Callahan:
    4.3: 13, 15
    4.4: 6, 11, 15, 16
    4.5: 1(a), (b), (c), (f)

    Log properties:

    Change of base:

    A power equation is an equation with a variable base:

    This also can be seen from the method for solving power equations:

    Log properties:

    A exponential equation is an equation with a variable exponent:

    Proof of the derivative of a base exponential equation:

    We can also find the antiderivative of a base exponential equation by integrating our equation for the derivative:

    The base determines the ratio that defines one unit on the log scale.  For an interval :

4.3: 13)

    a)  Newton's law of cooling.  Verify that is a solution to the initial value problem:

    We need to check to see if:

    So:

    To verify that the initial condition is satisfied we plug it in and check:

    So the initial condition is satisfied.

    What is the relationship between this formula and the one found in problem 4.2 11?

    They are the same, by use of the log rules for change of base:

    b)  Verify that is a solution to the initial value problem:    (the text has an error, below is the correct question)

    First check the initial value:

    Now see if the derivative of the function matches:

    So it is a solution!

    What is the relationship between this formula and the one found in problem 4.2 13?

    They are the same, by use of the log rules for change of base:

15)

4.4 6)  Suppose a beam of X-rays whose intensity is A rads falls perpendicularly on a heavy concrete wall.  After the rays have penetrated feet of the wall, the radiation intensity has fallen to:

    What is the radiation intensity 3 inches inside the wall?  18 inches?

    How far into the wall must the rays travel before their intensity is cut in half, to ?   ?

    To fall to , the intensity must once again be cut in half.

    But this is exactly 2 time the distance it took to fall to .  So we can see that every the intensity of the radiation falls to one half of what it was.

11)  Atmospheric pressure is a function of altitude.  Assume that at any given altitude the rate of change of pressure with altitude is proportional to the pressure there.  If the barometer reads 30 psi at see level and 24 psi at 6000 feet above see level, how high are you when the barometer reads 20 psi?

    First lets find a formula for , so we need to use separation of variables, so we can get each variable isolated on one side of the equation:

    Now we need to find the antiderivative of each side:

    To find our antiderivative we just think of these functions as a derivative of a function :

    Make our guess:   (p221)

    Now we simply set the derivative we found equal to and solve for :

    And since our derivative functions were equal, our antiderivatives will be within a constant of being equal:

    Now we just need to solve for :

    But is just another constant, say :

    Now using the initial conditions:

    So our specific solution for this initial condition is:

    To figure out how high we are you when the barometer reads 20 psi, we need to solve for :

15)  Find a solution to the differential equation:

    Now we find the antiderivative of our function:

    Make our guess:

    Now we simply set the derivative we found equal to and solve for :

    And substituting we find our antiderivative :

    Our antiderivative should be within a constant of our solution:

    Now using the initial condition we can determine the value of :

    So our specific solution for this initial condition is:

16)

4.5: 1(a), (b), (c), (f))

    c)  

    To find a formula for the solution, we find the antiderivative of the function:

    To find our antiderivative we just think of the function as a derivative of another function :

    Make our guess:

    Now we simply set the derivative we found equal to and solve for :

    And substituting we find our antiderivative :

    And so our antiderivative should be within a constant of :

    Now if we had an initial condition we could determine the value of .