty-but- For a heated discussion: elementary-functions
  8. 8. Continuous and Differentiable FunctionsIn this section we highlight some important facts aboutcontinuous functions and differentiable functions.Elementary FunctionsRecall that a formula that may be constructed from the“basic formulas” by applying the +, – , *, / and thecomposition operations in finitely many steps is calledan elementary formula. Most commonly used formulasare elementary formulas. These formulas are also theones whose derivatives can be computed easily.But under this definition, one may constructed anelementary function whose discontinuity is quitemessy. However there are theorems about thecontinuous functions and the differentiable functionswhich we may apply to elementary functions.
  9. 9. Continuous and Differentiable FunctionsContinuous Functions
  10. 10. Continuous and Differentiable FunctionsContinuous Functions
  11. 11. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.
  12. 12. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.I. Intermediate Value Theorem
  13. 13. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.I. Intermediate Value TheoremLet f(x) be a continuous function defined over theclosed interval [a, b] such that f(a) < f(b),
  14. 14. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.I. Intermediate Value TheoremLet f(x) be a continuous function defined over theclosed interval [a, b] such that f(a) < f(b), f(b) f(a) a b
  15. 15. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.I. Intermediate Value TheoremLet f(x) be a continuous function defined over theclosed interval [a, b] such that f(a) < f(b),let m be any number where f(a) < m < f(b), f(b) m f(a) a b
  16. 16. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.I. Intermediate Value TheoremLet f(x) be a continuous function defined over theclosed interval [a, b] such that f(a) < f(b),let m be any number where f(a) < m < f(b),then there exists at leastone c, i.e. one or more, f(b)where a < c < b, m f(a) a c b
  17. 17. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.I. Intermediate Value TheoremLet f(x) be a continuous function defined over theclosed interval [a, b] such that f(a) < f(b),let m be any number where f(a) < m < f(b),then there exists at leastone c, i.e. one or more, f(b)where a < c < b, mand that f(c) = m. f(a) a c b
  18. 18. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.I. Intermediate Value TheoremLet f(x) be a continuous function defined over theclosed interval [a, b] such that f(a) < f(b),let m be any number where f(a) < m < f(b),then there exists at least other c’sone c, i.e. one or more, f(b)where a < c < b, mand that f(c) = m. f(a) a c b
  19. 19. Continuous and Differentiable FunctionsContinuous FunctionsThere are two important facts about continuousfunctions.I. Intermediate Value TheoremLet f(x) be a continuous function defined over theclosed interval [a, b] such that f(a) < f(b),let m be any number where f(a) < m < f(b),then there exists at least other c’sone c, i.e. one or more, f(b)where a < c < b, mand that f(c) = m.We omit the proof here. f(a)Here is a link to its proof.http://en.wikipedia.org/wiki/Intermediate a c b_value_theorem
  20. 20. Continuous and Differentiable FunctionsRemarksI. Similarly, if f(a) > f(b) then there is some c witha < c < b such that f(c) = m for f(a) > m > f(b).
  21. 21. Continuous and Differentiable FunctionsRemarksI. Similarly, if f(a) > f(b) then there is some c witha < c < b such that f(c) = m for f(a) > m > f(b).
  22. 22. Continuous and Differentiable FunctionsRemarksI. Similarly, if f(a) > f(b) then there is some c witha < c < b such that f(c) = m for f(a) > m > f(b).II. If the condition is f(a) ≤ m ≤ f(b) then theconclusion is a ≤ c ≤ b.
  23. 23. Continuous and Differentiable FunctionsRemarksI. Similarly, if f(a) > f(b) then there is some c witha < c < b such that f(c) = m for f(a) > m > f(b).II. If the condition is f(a) ≤ m ≤ f(b) then theconclusion is a ≤ c ≤ b.One important application for this theorem is theexistence of roots.(Bozano’s Theorem) Let y = f(x) be a continuousfunction over the closed interval [a, b], then thereexists at least one c where a < c < b with f(c) = 0if the signs of f(a) and f(b) are different.
  24. 24. Continuous and Differentiable FunctionsRemarksI. Similarly, if f(a) > f(b) then there is some c witha < c < b such that f(c) = m for f(a) > m > f(b).II. If the condition is f(a) ≤ m ≤ f(b) then theconclusion is a ≤ c ≤ b.One important application for this theorem is theexistence of roots.(Bozano’s Theorem) Let y = f(x) be a continuousfunction over the closed interval [a, b], then thereexists at least one c where a < c < b with f(c) = 0if the signs of f(a) and f(b) are different.With this theorem, we conclude that f(x) = x3 – 3x2 – 5has a root between 2 < x < 5, as in 3.6 Example B,because f(2) and f(5) have opposite signs.
