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Lecture 6 Notes (MATH M-119; Brief Survey of Calculus I, Staff)

# Lecture 6 Notes (MATH M-119; Brief Survey of Calculus I, Staff)

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Unformatted text preview: 1.5 Exponential Functions and Compounding I What is the y-intercept‘? What happens as x —) oo , x —> —-oo? t r An exponential function has a constant ratio, while linear functions has a constant difference. 1 <7 1 2 3 4 5 6 r: 1- ,v 1 4 7 1o 13 16 19 13 AV Y Y' t 1 2 3 4 5 6 I \$3,, 3/ 2 6 18 54 162 486 [3“] " 1“ t ‘1 An exponential function has an equation: )2 = Paa‘ . What is an equation for the exponential function listed above? Exponential models are very useful for many models, including how money grows. Let’s take \$1000 and let it grow by 6% per year under different forms of compounding for 40 years. Simple Interest: no compounding / A =(Pr)t+P=mt+b \ Annual Compounding You get interest once per year and calculate the interest off the balance from the prior period. g! / [ lvaM-{ﬁ 94 WW tale-ii) ,_.. O. 0;.) ) Monthly Compounding You get interest once per month (but only 1i’12th of it) and calculate the interest off the balance from the prior period. {7:09 0‘43 K, x 0 ' ﬂ, ”minor?“ W W 15,. 0001.150?“ F 121* DUO (I‘D A=P[1+E] {1:20 1.6 The Natural Log Ex. 10" 210.00. This can be solved with any base of logarithm, each of which makes the am; or look different, but are mathematically equivalent. " Mm x:lasﬂDDli) y 3 luaHJOOT-B / '3 gives us now all 4 log rules: 1) logurB) = Iog(A) + log(B) 2) log(A/B) = log (A) — log (B) k 3) log(Aa) = Blog(A) 4) loga(3)=ﬂ=1°33 =1°gx 1 1a Ioga 103x“ [we Limo {maﬁa ”533 Mm Io)= {mace x: me- (Zyiszﬁ‘lﬁ) Solve the following using natural logs: / _‘.5-§”'Jj_ f: f 20-733“ , go , 3 ﬂ r'.‘ f” 1.15;- f7“ “/7 K K 7 f' "”' ‘2“ ‘ "f g " ‘r.’ ' (If 1000=8(5_)" M (£3;/> {; K J in {Jim} " 1?: {if-'3" {h (I A? vb Nora We, Vs . 3mm; ﬁqﬁé Q 0 Qﬁfwcwaiﬁmtw ﬂmwf Harpmpwb, Mb 8 7‘ l + Q I1 ‘/——-— r: (:9 [d G m9“?— {magmﬁmﬁ { 0G,}? 1‘4. Ci} MFG-12M 9&‘1 ﬁslﬁ‘i’lh’ut’gil} (a i f 5, “e , ﬂog) @Fﬂ Ff £46739 ZL/f, 6mm 0143-1?- 952 names-v’Wi-ﬁ 5M"! " 6/0 / L/r WAS : 3.2 ...
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