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3.7 applications of tangent lines

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3.7 applications of tangent lines

  1. 1. Applications of Tangent Lines
  2. 2. Applications of Tangent Lines In this section we look at two applications of the tangent lines.
  3. 3. Applications of Tangent Lines In this section we look at two applications of the tangent lines. Differentials and Linear Approximation
  4. 4. Applications of Tangent Lines In this section we look at two applications of the tangent lines. Differentials and Linear Approximation Let f(x) = x2 be as shown. y Its derivative f '(x) = dy = 2x. dx y = x2 x
  5. 5. Applications of Tangent Lines In this section we look at two applications of the tangent lines. Differentials and Linear Approximation Let f(x) = x2 be as shown. y Its derivative f '(x) = = 2x. dy dx The slope at the point (3,9) is f '(3) = 6. (3,9) y = x2 x
  6. 6. Applications of Tangent Lines In this section we look at two applications of the tangent lines. Differentials and Linear Approximation Let f(x) = x2 be as shown. y Its derivative f '(x) = = 2x. dy dx The slope at the point (3,9) is f '(3) = 6. (3,9) y = x2 y = 6x – 9 Hence the tangent line at (3, 9) is y = 6(x – 3) + 9 or y = 6x – 9. x
  7. 7. Applications of Tangent Lines In this section we look at two applications of the tangent lines. Differentials and Linear Approximation Let f(x) = x2 be as shown. y Its derivative f '(x) = = 2x. dy dx The slope at the point (3,9) is f '(3) = 6. (3,9) y = x2 y = 6x – 9 Hence the tangent line at (3, 9) is y = 6(x – 3) + 9 or y = 6x – 9. Note that in the notation we write that dx dy dx = 6 x = 3 dy dx = f '(a) x = a dy where in general x
  8. 8. In general the tangent line at x = a, i.e. at (a, f(a)), is T(x) = f'(a)(x – a) + f(a) with f'(a) = dy dx x = a tangent line T(x) = f'(a)(x – a) + f(a) (a,f(a)) y = f(x) x Applications of Tangent Lines
  9. 9. In general the tangent line at x = a, i.e. at (a, f(a)), is T(x) = f'(a)(x – a) + f(a) with f'(a) = dy dx x = a tangent line T(x) = f'(a)(x – a) + f(a) (a,f(a)) y = f(x) x Applications of Tangent Lines For any x value that is near the point x = a, say x = b, the function–values f(b) and T(b) are near each other, (a,f(a)) T(x) = y (b,f(b)) (b,T(b)) y = f(x)
  10. 10. In general the tangent line at x = a, i.e. at (a, f(a)), is T(x) = f'(a)(x – a) + f(a) with f'(a) = dy dx x = a tangent line T(x) = f'(a)(x – a) + f(a) (a,f(a)) y = f(x) x Applications of Tangent Lines For any x value that is near the point x = a, say x = b, the function–values f(b) and T(b) are near each other, (a,f(a)) T(x) = y (b,f(b)) (b,T(b)) y = f(x) f(b)
  11. 11. In general the tangent line at x = a, i.e. at (a, f(a)), is T(x) = f'(a)(x – a) + f(a) with f'(a) = dy dx x = a tangent line T(x) = f'(a)(x – a) + f(a) (a,f(a)) y = f(x) x Applications of Tangent Lines For any x value that is near the point x = a, say x = b, the function–values f(b) and T(b) are near each other, (a,f(a)) T(x) = y (b,f(b)) (b,T(b)) y = f(x) f(b) T(b)
  12. 12. In general the tangent line at x = a, i.e. at (a, f(a)), is T(x) = f'(a)(x – a) + f(a) with f'(a) = dy dx x = a tangent line T(x) = f'(a)(x – a) + f(a) (a,f(a)) y = f(x) x Applications of Tangent Lines For any x value that is near the point x = a, say x = b, the function–values f(b) and T(b) are near each other, hence we may use T(b) as an estimation for f(b), (a,f(a)) T(x) = y (b,f(b)) (b,T(b)) y = f(x) f(b) T(b)
  13. 13. In general the tangent line at x = a, i.e. at (a, f(a)), is T(x) = f'(a)(x – a) + f(a) with f'(a) = dy dx x = a tangent line T(x) = f'(a)(x – a) + f(a) (a,f(a)) y = f(x) x Applications of Tangent Lines For any x value that is near the point x = a, say x = b, the function–values f(b) and T(b) are near each other, hence we may use T(b) as an estimation for f(b), since T(x) is a linear function and most likely it’s easier to calculate T(b) than f(b). (a,f(a)) T(x) = y (b,f(b)) (b,T(b)) y = f(x) f(b) T(b)
  14. 14. Applications of Tangent Lines There are two ways to find T(b).
