Section 6.4: Derivatives of Logarithm Functions

Review

Logarithms

Since f(x)=(e^x) is a one-to-one function, it has an inverse function. That function is called the natural logarithm function and is denoted
	(f^(-1))(x)=ln(x)

Everything we know about ln(x) comes from the fact that
	y=ln(x) ⟷ x=(e^y)

In general f(x)=(a^x) is a one-to-one function as long as base a is a positive number other than 1. Since it is one-to-one, it has an inverse and that inverse is denoted
	(f^(-1))(x)=(Subscript[log, a])(x)
This means y=(Subscript[log, a])(x) and x=(a^y) are equivalent.

Questions

Let g(x)=2(e^x)-5x.

g[x_]:=2E^x-5x
g'[x]
soln=Log[5.0/2]
g''[x]
Show[
Plot[g[x],{x,-1,3}],
ListPlot[{{soln,g[soln]}}]
]

Find all critical numbers for g(x). Classify each as either a local maximum, a local minimum, or neither.

Determine where y=g(x) is concave up and where it is concave down.

Question

Let h(x)=x (e^(x/3)).

h[x_]:=x E^(x/3)
h'[x]
Plot[h[x],{x,-2,2}]

Find the absolute maximum value and the absolute minimum value of h(x) for -2≤x≤2.

Question (a little challenging...)

What is an equivalent way to write g(x)=((((x^2)+1))^(cos(x)))? What is its domain?

g[x_]:=(x^2+1)^Cos[x]
h[x_]:=E^(Cos[x]Log[x^2+1])
Plot[{(x^2+1),g[x],h[x],1/(x^2+1)},{x,-5,5}]

Derivative of ln(x)

Let y=ln(x), then x=(e^y). Use implicit differentiation to find ((d y)/(d x)).
	(d/(d x))(x)=(d/(d x))((e^y))
	⟶ 1=(e^y)((d y)/(d x))
	⟶ ((d y)/(d x))=(1/(e^y))=(1/x)

The derivative of ln(x) is
	(d/(d x))(ln(x))=(1/x)

Questions

Find any maximum or minimum points to f(x)=x ln(x).

f[x_]:=x Log[x]
f'[x]
Show[
Plot[f[x],{x,.01,3}],
ListPlot[{{E^(-1),f[E^(-1)]}}]
]

What is an equation for the tangent line to g(x)=(x^(sin(x))) when x=π/2?

f[x_]:=x^Sin[x]
x0=Pi/2
f'[x0](x-x0)+f[x0]
Show[
Plot[{f[x],f'[x0](x-x0)+f[x0]},{x,0,Pi}],
ListPlot[{{x0,f[x0]}}]
]

Let h(x)=ln(-x). What is its domain? What is its derivative?

f[x_]:=Log[-x]
Plot[f[x],{x,-3,-.1}]
f'[x]
f[x_]:=Log[Abs[x]]
Plot[f[x],{x,-3,3}]
f'[x]

What is ∫(x^(-1))d x ?

Integrate[1/x,x]

Find the area of the region bounded by the x-axis and y=(1/x) for 1≤x≤2.

Plot[{1/x,0},{x,1,2}]
Integrate[1/x,{x,1,2}]
N[%]

Find the area of the region bounded by the x-axis and y=(1/x) for -3≤x≤-1.

Plot[{1/x,0},{x,-3,-1}]
Integrate[1/x,{x,-3,-1}]
N[%]

Find the volume of the solid of revolution obtained by rotating the region bounded by the x-axis and y=1+(1/x) for 1/2≤x≤2.

f[x_]:=1+1/x
Plot[{f[x],-f[x]},{x,1/2,2}]
Integrate[Pi f[x]^2,{x,1/2,2}]
N[%]

Homework (in progress)

Weekly assignment 1 is coming due soon. Keep up with your homework assignment page for dates!

Imath problems on differentiating logarithmic functions.

Imath problems on logarithmic functions.

Imath problems on exponential functions.