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Announcements:
· Last day of class! · Review problems: · Ch. 7 Review (pg 573): 5,7,47,48 · Ch. 8 Review (pg 687): 1,3,9,13,18,22,49; · Ch. 9 Review (pg 765): 1,3,5,7,8,9,10,11,12,13,14,25,29,31,34,38; Schedule review session for Wed. Review for test. |
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Announcements:
· Quiz Thursday · HW 9-5: 3,11,12,13,15,19,21,23,27,43,45,47 Mistake on lab: How many 4-of-kind hands? Review counting: § Ps and Cs imply no duplications § multiple operations imply multiplication § Quintessential examples: coin,dice,cards,people,passwords,menu Review probability: § sample space: total possibilities: n(S) § event space: possibilities we want: n(E) § equally likely assumption: n(E)/n(S) Examples: § first 6 letters of your name on license plate? § rolling a 7 w/ 2 dice? § your being ranked #1 in class of 15? § you and friend choose same sandwich (2 of 5 meats, 1 of 5 breads, 2 of 4 condiments) § how many different IP numbers? § 5-card all face card hand? C_16,5 / C_52,5 § flush? |
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Lab 7 |
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· Quiz Thursday · HW-20 Ch. 9.4: 1,3,5,7,9,13,15,17,21,22,23,24,25,27,31,35,45; · Questions on Sequences/Series HW Counting Techniques: § examples of large sets: # of telephone numbers § gambling possibilities § lock combinations § enumerating often not practical § multiplication principle: #/N = N_1 * N_2 * . . . N_n Examples: § how many 6-place (alphanumeric) license plates? § how many 5-dice combo? § how many different class rankings with 15 people? § how many different 5-digit zip codes? § how many different IP numbers? Factorials: § represented by n! for some number n § n! = (n-1)! .... 1! = 1, 0! = 1 § Careful working with: 7!/ 6! = ? 3!+2!=? § Grow quickly: calculator can have problems Permutations: § "a particular arrangement of n objects w/o repetition" § Ordering matters! § Eg: a particular class ranking, § # of permutations of n objects: P_n,n = n! Advanced Permutations: § diff. SGA of officers? § diff. starting rotations (for spec. positions)? § # of permutations of n objects taken r at a time § P_n,r = n!/ (n-r)! Combinations: § ordering doesn't matter § # of combos of students for hall monitors § C_n,r equal to P_n,r except that ordering doesn't matter § how many ways to order r terms? § so P_n,r = r! * C_n,r § C_n,r = (n r) = P_n,r/r! = n! / r! / (n-r)! § Examples: set of officers, set of rotations § Consider Explore/Discuss 3 on pg 734 § Consider Example 9 on pg 735 More examples: § How many 5-card all face card hands? C_16,5 § Different sandwiches?: 5 breads; 3 meats, 4 condiments? 5*3*4 § Different 6-digit passwords? |
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Announcements:
· Quiz Thursday · HW-19 Ch. 7.1: 1,3,5,7,9,11; · 7.2: 3,5,9,11,13,15,33,39; · 7.5: 1,3,23,25; Return tests; discuss black numbers Start Ch. 7 Additional Topics in Trig Law of sines Law of cosines Polar Coordinates and Graphs § need to be able to verify identities |
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· Test Take test |
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· Test Thursday · HW-18 Review problems for test: · 3-Review: 1,2,3,5,7,9,10,11,15,33; · 4-Review: 9-37,39,43,46,51,52,54-58; · Review for test and go over any homework questions |
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· Test Thursday · No class Monday · Tuesday will go over any of these review problems · HW-17: 5-Review: 3,5,9,14,15,16,21,23,37,40,42,48,50,51,55; · 6-2: 1,3,9,13,15,31; · 6-Review: 1,6,18; · Hand back Lab 4 and go over Ch. 6-2: Trig Identities (pg 465) Ch. 6-3: Double Angle Identities (pg 472) Ch. 6-3: Half Angle Identities (pg 475) Ch. 6-3: Product Sum Identities (pg 480) Ch. 6-3: Sum Product Identities (pg 482) § need to be able to verify identities § need to be able to use to simplify |
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· No class tomorrow · HW-16: 5-5: 7,9,11,13,15,17,19,21,39; · 5-6: 3,15; · 5-7: 1,3,5,13,15,17,21,23,47; · 5-9: 1,13,15, 23,25,35,37; Hand back Quiz 6 Lab 4 |
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· Quiz & Survey today · No class Tuesday (but we do have class Monday) · HW-15: 5-2: 3,5,7,9,11,13,15,21,22,23,33,35; · 5-3:1,3,5,7,9,11,13,17,19,21; · 5-4:5,9,15,25,55,57; Hand back Lab 3 Questions Quiz Trig Functions § trig identities (pg 349) § complementary/supplementary angles § radians in terms of arclength: \theta = s/r § reference triangle within unit circle 5-5: Solving Right Triangles 5-6: Graphs of Trig Functions § periodic § periodic 5-7: Harmonic Analysis 5-8: Inverse Trig Fnts |
