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

Higher order derivatives Second derivative = rate of change of first derivative f ’<0 f ’>0 f ’’>0 f ’>0 f ’<0 f ’’<0

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

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

Proof (justification)

Spend 10-15 minutes doing odd # problems on p.121 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

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

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)) =

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

Candidates for y= f(g(x))

Answer

Calc Ch.3.4 Conceptest 2

Calc Ch.3.4 Conceptest 2 options

Calc Ch.3.4 Conceptest 2 answer

Calc Ch.3.4 Conceptest 3

Calc Ch.3.4 Conceptest 3 options

Calc Ch.3.4 Conceptest 3 answer

Spend 10-15 minutes doing odd # problems on p.126 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