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
1( )nn ndf d xIf f x then n xdx dx
Differentiating polynomials:
Integrating polynomials:
Slope of ex increases exponentially:
d/dx(ex) = ex
d/dx(ax) = ln(a) ax
1
1
pp p xIf g x then g dx x dx
p
Higher order derivatives
Second derivative = rate of change of first derivative
2
2
d f d dff slopeof slopeof fdx dx dx
dff slopedx
d dff slopeof slopedx dx
f ’<0 f ’>0
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 functionsEx: 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
dy dg dff gdx dx dg
dgdxdfdgdydx
Proof (justification)
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 functionsEx: 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
dy dy dzdx dz dx
dydzdzdxdydx
Proof (justification)Differentiating functions of functions: y(x) = f(g(x))See #17, p.154Derive 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
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