  15. 15. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly.
  16. 16. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly. For example, the tangent line of y = sin(x) at x = 0 is T(x) = x,
  17. 17. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly. For example, the tangent line of y = sin(x) at x = 0 is T(x) = x, hence for small x’s that are near 0, sin(x) ≈ x. (Remember that x must be in radian measurements).
  18. 18. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly. For example, the tangent line of y = sin(x) at x = 0 is T(x) = x, hence for small x’s that are near 0, sin(x) ≈ x. (Remember that x must be in radian measurements). (a,f(a)) (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a x=b 2. Find ΔT first and ΔT + f(a) = T(b).
  19. 19. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly. For example, the tangent line of y = sin(x) at x = 0 is T(x) = x, hence for small x’s that are near 0, sin(x) ≈ x. (Remember that x must be in radian measurements). (a,f(a)) (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a ΔT x=b 2. Find ΔT first and ΔT + f(a) = T(b).
  20. 20. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly. For example, the tangent line of y = sin(x) at x = 0 is T(x) = x, hence for small x’s that are near 0, sin(x) ≈ x. (Remember that x must be in radian measurements). 2. Find ΔT first and ΔT + f(a) = T(b). In practice, ΔT is denoted as dy, the y–differential, which is identified with the dy in dy/dx. (a,f(a)) (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a ΔT= dy x=b
  21. 21. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly. For example, the tangent line of y = sin(x) at x = 0 is T(x) = x, hence for small x’s that are near 0, sin(x) ≈ x. (Remember that x must be in radian measurements). 2. Find ΔT first and ΔT + f(a) = T(b). In practice, ΔT is denoted as dy, the y–differential, which is identified with the dy in dy/dx. And we set the x–differential dx = Δx, (a,f(a)) (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b
  22. 22. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly. For example, the tangent line of y = sin(x) at x = 0 is T(x) = x, hence for small x’s that are near 0, sin(x) ≈ x. (Remember that x must be in radian measurements). 2. Find ΔT first and ΔT + f(a) = T(b). In practice, ΔT is denoted as dy, the y–differential, which is identified with the dy in dy/dx. And we set the x–differential dx = Δx, and as before Δy = f(b) – f(a). (a,f(a)) (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy
  23. 23. Applications of Tangent Lines There are two ways to find T(b). 1. Find T(x) and evaluate T(b) directly. For example, the tangent line of y = sin(x) at x = 0 is T(x) = x, hence for small x’s that are near 0, sin(x) ≈ x. (Remember that x must be in radian measurements). which is identified with the dy in dy/dx. And we set the x–differential dx = Δx, and as before Δy = f(b) – f(a). These measurements are shown here. It’s important to (a,f(a)) (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy 2. Find ΔT first and ΔT + f(a) = T(b). In practice, ΔT is denoted as dy, the y–differential, “see” them because the geometry is linked to the algebra.
