MATH 234, Calculus for BusinessLecture 4, Textbook Section 2.6Logarithms, Growth and Decay;Mathematics of FinanceJanuary 30th, 2017Announcements:(1) Quiz 2 is tomorrow in section. The coverage is textbook Sections 2.1, 2.2, 2.3, and 2.4. In particular,Sections 2.5 (logarithms) and 2.6 (applications) will not be on the quiz. (2) HW2 is due Thursday 2/2,11:59pm; late HWs are not accepted.Review of Sections 2.4, 2.5Remark. The value of y = loga(x) is the number that solves ay= x.Theorem 1. Fix a, b. For variables x and u, the rules are listed as counter-parts(1) a1=(2) a0=(3) (ax)u=(4) ax∗ au=(5)axau=(6) axbx=(7) aloga(x)=(8) ax=(1) loga(a) =(2) loga(1) =(3) loga(xu) =(4) loga(x ∗ u) =(5) logaxu=(6) logab(x) =(7) loga(ax) =(8) loga(x) =1Example 2. Find as many mistakes per line as possible, and explain what went wrong.ln(x)2−ln(y)2ln(xy)=ln(x2) − 2 ln(y)ln(x) ln(y)=ln(x2) − 2ln(x)=ln(x2− 2)ln(x)= logx(x2− 2)Line 1:Line 2:Line 3:Line 4:The correct version of the computation isln(x)2−ln(y)2ln(xy)=Section 2.6, ApplicationsGrowth and DecayDefinition. Let y0be the initial quantity at time t = 0. A quantity grows or decays exponentiallyat rate k if the quantity at time t isy =If k > 0, then quantity , and k is the .If k < 0, then quantity , and k is the .2Remark. The value of e is the most commonly used base. Other bases also work:y = or y =are also exponential growth/decay functions.Example 3. Initially, there are 2 grams of yeast in a sugar solution. The amount of yeast grows exponentiallyto 7 grams after 12 hours. Determine the growth function, in terms of t hours.Solution. Step 1: Determine values of y0and k.The initial quantity is . Note that time gives quantity , soStep 2: Determine growth function using approximate values.Suppose the question wants k to 3 decimals, thenand so the growth function isAlternate Step 2: Determine growth function using exact values.The growth function isRemark. The exact solution is useful for verifying the calculations: at time t = 0,and at time t = 12 hours,3Mathematics of FinanceRemark. Principal is P dollars, interest is r percent, m is times compounded/year. In terms of growth rate:simple interest < compound interest < continuous compound interest< <Definition. Nominal rate is . The effective rate isExample 4. Alice puts $10 in an account with 8% interest, compounded quarterly. What are the nominaland effective rates?Solution. Step 1: Establish baseline, nominal rate.Nominal rate is . Simple interest grows account by :i.e., simple interest gives in 1 year.Step 2: Determine growth function and compute effective rate.Principal is , interest is , compounding speed is .A(t) =After t = 1 year, the account has A(1) = .The account grew by in 1 year. The effective rate isTheorem 5 (Effective Rate for Compound Interest). Fix the principal P , the interest r, and the compoundingspeed m. The effective rate isrE=Remark. Effective rate P . Recalling A(t) =rE=amount of growth in 1 yearinitial amount=4Example 6 (Revisiting the old example, and using the new formula). Alice puts $10 in an account with 8%interest, compounded quarterly. Using the effective rate formula:Theorem 7 (Effective Rate for Continuous Compounding). Fix the principal P , the interest r. The effectiverate isrE=Remark. Effective rate P . Recalling A(t) =rE=amount of growth in 1 yearinitial amount=Example 8. Alice puts $10 in an account with 8% interest, compounded continuously. So, r = andthe effective rate isExample 9. Alice puts $30,000 in an account earning 4.8% interest, and the interest is compounded quarterly.How long it will take for the account to have $45,000?Solution. Since P = , r = , m = , thereforeA(t) =To have $45,000, solve for t:45,000 = A(t) =Interest is added at . Alice must wait .5Example 10. Bob wants to have $20,000 in his savings account after 5 years. The bank pays an interest rateof 5.5% and compounds daily. How much money must Bob deposit today to achieve his savings goal?Solution. Interest rate is r = and m = . At t years, the account hasA(t) =Bob wants , so the principal P must satisfyBob must deposit today to achieve his savings
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