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Winter wk 4 – Tues.25.Jan.05

Winter wk 4 – Tues.25.Jan.05. Review: Polynomial rule for derivatives Differentiating exponential functions Higher order derivatives How to differentiate combinations of functions? Product rule (3.3) Quotient rule (3.4). Energy Systems, EJZ. Differentiating polynomials and e x.

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Winter wk 4 – Tues.25.Jan.05

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  1. Winter wk 4 – Tues.25.Jan.05 • Review: • Polynomial rule for derivatives • Differentiating exponential functions • Higher order derivatives • How to differentiate combinations of functions? • Product rule (3.3) • Quotient rule (3.4) Energy Systems, EJZ

  2. Differentiating polynomials and ex Differentiating polynomials: Integrating polynomials: Slope of ex increases exponentially: d/dx(ex) = ex d/dx(ax) = ln(a) ax

  3. Higher order derivatives Second derivative = rate of change of first derivative

  4. Ch.3.3: Products of functions If these are plots of f(x) and g(x) Then sketch the product y(x) = f(x).g(x) = f.g

  5. Differentiating products of functions Ex: We couldn’t do all derivatives with last week’s rules: y(x) = x ex. What is dy/dx? Write y(x) = f(x) g(x). Slope of y = (f * slope of g) + (g * slope of f) Try this for y(x) = x ex, where f=x, g=ex

  6. Proof (justification)

  7. Practice – Ch.3.3 Spend 10-15 minutes doing odd # problems on p.121 Pick one or two of these to set up together: 32, 38, 45

  8. Ch.3.4 Functions of functions Ex: We couldn’t do 3.1 #36 with last week’s rules: y =(x+3)½ What is dy/dx? Consider y(x) = f(g(x)) = f(z), where z=g(x). Try this for y =(x+3)½ , where z=x+3, f=z½

  9. Proof (justification) Differentiating functions of functions: y(x) = f(g(x)) See #17, p.154 Derive the chain rule using local linearizations: g(x+h) ~ g(x) + g’(x) h = f(z+k) ~ f(z) + f’(z) k = y’ = f’(g(x)) =

  10. Differentiating functions of Functions If these are plots of f(x) and g(x) Then sketch function y(x) = f(g(x)) = f(g)

  11. Candidates for y= f(g(x))

  12. Answer

  13. Calc Ch.3.4 Conceptest 2

  14. Calc Ch.3.4 Conceptest 2 options

  15. Calc Ch.3.4 Conceptest 2 answer

  16. Calc Ch.3.4 Conceptest 3

  17. Calc Ch.3.4 Conceptest 3 options

  18. Calc Ch.3.4 Conceptest 3 answer

  19. Practice – Ch.3.4 Spend 10-15 minutes doing odd # problems on p.126 Pick one or two of these to set up together: 52, 54, 62, 66, 68

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