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Chapter 5 Higher Derivatives

Chapter 5 Higher Derivatives. Mindmap of C5. See Patterns. Asymptotes to a Curve. C5 Derivatives. Higher Derivatives. Partial Fractions. Direct Evaluation. Leibniz's Rule. Differential Equations of High Orders. Points of Inflexion. Either Decreasing or Increasing. Extrema.

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Chapter 5 Higher Derivatives

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  1. Chapter 5 Higher Derivatives

  2. Mindmap of C5 See Patterns Asymptotes to a Curve C5 Derivatives Higher Derivatives Partial Fractions Direct Evaluation Leibniz's Rule Differential Equations of High Orders Points of Inflexion Either Decreasing or Increasing Extrema Curve Sketching Inequatities Decreasing and Increasing Greatest & Least Values Convexity of a Function Inequalities

  3. Definition They are also denoted by y(1), y(2),…, y(n). y(0) denotes y.

  4. xm sinx 1/x 1/(ax+b) lnx eax 1/(x-4)(x+2) Find the 10th derivatives of the following functions

  5. xm sinx 1/x 1/(ax+b) lnx eax 1/(x-4)(x+2) If n<=m, (xm)(n) =m(m-1)…(m-n+1)xm-n. If n>m, (xm)(n)= 0. (-1)nn!/xn+1 (-1)nann!/(ax+b)n+1 (-1)n-1(n-1)!/xn aneax Find the nth derivatives of the following functions Can you give formal proofs of your results?

  6. 1/(x2-2x-3) xlnx sinx exsinx ¼ (1/(x-3))(n) –¼(1/(x+1))(n) ¼(-1)nn![1/(x-3)n+1 – 1/(x+1)n+1] y’=1+lnx y(n)=(-1)n(n -2)!/xn-1 for n>=2. y’=cosx, y”=-sinx, y”’=-cosx, y(4)=sinx y(n)=sin(n/2+x) Difficult. See p.166 More nth derivatives

  7. (x+ex)(10)=? (xex)(10)=? (f + g)(n)=f(n) + g(n) (fg)(n)=f(n).g(n) ? No! So how? Questions

  8. Show time • (x2ex)(5)=?

  9. (fg)(n) • Find (fg)(n) for n = 1,2,3,4,5. • Guess the formula for (fg)(n).

  10. Leibniz’s Rule Proof :

  11. Example 1 • Let y = x2sinx, find y(80).

  12. Example 2 • If y = x3eax, find dny/dxn.

  13. Example 3 • Find the nth derivative of y = 2xlnx.

  14. Example 4 • Let y be function of x, which is differentiable any times. If (1 – x2)y”– xy’– a2y = 0, show that for any positive integer n, • (1 – x2)y(n+2)– (2n+1)xy(n+1)– (n2 + a2)y(n) = 0. CW Ex.5.2, 7

  15. Can you write down an inequality? Section 4 Extrema and inequalities • Can you write down an inequality? y=ln(1+x) – x + x2/2 y=ln(1+x) – x + x2/2 – x3/3 Can you combine these two inequalities?

  16. Example 4.4 • If x > 0, prove, without using graphs, that x – x2/2 < ln(1+x) < x – x2/2 + x3/3! • Steps: • Consider h(x) = ln(1+x) – x +x2/2 • Evaluate h’(x). • Consider whether it’s monotonic. • Conclusion

  17. Theorem: If f’(x) > g’(x) for x > a and f(a) = g(a), then f(x) < g(x). Steps : 1. Let h(x) = f(x) – g(x). 2. Then h’(x) = f’(x) – g’(x). 3. h’(x) > 0(< 0) , then h(x) is strictly increasing(decreasing). 4. h(x) > h(a)= 0(< h(a) for all x > a. 5. f(x) – g(x) > 0 for all x > a 6. f(x) > g(x) for all x. > a (Q.E.D.) Inequalities by monotonicity

  18. Section 6 Greatest and Least Values • e.g.6.2 Show that x – lnx >= 1 for all x > 0. • Proof: • Let f(x) = x – lnx , then • f’(x) = 1 – 1/x > 0 for x > 1 and • < 0 for 0< x < 1, • f(1) is the least value of f. • f(x)>=f(1)=1 • i.e. x – lnx >=1 • When does the equal sign hold?

  19. Section 8 Convexity of a Function • Definition 8.3 The following graph of a real-valued function f(x) is said to be convex downward. • Find a condition for it. • 1. f ”(x)

  20. Section 8 Convexity of a Function • Definition 8.4 The following graph of a real-valued function f(x) is said to be convex upward. • Find a condition for it.

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