Detailed Coverage Checklist for Math 143, 144, 147 revised 1-5-10 for Math 143, Math 144, Math 147 (Math 14x) X = mandatory H = helpful (cover these to make the course better than minimally acceptable) P = prerequisite knowledge (do not “teach”, just review “in passing”) “blank” = totally optional (do only after “mandatory” and “helpful” are covered) GCM = Graphing Calculator Manual from Math 108 T = a topic covered on test, quiz and homework Q = a topic covered on quiz and homework C = a topic covered in homework only Keep in mind that many of our Math 143 and Math 147 students gain entry into our classes by using their ACT/SAT scores or the COMPASS placement test. We may have only 10% - 25% of any one class that has taken Math 108 sometime in their college career, and this is not necessarily “last semester”. Thus, when you see a concept that has an X (mandatory) on it for Math 14x and a T on it (homework, quiz, and test) from Math 108, do not automatically assume your students are fine with that concept. In including the designation for Math 108 below, the concepts were compared, not actual problems. So, the level of difficulty may not be the same. In fact, starting with Chapter 2, we expect the level of difficulty to be greater in Math 147 as soon as you get away from the most basic idea. Math 143 - chapters 1, 2, 3, 4, 9 Math 144 - chapters 5, 6, 7, 8 Math 147 - all listed Chapter 1 Fundamentals (over set of real numbers) Do not get bogged down in this chapter. Try to stick carefully to the daily schedule guidelines. The students should have the basics of each section prior to this class. We want to expose them to the more difficult problems that are a “step up” from Math 108 problems. We want to give the students the opportunity to review/learn these concepts, but not to the detriment of rushing through or skipping the remaining ideas we need to cover in the rest of the chapters. Don’t assign all the problems listed here, but concentrate on these in your assignment and class presentation. Day 1 Assign Section 1.1 to be read (but not covered in class), and assign some problems from 21-34, 41-54, 63-70, 76. Cover Section 1.2 in class, and concentrate on problems from 1-22, 27-70, 83-86. Assign the Preface of the Graphing Calculator Manual (GCM) to be read. 108 14x T P interval notation, absolute value as a distance evaluating expression for a < 0, b > 0, book (problem 76) and GCM T T T P X X X laws of exponents rational exponents rationalize denominators such as √32 √ n an = |a| for n even entering expressions (rational exponents) in calculator, read as indicated book 1.1 1.1 GCM 1.1 1.2 1.2 1.2 1.2 1.2 Day 2 Cover Section 1.3. Concentrate on problems from 25-32, 37, 53-104. Give a reading assignment for GCM Section 0.1, and assign a few exercises from GCM Section 1.2. This will help review the previous class and get the students started with the calculator manual. 108 T C T T C 14x P X P P H X X H add, subtract, multiply polynomials multiply expressions, such as problems 17, 18, 30-32, 37 factor ax2 + bx + c, a = 1, a 6= 1 factor a2 − b2 factor a3 ± b3 factor by grouping (covered extensively in Math 025) 3 1 −1 −1 factor with rational exponents, such as a 2 − a 2 or (x2 + 1) 2 + 2(x2 + 1) 2 start work with GCM , assign reading as indicated, choose a few problems Day 3 Cover Section 1.4. Concentrate on problems from 23-77, 85-92. Assign reading of GCM Section 0.2, and a few exercises from GCM Section 1.3, which will review factoring and start the students on graphing with their calculator. 108 T T Q T T 14x H X X X X find the domain of an expression simplify rational expressions simplify complex fractions add, subtract rational expressions rationalize denominators, such as 2+2√x H rationalize numerators, such as x+2 5 assign GCM reading as indicated, choose a few problems √ 2 √ book 1.4 1.4 1.4 1.4 1.4 GCM 1.4 1.4 book 1.3 1.3 1.3 1.3 1.3 1.3 GCM 1.3 1.3 Days 4 and 5 Cover Section 1.5. Concentrate on problems from 23-63, 69-93. Day 4: Assign reading of GCM Section 0.3.1, and a few exercises from GCM Section 1.4. Day 5: Assign reading of GCM Section 0.9 intro, Section 0.9.1 x−intercept method and Section 0.3.4. Assign a few exercises from GCM Section 1.5 using x−intercept method. 108 T T Q Q Q T T T 14x P X X X X X X X X H H solve linear equations 2 4 − x−1 = x25−1 solve rational equations, such as x+1 solve literal equations, such as P = 2L + 2W for W solve equations by factoring solve quadratic equations by completing the square solve equations using quadratic formula √ solve equations involving radicals,√such as 2x + 1 + 1 = x (Math 108 sees it in the form 2x + 1 = x − 1) 1 −1 −3 solve equations, such as x 2 + 3x 2 = 10x 2 use discriminant to determine number and type of solutions of quadratic without solving: 1 (repeated) real, 2 distinct real, 0 real recognize number and type of solutions for quadratic from graph (and state sign of discriminant from graph) solve equations graphically (calculator), such as Day 5: set = 0, graph, find x−intercepts Day 6: graph left side, right side, find x−value of points of intersection assign GCM reading as indicated, choose a few problems These are major techniques that will be used often. application problems for various types of equations book 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 Day 6 Cover Section 1.6. Concentrate on problems from 35-50, 73-79. These are like examples 3-5 in the text (the other types of word problems in the section are M108 exercises). Assign reading of GCM Section 0.9.2 (points of intersection method for solving equations by graphing). Assign a few exercises from GCM Section 1.5 using points of intersection method. (This is the second technique for solving equations by graphing.) 