  24. 24. Applications of Tangent Lines (a,f(a)) (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy dy dx The derivative notation is inherited from the notation of slopes as ratios. Δy Δx
  25. 25. Applications of Tangent Lines dy dx The derivative notation is inherited from the notation of slopes as ratios. Suppose that = f '(a) Δy = L (= lim ), Δx0 (a,f(a)) (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy dy dx x = a Δx Δy Δx
  26. 26. Applications of Tangent Lines dy dx The derivative notation is inherited from the notation of slopes as ratios. Suppose that = f '(a) Δy = L (= lim ), Δx0 then for Δx close to 0, (a,f(a)) Δy Δx (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy dy dx x = a Δx ≈ L = slope at (a, f(a)) Δy Δx
  27. 27. Applications of Tangent Lines dy dx The derivative notation is inherited from the notation of slopes as ratios. Suppose that = f '(a) Δy = L (= lim ), Δx0 dy dx x = a then for Δx close to 0, (a,f(a)) Δy Δx (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy Δy ≈ L(Δx) Δx ≈ L = slope at (a, f(a)) so Δy Δx
  28. 28. Applications of Tangent Lines The derivative notation is inherited from the notation of slopes as ratios. Suppose that = f '(a) Δy = L (= lim ), Δx0 dy dx x = a Δx then for Δx close to 0, (a,f(a)) Δy Δx (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy ≈ L = slope at (a, f(a)) so Δy ≈ L(Δx) = L dx = dy dy dx Δy Δx
  29. 29. Applications of Tangent Lines dy dx The derivative notation is inherited from the notation of slopes as ratios. Suppose that = f '(a) Δy = L (= lim ), Δx0 dy dx x = a Δx then for Δx close to 0, (a,f(a)) Δy Δx (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy ≈ L = slope at (a, f(a)) so Δy ≈ L(Δx) = L dx = dy The accuracy of this approximation may be formulated mathematically. Δy Δx
  30. 30. Applications of Tangent Lines dy dx The derivative notation is inherited from the notation of slopes as ratios. Suppose that Δy Δx = f '(a) Δy = L (= lim ), Δx0 dy dx x = a Δx then for Δx close to 0, (a,f(a)) Δy Δx ≈ L = slope at (a, f(a)) so (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy Δy ≈ L(Δx) = L dx = dy So what have we done? We have clarified the meaning of dy and dx, and made it legal to treat dy/dx as a fraction both algebraically and geometrically.
  31. 31. Applications of Tangent Lines dy dx The derivative notation is inherited from the notation of slopes as ratios. Suppose that Δy Δx = f '(a) Δy = L (= lim ), Δx0 dy dx x = a Δx then for Δx close to 0, (a,f(a)) Δy Δx ≈ L = slope at (a, f(a)) so (x. T(x)) (b,f(b)) (b,T(b)) y = f(x) x=a Δx=dx ΔT= dy x=b Δy Δy ≈ L(Δx) = L dx = dy So what have we done? We have clarified the meaning of dy and dx, and made it legal to treat dy/dx as a fraction both algebraically and geometrically. To summarize, given dy dx y = f(x) and = f '(x), then Δy ≈ dy = f '(x)dx.
  32. 32. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx.
  33. 33. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½
  34. 34. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx
  35. 35. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx b. Find the specific dy in terms of dx when x = 4.
  36. 36. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx b. Find the specific dy in terms of dx when x = 4. When x = 4 we get that dy = ¼ dx.
  37. 37. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx b. Find the specific dy in terms of dx when x = 4. When x = 4 we get that dy = ¼ dx. (This says that for small changes in x, the change in the output y is approximate ¼ of the given change in x, at x = 4.)
  38. 38. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx b. Find the specific dy in terms of dx when x = 4. When x = 4 we get that dy = ¼ dx. (This says that for small changes in x, the change in the output y is approximate ¼ of the given change in x, at x = 4.) c. Given that Δx = dx = 0.01, find dy at x = 4. Use the result to approximate √4.01.
  39. 39. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx b. Find the specific dy in terms of dx when x = 4. When x = 4 we get that dy = ¼ dx. (This says that for small changes in x, the change in the output y is approximate ¼ of the given change in x, at x = 4.) c. Given that Δx = dx = 0.01, find dy at x = 4. Use the result to approximate √4.01. Given that Δx = dx = 0.01 and that dy = ¼ dx at x = 4,
  40. 40. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx b. Find the specific dy in terms of dx when x = 4. When x = 4 we get that dy = ¼ dx. (This says that for small changes in x, the change in the output y is approximate ¼ of the given change in x, at x = 4.) c. Given that Δx = dx = 0.01, find dy at x = 4. Use the result to approximate √4.01. Given that Δx = dx = 0.01 and that dy = ¼ dx at x = 4, we have dy = ¼ (0.01) = 0.0025 ≈ Δy.