  25. 25. Continuous and Differentiable FunctionsThe other important fact about continuous functionsis the existence of extrema over a closed interval.
  26. 26. Continuous and Differentiable FunctionsThe other important fact about continuous functionsis the existence of extrema over a closed interval.II. Extrema Theorem for Continuous Functions
  27. 27. Continuous and Differentiable FunctionsThe other important fact about continuous functionsis the existence of extrema over a closed interval.II. Extrema Theorem for Continuous FunctionsLet y = f(x) be a continuous function defined over aclosed interval V = [a, b], then both the absolute max.and the absolute min. exist in V.
  28. 28. Continuous and Differentiable FunctionsThe other important fact about continuous functionsis the existence of extrema over a closed interval.II. Extrema Theorem for Continuous FunctionsLet y = f(x) be a continuous function defined over aclosed interval V = [a, b], then both the absolute max.and the absolute min. exist in V.In particular if M is the maximum and m is the minimum,then m ≤ f(x) ≤ M for every x in the interval [a, b].
  29. 29. Continuous and Differentiable FunctionsThe other important fact about continuous functionsis the existence of extrema over a closed interval.II. Extrema Theorem for Continuous FunctionsLet y = f(x) be a continuous function defined over aclosed interval V = [a, b], then both the absolute max.and the absolute min. exist in V.In particular if M is the maximum and m is the minimum,then m ≤ f(x) ≤ M for every x in the interval [a, b].So a continuous function defined over a closedinterval V is always bounded.
  30. 30. Continuous and Differentiable FunctionsThe other important fact about continuous functionsis the existence of extrema over a closed interval.II. Extrema Theorem for Continuous FunctionsLet y = f(x) be a continuous function defined over aclosed interval V = [a, b], then both the absolute max.and the absolute min. exist in V.In particular if M is the maximum and m is the minimum,then m ≤ f(x) ≤ M for every x in the interval [a, b].So a continuous function defined over a closedinterval V is always bounded.Corollary. Let y = f(x) be an elementary functiondefined over a closed interval V = [a, b], then f(x) iscontinuous over [a, b], hence both the absolute max.and the absolute min. exist in V.
  31. 31. Continuous and Differentiable FunctionsDifferentiable Functions
  32. 32. Continuous and Differentiable FunctionsDifferentiable FunctionsWe state the following important theorems aboutdifferentiable functions without proofs.
  33. 33. Continuous and Differentiable FunctionsDifferentiable FunctionsWe state the following important theorems aboutdifferentiable functions without proofs.Differentiability is a lot stronger condition thancontinuity at a point on a graph.
  34. 34. Continuous and Differentiable FunctionsDifferentiable FunctionsWe state the following important theorems aboutdifferentiable functions without proofs.Differentiability is a lot stronger condition thancontinuity at a point on a graph.Theorem. If f(x) is differentiable at x = a then it iscontinuous at x = a.
  35. 35. Continuous and Differentiable FunctionsDifferentiable FunctionsWe state the following important theorems aboutdifferentiable functions without proofs.Differentiability is a lot stronger condition thancontinuity at a point on a graph.Theorem. If f(x) is differentiable at x = a then it iscontinuous at x = a.Differentiability means the rate of change may bemeasured.This observation leads to Rolle’s Theorem whichgives the existence of a point c where f(c) = 0.
  36. 36. Continuous and Differentiable FunctionsDifferentiable FunctionsWe state the following important theorems aboutdifferentiable functions without proofs.Differentiability is a lot stronger condition thancontinuity at a point on a graph.Theorem. If f(x) is differentiable at x = a then it iscontinuous at x = a.Differentiability means the rate of change may bemeasured. The rate of change at an extremum x = cof f(x) must be 0 because it can’t be + (increasing)or – (decreasing) hence f(c) = 0.
  37. 37. Continuous and Differentiable FunctionsDifferentiable FunctionsWe state the following important theorems aboutdifferentiable functions without proofs.Differentiability is a lot stronger condition thancontinuity at a point on a graph.Theorem. If f(x) is differentiable at x = a then it iscontinuous at x = a.Differentiability means the rate of change may bemeasured. The rate of change at an extremum x = cof f(x) must be 0 because it can’t be + (increasing)or – (decreasing) hence f(c) = 0.This observation leads to Rolle’s Theorem whichgives the existence of a point c where f(c) = 0.
