Transcript Winter wk 6 – Tues.8.Feb.05 • Calculus Ch.3 review:

Winter wk 6 – Tues.8.Feb.05
• Calculus Ch.3 review:
– Polynomial rule for derivatives
– Differentiating exponential functions
– Chain rule and product rule
• 3.5 Trigonometric functions
• 3. 6 Applications of chain rule
Energy Systems, EJZ
Differentiating polynomials and ex
Differentiating polynomials:
n
df
d
(
x
)
n
If f  x then

 n x n 1
dx
dx
Integrating polynomials:
p 1
x
If g  x p then  g dx   x p dx 
p 1
Slope of ex increases exponentially:
d/dx(ex) = ex
d/dx(ax) = ln(a) ax
Review chain rule
Differentiate exponential function y=eax
y  eax  e z where z  ax
dy
dz
 ___________ ,
 ______________
dz
dx
dy dy dz


dx dz dx
Differentiate y=ex2=ez
z=x2
dy
dz
 ___________ ,
 ______________
dz
dx
dy dy dz


dx dz dx
Review product rule and quotient rule
If y(x) = f(x) g(x) then
Practice:
dy
dg
df
 f
g
dx
dx
dx
y = x2 e-3x=f.g where f=______, g=______
df dx 2
dg de-3x
=
 ____________, =
 _____________
dx
dx
dx dx
dy

dx
If y(x) = f(x) / g(x) then
f 'g  g' f
y' 
g2
Differentiating trig functions
Sketch the slope of y=sin(x)
Does this look familiar?
d
sin( x)  ___________
dx
Differentiating cosine
Sketch the slope of y=cos(x)
Does this look familiar?
d
cos( x)  ___________
dx
Conceptest 1
Conceptest 1 solution
Conceptest 2
Conceptest 2 solution
Conceptest 3
(Hint: where is df/dx=0?)
Conceptest 3 solution
Conceptest 4
(Hint: calculate the slope)
Conceptest 4 solution
Differentiating tangent
tan(x) = sin(x)/cos(x)
1. Use identity sin2x+cos2x=1 to derive tan2x+1=_____
2. Use product or quotient rule to find d(tan(x))/dx
Practice differentiating trig functions
Drill on Ch.3.5 odd # problems through 41 (p.131)
(skip #19)
#5: y = sin(3x) = sin z where z=__________
dy d (sin z )
dz d (3 x)

___________ ,

 ______________
dz
dz
dx
dx
d (sin(3 x)) dy dy dz


 _______________________
dx
dx dz dx
3.6: Applications of the chain rule
Finding the derivative of an inverse function (133)
What is df/dx if f=x½? Trick: Write f2=x
d
d
2
Differentiate
 f ( x)   x
dx
dx
Solve for df/dx=
(Compare to result from polynomial rule.)
Using chain rule to find d/dx(lnx)
Recall that
elnx=x.
Differentiate
d
d ln x
e 
x
dx
dx
v
d ln x
d v
d (e ) dv
e   e  
where v  ln x
dx
dx
dv dx
d (e v )
dv
 ______________
 ________________
dv
dx
dx
d ln x
 ____  e   ______________________
dx
dx
d
ln x  
dx
Practice p.136 on odd problems through #15
Using chain rule to find d/dx(ax)
d
x
(ln
a
)
Recall that
ln a. Differentiate
dx
d
d
d (ln v) dv
x
x
ln a    ln v  
where v  a
dx
dx
dv dx
d (ln v)
dv
 ______________
 ________________
dv
dx
d
d
x
ln a   ___________   x ln a   ______________
dx
dx
d x
Solve for
a 

dx
ln(ax)=x
Practice p.136 odd problems thru #35 (skip arc_ probs)
Chain rule for related rates
Practice p.137 #46, 48