Download MATH 1113 Review Sheet for Test #1 (Chapter 2, Section 3

MATH 1113 Review Sheet for Test #2 (Chapter 4, Sections 4.1-4.7)
Section 4.1 Composition
 Defining a composition of functions in words
 Intuitive idea of "chaining" functions together
 Finding the composition of functions using formula descriptions
 Determining the domain of the composition
 Evaluating the composition of functions at a point
 Decomposing a composition into component functions
Section 4.2 Inverse Functions
 Defining Inverse of a function
 Inverse Function of a function; connection between domains and ranges of these functions
 Defining the terms one to one and one to one function
o Intuitively, one to one means no partner sharing
o Determining when a graph that represents a function is one to one; the horizontal line test
 Relationship between a one to one function and its inverse function in terms of composition
 Relationship between the graphs of a one to one function and its inverse function
 Reading the domain and range of a function from the graph
 Finding the inverse function of a one to one function using formula descriptions
 Finding the range of a one to one function by finding the domain of its inverse function
Section 4.3 Exponential Functions
 Laws of exponents; add this law ax-y = ax/ay for all positive real numbers a, and for all real numbers x and y.
 Definition of an exponential function; restrictions on the bases we consider; all have vertical intercept (0,1)
 Graphs of exponential functions: two cases (0 < a < 1 (exponential decay) ; a >1 (exponential growth))
 Transformations (scaling, reflecting, shifting) of exponential functions
 Definition of the irrational number e in terms of a limit, e is considered the natural base
 Characterization of exponential functions:
o If E(x)=ax is an exponential function, then E(d)/E(c) = ad-c for all real numbers c and d.
 Exponential functions are one to one
 Simple exponential equations – strategy: write each side as an exponential expression with the same base
Section 4.4 Logarithmic Functions
 Definition of logarithm; logarithmic form and exponential form; restrictions on the bases we consider
o Intuitively logarithm asks a question
o logb(a) asks "What power of b is a?"
 Definition of logarithmic functions; all have horizontal intercept (1,0)
 Inverse Function relationship between an exponential function and the corresponding logarithmic function
 Transformations (scaling, reflecting, shifting) of logarithmic functions
 Common logarithm (base 10; sometimes base suppressed)
 Natural logarithm (base e, usually written as ln(x))
 Simple logarithmic equations: sometimes simply rewrite using exponential form will help us solve these
Section 4.5 Properties of Logarithms
 Based on Exponential and Logarithmic Functions as Inverses (4 properties)
 Based on Rules of Exponents (3 properties) AND Change of Base Relationship
 Applying these properties (write as a single logarithm, expand to logarithms of "simple" expressions)
 Using change of base to convert to base 10 or base e supported by the technology
Section 4.6 Logarithmic and Exponential Equations
 Using Properties of Logarithms and Rules of Exponents and the facts that exponential and logarithmic
functions are one to one to find exact solutions to equations
 Using a graphing calculator to approximate a solution to an exponential or logarithmic equation
Section 4.7 Compound Interest
 Simple Interest, Compound Interest, Continuously Compound Interest
 Future Value A, Present Value P, Number of compoundings in one year n, Time of investment in years t
 Nominal Annual Interest Rate expressed as a decimal r, so for example 8.347% corresponds to r = 0.08347
 Terms for compounding frequencies: annually, semiannually, quarterly, monthly, weekly, daily
 Solving for various parameters given values for the others: Solving for A, r, t, P; Word problems
 Calculating and defining Effective Rate – comparing investments
 Doubling Time (how long to double?) and generalize -- how long will it take to grow to a given size?
Suggested Problem List
4.1: in text 9, 19, 31, 33, 43
others 7, 13, 15, 17, 21, 23, 25, 35, 39, 45, 47, 49, 51, 57, 59
4.2: in text 9, 13, 17, 23, 29, 41, 55
others 11, 15, 19, 25, 31, 33, 35, 37, 43, 45, 49, 53, 57, 63
4.3: in text 11, 21, 37, 45, 53
others 23, 27, 29, 31, 33, 35, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 91, 93
4.4: in text 9, 21, 33, 47, 53, 63, 75, 85, 91, 103
others 11, 13, 15, 17, 23, 25, 27, 35, 37, 39, 41, 43, 45, 51, 55, 61, 77, 81, 83, 89, 93, 95, 97, 101, 105, 107, 109
4.5: in text 9, 13, 17, 45, 51, 65
others 7, 11, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 37, 41, 47, 49, 53, 55, 57, 59, 61, 63, 67, 69, 71
4.6: in text 5, 9, 17, 25
others 1, 3, 7, 11, 13, 15, 19, 21, 23, 27, 29, 31, 33, 35, 39, 41, 45, 51, 53, 55, 57
4.7: in text 3, 11, 13, 23, 25, 31
others 5, 7, 17, 19, 27, 29, 33, 35, 37, 39, 41, 43, 45, 47, 49
Properties of Exponents: For all positive real numbers a and b and for all real
numbers s and t:
(E1) asat = as+t (E2) (as)t = ast (E3) asbs = (ab)s (E4) a-s = 1/(as) = (1/a) s
(E5) 1s = 1 (E6) a0 = 1 (E7) as/at = as-t
Properties of Logarithms: For all positive real numbers a and b such that a ≠ 1 and
b ≠ 1 and for all positive real numbers M and N and for all real numbers r:
(L1) logb(br) = r (L2) blogb(M) = M (L3) logb(b) = 1 (L4) logb(1) = 0
(L5) logb(MN) = logb(M) + logb(N)
(L6) logb(M/N) = logb(M)  logb(N)
(L7) logb(Mr) = r logb(M)
(L8) logb(M) = loga(M)/ loga(b)
Banking Problems
(B1) A = P + Prt
(B2) A = P(1+ r/n)nt
(B3) A = Pert
Other Facts:
(F1) For all real numbers x and y and all positive real numbers a satisfying a ≠ 1,
if x = y, then ax = ay
(F2) For all real numbers x and y and all positive real numbers a satisfying a ≠ 1,
if ax = ay then x = y.
(F3) For all positive real numbers x and y and all positive real numbers b satisfying b ≠ 1,
if logb(x) = logb(y) then x= y
(F4) For all positive real numbers x and y and all positive real numbers b satisfying b ≠ 1,
if x= y then logb(x) = logb(y)