  38. 38. Continuous and Differentiable FunctionsRolle’s TheoremLet f(x) be a differentiable function defined over theclosed interval [a, b] with a < b and that f(a) = f(b),
  39. 39. Continuous and Differentiable FunctionsRolle’s TheoremLet f(x) be a differentiable function defined over theclosed interval [a, b] with a < b and that f(a) = f(b), f(a)=f(b) x a c b x
  40. 40. Continuous and Differentiable FunctionsRolle’s TheoremLet f(x) be a differentiable function defined over theclosed interval [a, b] with a < b and that f(a) = f(b),then there is at least one c where a < c < bsuch that f(c) = 0. other c’s f(c) = 0 f(a)=f(b) x a c b x
  41. 41. Continuous and Differentiable FunctionsRolle’s TheoremLet f(x) be a differentiable function defined over theclosed interval [a, b] with a < b and that f(a) = f(b),then there is at least one c where a < c < bsuch that f(c) = 0. other c’sProof. Consider the following f(c) = 0two cases. f(a)=f(b) x a c b x
  42. 42. Continuous and Differentiable FunctionsRolle’s TheoremLet f(x) be a differentiable function defined over theclosed interval [a, b] with a < b and that f(a) = f(b),then there is at least one c where a < c < bsuch that f(c) = 0. other c’sProof. Consider the following f(c) = 0two cases.1. The function f(x) is a f(a)=f(b)constant function, i.e. x a c b xf(x) = f(a) = k, then f(x) = 0
  43. 43. Continuous and Differentiable FunctionsRolle’s TheoremLet f(x) be a differentiable function defined over theclosed interval [a, b] with a < b and that f(a) = f(b),then there is at least one c where a < c < bsuch that f(c) = 0. other c’sProof. Consider the following f(c) = 0two cases.1. The function f(x) is a f(a)=f(b)constant function, i.e. x a c b xf(x) = f(a) = k, then f(x) = 0.Any number c where a < c < b would satisfy f(c) = 0.
  44. 44. Continuous and Differentiable FunctionsRolle’s TheoremLet f(x) be a differentiable function defined over theclosed interval [a, b] with a < b and that f(a) = f(b),then there is at least one c where a < c < bsuch that f(c) = 0. other c’sProof. Consider the following f(c) = 0two cases.1. The function f(x) is a f(a)=f(b)constant function, i.e. x a c b xf(x) = f(a) = k, then f(x) = 0Any number c where a < c < b would satisfy f(c) = 0.2. If the function f(x) is not a constant function,then there exists an extremum c between a and b, withf(c) ≠ f(a) and f(c) ≠ (b) and we must have f(c) = 0.
  45. 45. Continuous and Differentiable FunctionsThe average rate of change of y = f(x) from x = a to bis f(b) – f(a) = Δy = slope of the chord as shown. Δx b–a Avg. rate of change y = f(x) = slope of the chord (b, f(b)) Δy = Δx Δy Δx (a, f(a)) a b
  46. 46. Continuous and Differentiable FunctionsThe average rate of change of y = f(x) from x = a to bis f(b) – f(a) = Δy = slope of the chord as shown. Δx b–aThe graph y = f(x) is the rotation of the graph of somefunction y = g(x) defined over some interval [A, B] Avg. rate of change y = g(x) y = f(x) = slope of the chord (b, f(b)) g(A) = g(B) Δy = Δx(A, g(A)) (B, g(B)) Δy Δx (a, f(a)) Rotate A C B a b
  47. 47. Continuous and Differentiable FunctionsThe average rate of change of y = f(x) from x = a to bis f(b) – f(a) = Δy = slope of the chord as shown. Δx b–aThe graph y = f(x) is the rotation of the graph of somefunction y = g(x) defined over some interval [A, B]with g(A) = g(B) with the rotation taking(A,g(A)) to (a, f(a)) and (B,g(B)) to (b, f(b)) as shown. Avg. rate of change y = g(x) y = f(x) = slope of the chord (b, f(b)) g(A) = g(B) Δy = Δx(A, g(A)) (B, g(B)) Δy Δx (a, f(a)) Rotate A C B a b
  48. 48. Continuous and Differentiable FunctionsIf in addition y = g(x) is differentiable over an intervalthat contains [A, B], then Rolle’s Theorem implies theexistence of at least one C where A < C < B such thatthe tangent line at (C, g(C)) is a horizontal. y = f(x)Rolle’s Theorem y = g(x) (b, f(b)) g(A) = g(B)(A, g(A)) (B, g(B)) (a, f(a)) There exists a C where g(C) = 0 Rotate A C B a b