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· Quiz Thursday · HW-14: 4-6: 1,3,5,9,13, 33,35; · 4-7: 1,3,5, 13,19,21; Review of Ch. 4: § LOG vs LN § Understanding log_b a = c 4-7 Exp/Log Equations § Solving: 2^(4x+1) = 7 § How long to make a certain amount of money? § log(x+1) + log x = 2 § Change of base: log_4 4.1 |
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· Lab 3 |
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· HW-13: 4-3: 71,73 · 4-4:1,3,5,9,21,33,35 · 4-5:1,5,13,17,19,21,23,27,31,35,39 Quiz, Continue w/ Ch. 4 |
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· Quiz Thursday · HW-12: 4-3: 1,19,21,25 Questions on HW Review examples of: § operations on functions § composition of functions § 1-to-1 functions § inverse functions 4.3: Exponential Functions § f(x) = b^x b>0 b!=1 § graph of exponential function: b<1 and b>1 4.4: Base e exponentials § f(x) = b^x b>0 b!=1 |
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· Quiz Thursday · HW-11: 4-1: 11,13,15,17,19,21,23,27,33,45,47 · 4-2: 1-4, 9-13,17,19,23,35,39,41,43 Ch. 4.1: Operations on Functions § functions give a real number that depends on the input x § Therefore, we can operate on the values of the functions themselves § Calculators can graph: y3=y1+y2 § let f(x) = 2+x; g(x) = 3x-1; find f(x)+/-g(x) f(x)*//g(x) Composition of functions § functions can operate on output of another function § f(g(x)) means to compute g(x) and substitute for x in f(x) § can picture domain and ranges of functions as diagram One-to-One functions § one-to-one functions: each x produces a *unique* f(x) § one-to-one functions: passes horizontal line test § one-to-one functions: everywhere increasing or decreasing functions Inverse Functions § functions give a new number f(x) which depends on an input number x § inverse functions simply reverse this process, when possible § Inverse functions only exist when f(x) is one to one § Notation: f-1(x) is inverse of f(x) § Reverse ordered pairs § Swap domain and range § f-1(f(x)) = x Finding Inverses § verify f is one-to-one. If not, inverse doesn't exist § solve y = f(x) for x. § Interchange x and y. § Write as f-1(x) = .... § Check f-1(f(x))=x |
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· Quiz Today · HW-10: 3-4: 5,7,9,11,13,17,19,21,25,63; Chapter 3 Review (pg. 239): 1,3,5,7,30,36,38; HW Questions Quiz Review Rational Functions |
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· Quiz Thursday Review synthetic division Properties of an nth-degree polynomial function: § continuous and smooth for all reals § at most n x-intercepts (zeros) § at most n-1 extrema § for n-even, goes to same infinity for x->-infty and x->infty § for n-odd, goes to different infinity for x->-infty and x->infty 3-3: Approximating Real Zeros § location theorem: if f(a) is opposite sign of f(b), then zero in (a,b) § the bisection method 3-4: Rational Functions § Defn: can be expressed as f(x)=n(x)/d(x) where n,d polynomials § If n(x)=0 AND d(x), not in lowest terms § If d(x)=0, function has a hole/break § If n(x)=0, zero of f(x) § Questions: domain and zeros of f(x) § Vertical asymptote where d(x)->0 § Horizontal asymptote when f(x)->b as x->+/-infty § Graphing rational functions |
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· Quiz Thursday · HW-9: 3-1: 1-4, 9,11,13,15,25,33,35,45,49; 3-2: 1,3,5,11,83 3-3: 35,53 Hand back Tests: go over, grades, black numbers Ch. 3-1: Polynomial Functions § "zero of the function": f(x)=0; x-interecept § synthetic division: divide Polynomial by x-r § remainder theorem: P(x=r) = R (where R is remainder dividing by x-r) Properties of an nth-degree polynomial function: § continuous and smooth for all reals § at most n x-intercepts (zeros) § at most n-1 extrema § for n-even, goes to same infinity for x->-infty and x->infty § for n-odd, goes to different infinity for x->-infty and x->infty 3-3: Approximating Real Zeros § location theorem: if f(a) is opposite sign of f(b), then zero in (a,b) § the bisection method |
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· Test I |
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· Test Thursday Hand back Lab 2 HW Questions Review for test: Algebra Review: § Sets § Real Numbers: order of ops, division by zero, etc § Polynomials: simplifying, operating on (+/-,*//) § Factoring polynomials § Rational Expressions: simplifying,LCD,compound fractions § Integer Exponents § Scientific Notation § Rational Exponents § Radicals § Linear Equations § Inequalities and Intervals Ch. 1: § Cartesian coordinates § Functions: domain,range,vertical line test § Functions Graphs: inc/dec,max/min § Transformations of functions § Even and Odd Functions Ch. 2: § Linear Functions and Intercepts § Slope-intercept form (y=mx+b) and Point-slope form y-y_1=m(x-x_1) § Parallel (m_1=m_2) and perpendicular (m_1*m_2=-1) lines § Solving Linear Equations and Inequalities § Quadratic Functions: parabola, domain § Completing the square and standard form a(x-h)^2+k § Quadratic Equations and Inequalities |
