5.1 Exponential Functions Hwk: Std(1, 6, 9, 14, 17, …), +60b, -57
Basic graphs (point plotting, graphing with transformations)
y = 2^x, 3^-x, 4^(-x^2), 2^(x-5), (1/3)^x
Key points: (exp,y) = (0,1), (1,base)
Basic models:
Ex 3, 4, 5 use f(x) = y0(1+r)^t, where the vars are initial qty, growth/decay rate(dec fraction increase in each time period), time (in nr of periods).
e (about 2.718) Basic exp functions can be written in form f(x) = Pekx , P = initial, k=growth rate
Ex 6 Solutions by sub or by grapher (for now)
Point out variations (Ex 7, also pg 354 catenary)
5.2 Applications of Exponential Functions Hwk: 1, 11, 12, 13, 23
Compound interest: A = P(1+r)t, where r = annual rate/compounding periods; t = years * comp periods
Example: X24
e: let amt = $1, annual rate=100%; time= 1 year; comp periods varies; graph (1+1/x)^x on [0,365]
trace to see amt for comp annually, quarterly, monthly, daily.
note A approaches e, 2.718…
Continuous compound int.: A = Pert.
Example: X24 but with continuous compounding.
5.3 Common and Natural Logarithmic Functions Hwk: Std(1, 6, 9, 14, 17, …), -77
log = exp: log1000 = exp of 10 needed to get 1000 etc: log (1/100,000)
Find log 10000, then use calc to get log 5000
Note there is a base involved: (and base 10 logs are "common logs"
Conversion between log and exp: log v = u exactly when 10u = v
Ex: convert log(18)=1.255, and convert 101/2=3.16552
Solve
log x = 5
10x = 1/100
Same discussion for e
Properties for both
Domain of definition
Range
log 1, ln 1
log 10, ln e
log 10k, ln ek
10 log(v), e ln(v) Let y = e ln(v), convert to logs, then solve for y
5.4 Properties of Logarithms Hwk: Std(1, 6, 9, 14, 17, …),
Product/quotient laws for logs: log(vw) = log(v) + log (w) etc.
Proof of prod law: let a=log(v), b=log(w), so 10^a=v, 10^b=w [also do with e]
Power Law for logs: log(v^k) = k log(v) Proof: same technique
Apps: write as a single log (x1-10)
expand and sub (X11-16)
Richter Scale See Example 9 R(i) = log(i/i0)
i = ground motion of the quake, i0 = grnd motion of "zero quake", <1 micron
5.4a Log Funcs to Other Bases Hwk: Std(1, 6, 9, 14, 17, …) to 69, -1, 6, +63, 65
Write the base as a subscript
Usual rules apply
Change of base thm: logbv = ln(v)/ln(b) = log(v)/log(b)
5.5 Algebraic Solutions of Exponential and Log Equations Hwk: Std(1, 6, 9, 14, 17, …) to 41, +49
Exp eqns: Cases: same bases; diff bases (1 exponent); diff bases (both w exponents)
Log eqns: Cases:same bases both sides; only one side w log expr; multiple log terms
Examples: X27, 6.2=log(i/1 micron) [How much grnd movement in a 6.2 quake?], X38
Find half-life: X44
5.6
Exp, Log and Other Models Hwk:
Std(1, 6, 9, 14, 17, …), OMIT
Exp
Log
Power
Log
5.7 Inverse Functions Hwk: Std(1, 6, 9, 14, 17, …), +23, 31, -54
Inv funcs
Informal def: reverse effect of orig func
Ex: f=x^3+1, g=(x-1)^(1/3): f(4)=65, g(65)=4; f(3)=28, g(28)=3, etc.
Formal: Let f be func. g is an inv func if g(y)=x exactly when f(x)=y
Alt def: Inv is a relation which reverses the members of every ordered pair (swap x's and y's)
(4,65) is in f, so (65,4) is in g.
Round Trip Thm: compos of func w inverse yields the arg of the orig func. g(f(4))=g(65)=4
This is the std test to determine whether 2 funcs are inverses of ea other
Apply to example
Finding inverses algebraically
Ex: like 12
Ex: f(x)=e3x-1 Important note: exp and log funcs are inverses
1-1 Funcs: [every func has an inverse relation, but the inverse may not be a function]
Def: given a y-value (value in range), only one x (elt in domain) can correspond to it
Signif: One to one funcs == invertible funcs.
HLT
Contrast w def of func, and VLT
Graphs of inverse funcs
Symmetric about line y=x Why: alt def.
To apply, reverse x and y coords of key points (and reverse the concavity)
Return to: Merced College; Don Power Updated 08/16/05 by Don Power