  41. 41. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx b. Find the specific dy in terms of dx when x = 4. When x = 4 we get that dy = ¼ dx. (This says that for small changes in x, the change in the output y is approximate ¼ of the given change in x, at x = 4.) c. Given that Δx = dx = 0.01, find dy at x = 4. Use the result to approximate √4.01. Given that Δx = dx = 0.01 and that dy = ¼ dx at x = 4, we have dy = ¼ (0.01) = 0.0025 ≈ Δy. Hence f(4 + 0.01) =√4.01 ≈ √4 + dy
  42. 42. Applications of Tangent Lines Example A. Let y = f(x) = √x. a. Find the general formula of dy in terms of dx. dy dx = f '(x) = ½ x–½ or dy = ½ x–½ dx b. Find the specific dy in terms of dx when x = 4. When x = 4 we get that dy = ¼ dx. (This says that for small changes in x, the change in the output y is approximate ¼ of the given change in x, at x = 4.) c. Given that Δx = dx = 0.01, find dy at x = 4. Use the result to approximate √4.01. Given that Δx = dx = 0.01 and that dy = ¼ dx at x = 4, we have dy = ¼ (0.01) = 0.0025 ≈ Δy. Hence f(4 + 0.01) =√4.01 ≈ √4 + dy = 4.0025. (The calculator answer is 002498439..).
  43. 43. Applications of Tangent Lines Your turn. Do the same at x = 9, and use the result to approximate √8.995. What is the dx?
  44. 44. Applications of Tangent Lines Your turn. Do the same at x = 9, and use the result to approximate √8.995. What is the dx? Hence we have the terms the “differentials” or the “small differences” as opposed to the “differences”
  45. 45. Applications of Tangent Lines Your turn. Do the same at x = 9, and use the result to approximate √8.995. What is the dx? Hence we have the terms the “differentials” or the “small differences” as opposed to the “differences” and the symbols are dx and Δx respectively.
  46. 46. Applications of Tangent Lines Your turn. Do the same at x = 9, and use the result to approximate √8.995. What is the dx? Hence we have the terms the “differentials” or the “small differences” as opposed to the “differences” and the symbols are dx and Δx respectively. The differentials are used extensively in numerical problems and the following is one example.
  47. 47. Applications of Tangent Lines Your turn. Do the same at x = 9, and use the result to approximate √8.995. What is the dx? Hence we have the terms the “differentials” or the “small differences” as opposed to the “differences” and the symbols are dx and Δx respectively. The differentials are used extensively in numerical problems and the following is one example. Newton’s Method for Approximating Roots
  48. 48. Applications of Tangent Lines Your turn. Do the same at x = 9, and use the result to approximate √8.995. What is the dx? Hence we have the terms the “differentials” or the “small differences” as opposed to the “differences” and the symbols are dx and Δx respectively. The differentials are used extensively in numerical problems and the following is one example. Newton’s Method for Approximating Roots The Newton’s Method of approximating the location of a root depends on the geometry of the tangent line and the x–axis.
  49. 49. Applications of Tangent Lines Your turn. Do the same at x = 9, and use the result to approximate √8.995. What is the dx? Hence we have the terms the “differentials” or the “small differences” as opposed to the “differences” and the symbols are dx and Δx respectively. The differentials are used extensively in numerical problems and the following is one example. Newton’s Method for Approximating Roots The Newton’s Method of approximating the location of a root depends on the geometry of the tangent line and the x–axis. If the geometry is right, the successive x–intercepts of tangent–lines approach a root not unlike a ball falls into a crevice. We demonstrate this below.
  50. 50. Applications of Tangent Lines y = f(x) x = r Suppose that we know there is a root in the shaded region and we wish to calculate its location, and let’s say it’s at x = r.
  51. 51. Applications of Tangent Lines y = f(x) x = r x1 Suppose that we know there is a root in the shaded region and we wish to calculate its location, and let’s say it’s at x = r. We select a starting point x = x1 as shown.
  52. 52. Applications of Tangent Lines (x1,f(x1)) y = f(x) x = r x1 Suppose that we know there is a root in the shaded region and we wish to calculate its location, and let’s say it’s at x = r. We select a starting point x = x1 as shown. Draw the tangent line at x1.
  53. 53. Applications of Tangent Lines (x1,f(x1)) y = f(x) x = r x1 x2 Suppose that we know there is a root in the shaded region and we wish to calculate its location, and let’s say it’s at x = r. We select a starting point x = x1 as shown. Draw the tangent line at x1. Let x2 be the x–intercept of this tangent.
  54. 54. Applications of Tangent Lines (x1,f(x1)) y = f(x) x = r (x2,f(x2)) x1 x3 x2 Suppose that we know there is a root in the shaded region and we wish to calculate its location, and let’s say it’s at x = r. We select a starting point x = x1 as shown. Draw the tangent line at x1. Let x2 be the x–intercept of this tangent. Then we draw the tangent line at x2, and let x3 be the x–intercept of the tangent line at x2.