  49. 49. Continuous and Differentiable FunctionsIf in addition y = g(x) is differentiable over an intervalthat contains [A, B], then Rolle’s Theorem implies theexistence of at least one C where A < C < B such thatthe tangent line at (C, g(C)) is a horizontal. Under therotation this horizontal line rotates into a tangent linethat is parallel to the chord from (a, f(a)) to (b, f(b)) y = f(x)Rolle’s Theorem y = g(x) (b, f(b)) g(A) = g(B)(A, g(A)) (B, g(B)) (a, f(a)) There exists a C There exists a C where where g(C) = 0 Rotate f (C) = Avg. rate of change A C B a b
  50. 50. Continuous and Differentiable FunctionsIf in addition y = g(x) is differentiable over an intervalthat contains [A, B], then Rolle’s Theorem implies theexistence of at least one C where A < C < B such thatthe tangent line at (C, g(C)) is a horizontal. Under therotation this horizontal line rotates into a tangent linethat is parallel to the chord from (a, f(a)) to (b, f(b))which gives us the Mean Value Theorem. y = f(x)Rolle’s Theorem Mean Value Theorem y = g(x) (b, f(b)) g(A) = g(B) (A, g(A)) (B, g(B)) (a, f(a)) There exists a C There exists a C where where g(C) = 0 Rotate f (C) = Avg. rate of change A C B a b
  51. 51. Continuous and Differentiable FunctionsMean Value TheoremLet y = f(x) be a differentiable function over an intervalthat contains [a, b], then there is at least one c wherea < c < b such that y = f(x)f (c) = f(b) – f(a) . slope = Avg. rate of change (b, f(b)) b–a = f(b) – f(a) b–a . (a, f(a)) a c b There exists a c where f(b) – f(a) f (c) = Avg. rate of change = b–a .
  52. 52. Continuous and Differentiable FunctionsMean Value TheoremLet y = f(x) be a differentiable function over an intervalthat contains [a, b], then there is at least one c wherea < c < b such that y = f(x)f (c) = f(b) – f(a) . slope = Avg. rate of change (b, f(b)) b–a = f(b) – f(a) b–a .Remarks1. The precise conditionfor the theorem is that (a, f(a))f(x) is continuous in [a, b],and differentiable in (a, b).The condition above isstronger but is sufficient a c bfor our purposes. There exists a c where f(b) – f(a) f (c) = Avg. rate of change = b–a .
  53. 53. Continuous and Differentiable Functions2. In statistics, the word “Mean” denotes the “average”.However, the statement of, the “Mean Value” in theMean Value Theorem refers to the average value ofthe rate–of–change, y = f(x) slope = Avg. rate of change (b, f(b)) f(b) – f(a) = b–a . (a, f(a)) a c b There exists a c where f(b) – f(a) f (c) = Avg. rate of change = b–a .
  54. 54. Continuous and Differentiable Functions2. In statistics, the word “Mean” denotes the “average”.However, the statement of, the “Mean Value” in theMean Value Theorem refers to the average value ofthe rate–of–change, y = f(x)not the “mean value” slope = Avg. rate of change (b, f(b)) f(b) – f(a)or the average value of = b–a .the function f(x) over theinterval [a, b]. (a, f(a)) a c b There exists a c where f(b) – f(a) f (c) = Avg. rate of change = b–a .
  55. 55. Continuous and Differentiable Functions2. In statistics, the word “Mean” denotes the “average”.However, the statement of, the “Mean Value” in theMean Value Theorem refers to the average value ofthe rate–of–change, y = f(x)not the “mean value” slope = Avg. rate of change (b, f(b)) f(b) – f(a)or the average value of = b–a .the function f(x) over theinterval [a, b].The average value of (a, f(a))the function f(x) over theinterval [a, b] is definedby integrals which are cour next topics. a b There exists a c where f(b) – f(a) f (c) = Avg. rate of change = b–a .
  56. 56. Continuous and Differentiable FunctionsExample A.We left Los Angeles at 12:00 PM arrived at SanFrancisco at 3:30 PM that same afternoon covering adistance of 350 miles.
  57. 57. Continuous and Differentiable FunctionsExample A.We left Los Angeles at 12:00 PM arrived at SanFrancisco at 3:30 PM that same afternoon covering adistance of 350 miles. Therefore our average rate is350 / 3½ miles/hr = 100 mph.