108 14x X application geometric problems involving area and perimeter, such as T - rectangular gardens, - farmer building 3 corrals Q - fixed width sidewalk around a pool Q X application problems, involving similar triangles X application problems, such as dimensions of a box made from sheet of cardboard, volume of cube with one side twice side x, etc 3 book 1.6 1.6 1.6 GCM GCM 1.6 1.6 Days 7 & 8 In Section 1.7, it is helpful to start with compound inequalities as in example 2 (problems 21-28). Cover nonlinear inequalities, as in examples 3-5 in the text (problems from 29-62, 87-90); and absolute value inequalities as in examples 6-7 in the text (problems from 63-86). Note: most odd absolute value inequality exercises are “less than”. Some “greater than” problems need to be added. Day 7, assign 1 or 2 exercises from GCM Section 1.6. Day 8, assign reading of GCM Section 0.10 and Section 0.3.2, and a few exercises from GCM Section 1.7. 108 T T T T 14x P H X X X X H T solve linear inequalities solve compound inequalities (Math 108 - as needed with absolute value) use interval notation as appropriate from now on solve product inequalities, such as (x + 2)(x − 3) > 0 and problem 87 x+2 solve quotient inequalities, such as x−3 > x2 and problem 89 solve absolute value inequalities solve inequalities by graphing (calculator) - same idea as equations using x−intercepts or points of intersection assign GCM reading as indicated, choose a few problems application problems (we did not compare types of applications) book 1.7 1.7 1.7 1.7 1.7 1.7 1.7 GCM 1.7 1.7 Day 9 Cover only part of Section 1.8: circles (problems 81-94) and symmetry (71-80). You can wait with the graphing material (51-70) until Section 2.4, since our aim is to leave point plotting behind as a graphing method and replace it with transformations. Assign reading of GCM Section 0.3.3, and a few exercises from GCM Section 1.8. Omit Stewart Section 1.9 - it has already been covered in GCM Sections 1.3 - 1.8. 108 14x Q P distance formula, midpoint formula X algebraically test equation for symmetry wrt x-axis, y−axis, origin H assign GCM reading as indicated, choose a few problems H decide symmetry wrt x-axis, y−axis, origin using calculator graph T X put circle equation in form (x − h)2 + (y − k)2 = r2 find center, radius book 1.8 1.8 1.8 1.8 1.8 Day 10 Cover the more difficult multi-step problems from Section 1.10 similar to problem 57, but assign problems from 19-36, 47-51, 57, 58, 60 and 60+. (The focus should be multi-step problems, not the 108 type of basic problems.) Cover Section 1.11 briefly. Concentrate on problems from 23-42, including at least one from 24, 31, 40-42. 108 14x book T P slope, slope-intercept form, point-slope form 1.10 T P parallel lines, perpendicular lines 1.10 slope-intercept, parallel, perpendicular problems X multiple-step problems, such as problem 57 1.10 H variation - direct, inverse, joint 1.11 4 GCM 1.10 GCM Chapter 2 Functions (over set of real numbers) 108 14x X C Q T X H T X X T X H C T C T T P X P X H H X X H H H X X X X H evaluate a function, f (2), f (a), f (x + 1), f (x + h) f (2), f (a), f (x + h) f (x + 1) use variable other than x in notation, f (t), f (r), f (s) (x) difference quotient f (x+h)−f h application problems for functions, such as problems 59 - 68 evaluating piecewise defined functions at a value algebraically find domain of a function especially involving radicals, rational expressions find domain of a function from calculator graph (see section 2.2 below) book 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.2 2.2 X X X X sketch function using tables of values graph without a table of values: Power functions f (x) = xn page 166 graphs y = x2 y = x3 y = x4 Note the pattern for xn , n even y = x5 Note the pattern for xn , n odd √ graph without a table of values: Root functions f (x) = n x page 166 graphs √ y= x √ y= 3x √ y = 4 x Note the pattern for the graphs √ y = 5 x Note the pattern for the graphs graph without a table of values: Reciprocal functions f (x) = x1n p.166 graphs y = x1 y = x12 graph without a table of values: y = |x| p.166 graphs graph without a table of values: greatest integer function, p.166 graphs problems p.170(92, 93) read a given graph to find f (2), domain, range book page 167(23, 24) read a given graph to find x where f (x) = g(x) page 167 (25) read a given graph to find x for which f (x) = 3 page 167 (26) sketch graph of piecewise defined function X X X H determine intervals of increasing, decreasing from a given graph determine intervals of increasing, decreasing from a calculator graph find average rate of change of f (x) applications of average rate of change 2.3 2.3 2.3 2.3 5 GCM 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.3 Math 108 does not use “transformation” terminology, and they work only with parabolas of the form y = a(x − c)2 . They answer questions such as how does the vertex change, is the parabola wider/narrower, and does the parabola open up/down? The students will probably not recognize the connection between the different terminologies. Also, some textbooks use the term “translations” instead of “transformations”. 