Other Bases

I.  Quiz

II.  Homework

III.  Exponentials With Other Bases

    Definition:  Let a > 0 then

              

ax = exlna

    Exercises:  Find the derivative of the following functions

        A)  f(x) = 2^x      B)  f(x) = 3^sinx      C)  x^x        D)  x^{2x + 1}          E)  x⁴  

IV.  Logs With Other Bases

    Definition:  

         

loga x = lnx/lna

    Exercises:  Find the derivative of the following functions

        A)  log₄ x        B)  log (3x + 4)        C)  xlog(2x) 

V.  Integration

    Exercises:  Find the integral of the following functions

        A)  f(x) = 2^x        B)  f(x) = 3^{(2x + 1)}   

    Application:  Compound Interest

        Recall that the interest formula is given by:

            A = P(1 + r/n)ⁿ 

        where n is the number of total compounds before we take the money out, r is the interest rate,
        P is the Principal and A is the amount the account is worth at the end.  
        If we consider continuous compounding, we take the limit as n approaches infinity we arrive at

            A = Pe^rt  

    Application:  Students are given an exam and retake the exam later.  
        The average score on the exam is

            S = 80 - 14ln(t+ 1) 

        where t is the number of months after the exam that the student retook the exam.  
        At what rate is the average student forgetting the information after 3 months?