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· Test Thursday · HW-8: Ch 2 Rev (pg 168) 1, 3, 5, 7, 11, 17, 21, 23, 39, 44 Hand back Lab 1 Lab 2 |
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· No class Thursday · Omit 2-4 · HW-7: 2-1: 23,35,55,57,83,85 2-2: 11,27,59,73 2-3: 1,7,19,25,29,33,43,55 2-5: 5,11,23,29,35,49,57 Questions about homework 2-2: Linear Equations § Solving by intersection: 3x +4 = x-3 § Variable in the denominator: 3/x - 1 = 4 - 2/x § Solving linear inequalities: x+1 <= 0 § Inequalities w/ Absolute Value: graphing 2-3: Quadratic Equations § f(x) = ax^2 + bx + c § Domain: all reals § Completing the square: add square of 1/2 coeff. of x § Transform to: a(x-h)^2 + k § Plotting this form § Find parabola which has maximum at x=2 y=2 and zero at x=0 2-5: Quadratic Equations § Solve by factoring § Solve by completing the square § Solve by quadratic formula § Solve by graphing |
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· No class Thursday · Tutor: Amy Littlefield Hours: M 3-5pm (332 Pratt) Hours: W 5-7pm (221 Pratt) Hours: TH 4-6pm (221 Pratt) Hand back Quizzes. Lab 1 |
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· Quiz Today · HW-6: 1-Rev: 5, 6, 7, 10,11,12,13,14,34,38 2-1: 1,3,17,21,25,27,31,33,45,47,49,67,79 2-2: 5,7,13,15,23,39,63 Review HW questions Hand back Quizzes and discuss grading scheme 2-1: Linear Functions § f(x) = mx + b § finding y intercepts: f(0) § finding x intercepts: f(x)=0 § linear equation in two variables: Ax + By=C § slope of a line (positive,negative,zero,not defined) § parallell (m_1 = m_2) and perpendicular (m_1 * m_2 = -1) lines 2-2: Linear Equations § Solving by intersection: 3x +4 = x-3 § Variable in the denominator: 3/x - 1 = 4 - 2/x § Solving linear inequalities: x+1 <= 0 § Inequalities w/ Absolute Value: graphing |
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· Quiz Thursday over Ch. 1 Review HW questions |
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· No class Monday · Quiz today · HW-5: 1-3: 1,3,9,15,19,29,33,35,47,61,65 1-4: 1,7,17,23,29,49 1-5: 1,3,11,15,23,27,73 Review HW questions Quiz Finding domains: f(x)=3x/(sqrt(x)-1) Difference quotients: f(x+h)-f(x) Local extremums: min & max Greatist Integer Function: [[x]] Transformations of functions: § vertical and horizontal shifts § reflections, expansions, and contractions § summary on page 6 § even and odd functions |
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· HW-4: A-8: 1,5,7,9,13,19,21,23,33,35,39,41,57 1-1: 5,9,17,21,27,31,55 1-2: 21,25,29,39 Review HW questions A-8: Linear Equations § standard form: ax+b=0 § intervals, union, intersection § linear inequalities Ch. 1: Functions & Graphs § Cartesian coordinates § Distance between points § Graphing § Functions: vertical line test § Functions: domain range § Functions: notation f(x) |
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· HW-3: A-3: 3,21,73 A-4: 17, 37 A-5: 15,33,39,61 A-6: 17,27,29,35,39 A-7: 1,3,11,13,23,25,33,37,53 Review HW questions Lab-1 |
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· (Practice) Quiz today · HW-2: A-4: 1,7,21,29,41 A-5: 7,9,13,17,19,29,31 A-6: 3,5,7,15,23,31 Review HW questions (Practice) quiz Integer Exponents · properties of integer exponents · scientific notation Rational Exponents · roots of real numbers · fractional exponents Radicals · nth root radical · properties of radicals · simplifying radicals · sums and differences · products · rationalizing operations |
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· (Practice) Quiz Thursday · HW-1: A-1: 1,3,5,27,29,42,53 A-2: 1,3,5,7,9,15,27,29,31,43,51,57 A-3: 1,5,19,35,45 FOIL Method Special Products to remember Order of Operations Factoring Polynomials: · factoring a number/polynomial -- find numbers/expressions which mutiply to yield the given number/polynomial · prime factorization · common factors · factoring by grouping · factoring 2nd degree polynomials Rational Expressions: Basic Operations · reducing to lowest terms · multiplying/dividing rational expressions · adding and subtracting with LCDs · compound fractions Integer Exponents · properties of integer exponents · scientific notation |
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Announcements:
· Beginning of class. Take attendance, learn names. Review appendix of text up to polynomial multiplication. |