  58. 58. Continuous and Differentiable FunctionsExample A.We left Los Angeles at 12:00 PM arrived at SanFrancisco at 3:30 PM that same afternoon covering adistance of 350 miles. Therefore our average rate is350 / 3½ miles/hr = 100 mph. By the Mean ValueTheorem, our speedometer must display the speed atexactly 100mph at some point in time during our trip.
  59. 59. Continuous and Differentiable FunctionsExample A.We left Los Angeles at 12:00 PM arrived at SanFrancisco at 3:30 PM that same afternoon covering adistance of 350 miles. Therefore our average rate is350 / 3½ miles/hr = 100 mph. By the Mean ValueTheorem, our speedometer must display the speed atexactly 100mph at some point in time during our trip.Example B. Find the x and y intercepts of the linethat is tangent to the graph y = √2x + 1 and is parallelto the chord connecting the points with x = 0 andx = 4. Use a graphing software to draw the graph toconfirm your answers.
  60. 60. Continuous and Differentiable FunctionsExample A.We left Los Angeles at 12:00 PM arrived at SanFrancisco at 3:30 PM that same afternoon covering adistance of 350 miles. Therefore our average rate is350 / 3½ miles/hr = 100 mph. By the Mean ValueTheorem, our speedometer must display the speed atexactly 100mph at some point in time during our trip.Example B. Find the x and y intercepts of the linethat is tangent to the graph y = √2x + 1 and is parallelto the chord connecting the points with x = 0 andx = 4. Use a graphing software to draw the graph toconfirm your answers.The end points in question are (0, 1) and (4, 3) sothe chord connecting them has slope ½.
  61. 61. Continuous and Differentiable FunctionsThe function y = √2x + 1 is differentiable and satisfiesthe condition of the Mean Value Theorem, thereforethere is some x = c where 0 < c < 4 and y(c) = ½ .
  62. 62. Continuous and Differentiable FunctionsThe function y = √2x + 1 is differentiable and satisfiesthe condition of the Mean Value Theorem, thereforethere is some x = c where 0 < c < 4 and y(c) = ½ .We may solve for c directly.
  63. 63. Continuous and Differentiable FunctionsThe function y = √2x + 1 is differentiable and satisfiesthe condition of the Mean Value Theorem, thereforethere is some x = c where 0 < c < 4 and y(c) = ½ .We may solve for c directly.y = √2x + 1 → y(x) = 1/√2x + 1
  64. 64. Continuous and Differentiable FunctionsThe function y = √2x + 1 is differentiable and satisfiesthe condition of the Mean Value Theorem, thereforethere is some x = c where 0 < c < 4 and y(c) = ½ .We may solve for c directly.y = √2x + 1 → y(x) = 1/√2x + 1Set 1 = 1 √2x + 1 2
  65. 65. Continuous and Differentiable FunctionsThe function y = √2x + 1 is differentiable and satisfiesthe condition of the Mean Value Theorem, thereforethere is some x = c where 0 < c < 4 and y(c) = ½ .We may solve for c directly.y = √2x + 1 → y(x) = 1/√2x + 1Set 1 = 1 √2x + 1 2 √2x + 1 = 2
  66. 66. Continuous and Differentiable FunctionsThe function y = √2x + 1 is differentiable and satisfiesthe condition of the Mean Value Theorem, thereforethere is some x = c where 0 < c < 4 and y(c) = ½ .We may solve for c directly.y = √2x + 1 → y(x) = 1/√2x + 1Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2
  67. 67. Continuous and Differentiable FunctionsThe function y = √2x + 1 is differentiable and satisfiesthe condition of the Mean Value Theorem, thereforethere is some x = c where 0 < c < 4 and y(c) = ½ . We may solve for c directly. y = √2x + 1 → y(x) = 1/√2x + 1Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2Therefore the tangent line passes through (3/2, 2) andit has slope ½.
  68. 68. Continuous and Differentiable FunctionsThe function y = √2x + 1 is differentiable and satisfiesthe condition of the Mean Value Theorem, thereforethere is some x = c where 0 < c < 4 and y(c) = ½ . We may solve for c directly. y = √2x + 1 → y(x) = 1/√2x + 1Set 1 = 1 √2x + 1 2 √2x + 1 = 2 2x + 1 = 4 x = 3/2Therefore the tangent line passes through (3/2, 2) andit has slope ½. So the equation of the tangent line inquestion is y = ½ (x – 3/2) + 2 or y = x/2 + 5/4.Its x intercept is at –5/2 and the y intercept is at 5/4.

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