  55. 55. Applications of Tangent Lines (x1,f(x1)) y = f(x) x = r (x2,f(x2)) (x3,f(x3)) x1 x3 x2 Suppose that we know there is a root in the shaded region and we wish to calculate its location, and let’s say it’s at x = r. We select a starting point x = x1 as shown. Draw the tangent line at x1. Let x2 be the x–intercept of this tangent. Then we draw the tangent line at x2, and let x3 be the x–intercept of the tangent line at x2. Continuing in this manner, the sequence of x’s is funneled toward the root x = r.
  56. 56. Applications of Tangent Lines The successive intercepts may be calculated easily. We give the formula here: xn+1 = xn – f(xn) f '(xn)
  57. 57. Applications of Tangent Lines The successive intercepts may be calculated easily. We give the formula here: xn+1 = xn – f(xn) f '(xn) Example B. Let y = f(x) = x3 – 3x2 – 5, its graph is shown here. Assume that we know that it has a root between 2 < x < 5. Starting with x1 = 4, approximate this roots by the Newton’s method to x3. Compare this with a calculator answer. 2 4 y x f(x) = x3 – 3x2 – 5
  58. 58. Applications of Tangent Lines We have that f '(x) = 3x2 – 6x = 3x(x – 2) Starting with x1 = 4, 2 x1=4 x f(x) = x3 – 3x2 – 5 The geometry of the Newton’s Method for example B. Back to math–265 pg
  59. 59. Applications of Tangent Lines We have that f '(x) = 3x2 – 6x = 3x(x – 2) Starting with x1 = 4, 2 x1=4 x f(x) = x3 – 3x2 – 5 The geometry of the Newton’s Method for example B. Back to math–265 pg
  60. 60. Applications of Tangent Lines We have that f '(x) = 3x2 – 6x = 3x(x – 2) Starting with x1 = 4, x2 = x1 – f(x1) f '(x1) 2 x1=4 x f(x) = x3 – 3x2 – 5 The geometry of the Newton’s Method for example B. Back to math–265 pg x2
  61. 61. Applications of Tangent Lines We have that f '(x) = 3x2 – 6x = 3x(x – 2) Starting with x1 = 4, x2 = x1 – f(x1) f '(x1) = 4 – f(4) f '(4) 2 x1=4 x f(x) = x3 – 3x2 – 5 The geometry of the Newton’s Method for example B. Back to math–265 pg x2
  62. 62. Applications of Tangent Lines We have that f '(x) = 3x2 – 6x = 3x(x – 2) Starting with x1 = 4, x2 = x1 – f(x1) f '(x1) = 4 – f(4) f '(4) = 85/24 2 x1=4 x f(x) = x3 – 3x2 – 5 ≈ 3.542 The geometry of the Newton’s Method for example B. Back to math–265 pg x2
  63. 63. Applications of Tangent Lines We have that f '(x) = 3x2 – 6x = 3x(x – 2) Starting with x1 = 4, x2 = x1 – f(x1) f '(x1) = 4 – f(4) f '(4) = 85/24 x3 = 85/24 – f(85/24) f '(85/24) 2 x1=4 x f(x) = x3 – 3x2 – 5 ≈ 3.542 x3 x2 The geometry of the Newton’s Method for example B. Back to math–265 pg
  64. 64. Applications of Tangent Lines We have that f '(x) = 3x2 – 6x = 3x(x – 2) Starting with x1 = 4, x2 = x1 – f(x1) f '(x1) = 4 – f(4) f '(4) = 85/24 x3 = 85/24 – f(85/24) f '(85/24) ≈ 3.432 2 x1=4 x f(x) = x3 – 3x2 – 5 ≈ 3.542 x3 x2 The geometry of the Newton’s Method for example B. Back to math–265 pg
  65. 65. Applications of Tangent Lines We have that f '(x) = 3x2 – 6x = 3x(x – 2) Starting with x1 = 4, x2 = x1 – f(x1) f '(x1) = 4 – f(4) f '(4) = 85/24 x3 = 85/24 – f(85/24) f '(85/24) ≈ 3.432 2 x1=4 x f(x) = x3 – 3x2 – 5 ≈ 3.542 x3 xThe software answer is 2 ≈ 3.425988.. The geometry of the Newton’s Method for example B. Back to math–265 pg

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