108 14x X X X X X X H X H Q X T Q T T Q H X X H X X X using transformations, describe the graph and graph f (x) with vertical, horizontal shifts reflections over x, over y vertical stretch/shrink horizontal stretch/shrink algebraically determine if f (x) is even, odd, neither determine if f (x) is even, odd, neither using a calculator graph work with even/odd functions with a graph sketch graph of f (x), use that graph to sketch |f (x)| put quadratic into f (x) = a(x − h)2 + k form by completing the square find vertex, sketch graph b b find vertex of quadratic function by using − 2a , f (− 2a ) find maximum/minimum value of f (x) from standard form f (x) = a(x − h)2 + k b using f (− 2a ) from a given graph from a calculator graph application problems for maximum/minimum (given the function) book 2.4 GCM 2.4 2.4 2.4 2.4 2.4 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 Additional note about section 2.6: Concentrate on setting up functions and the simpler maximum/minimum problems without a calculator. The purpose here is be familiar with the concepts, but to not get bogged down in the problems that involve several steps (such as problems 7, 8, 15). 108 14x book X modeling information with quadratic functions by finding the function, 2.6 maximum/minimum values and where they occur p.210(1-25, omit 7, 8, 15) − algebraically finding extremum value of a function, such as A(x) = −2x2 + 2400x or − using calculator to find extremum value of function, such as S(x) = x2 + 48 x 6 GCM 108 14x T X given function definitions, find f + g, f − g, f · g, fg X given graphs of f and g, find f + g by graphical addition X find domain of f + g, f − g, f · g, fg using function definitions X find domain of f + g, f − g, f · g, fg from a given graph of f (x) and g(x) X given function definitions, find T - (f ◦ g)(x), (g ◦ f )(x), g(f (3)), (f ◦ g)(5), ... - (f ◦ f )(x), (g ◦ g)(x) X find domain of composite function X given graphs of f and g, find (f ◦ g)(3) X given graphs of f and g, find x such that (f ◦ g)(x) = 5 H given F (x) = (x − 9)5 , decompose F to find f, g such that F (x) = (f ◦ g)(x) H given f (x), g(x), h(x), find f ◦ g ◦ h H application problems for composite function T T X X X X T X H T H H determine if function is one-to-one from graph or function definition use Inverse Function Property to show f and g are inverses given the graph of f (x), draw the inverse function by reflecting over y = x given the definition of f (x), algebraically find the inverse function (Math 108 does this for very simple functions) restrict f (x) when not 1-1 so f (x) has an inverse and find that inverse application problems for inverse functions, such as given v(r) formula for velocity v of blood through a vein or artery, find v −1 (x), what does v −1 (x) represent?, find v −1 (30) and what does it represent? 7 book 2.7 2.7 2.7 GCM 2.7 2.7 2.7 2.7 2.7 2.7 2.7 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.8 2.7 2.7 Chapter 3 Polynomial and Rational Functions The students need a basic understanding of what the graphs of polynomial and rational functions look like and how they behave before we graph them by hand or with a calculator. (The graphing emphasis here is to “not” use a table-of-values. When the author plots “additional” points or “test” points, feel free to not do so. Just use the information already obtained.) 108 14x X graph polynomials transformed from y = xn X find end behavior of polynomial function, such as x → ∞, y → −∞ X factor, then graph polynomials using degree, end behavior, x−intercepts, and if the factor is “linear”, “quadratic”, “cubic”, etc. H using calculator graph, find all local extrema (or number of local extrema) H application problems with polynomial functions (using calculator graphs), such as find maximum volume of open box made from sheet of cardboard of given dimensions H application problems with polynomial functions using calculator graphs X H X X X X X H X X X divide polynomials using long division divide polynomials using long division (exercises 1 or 2) divide polynomials using synthetic division find quotient, remainder, and use Division Algorithm to write P (x) = D(x)Q(x) + R(x) find quotient, remainder, and write in form P (x)/D(x) = Q(x) + R(x)/D(x) use synthetic division and Remainder Theorem to find f (c) use Factor Theorem to determine if x − c is a factor of P (x) use Factor Theorem to determine if x − c is a factor of P (x) (exercises 3 or 4,7 or 8) given integer zeros of a polynomial, find P (x) given a polynomial graph, find a possible P (x) using the x−intercepts, end behavior, and behavior at x−intercepts (linear, quadratic, cubic) given a polynomial graph, find a possible P (x) using the x−intercepts, end behavior, and behavior at x−intercepts (linear, quadratic, cubic) (exericses 5 or 6) 8 book 3.1 3.1 3.1 GCM 3.1 3.1 3.1 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 3.2 Sections 3.3 and 3.5 There are two suggested approaches to sections 3.3 and 3.5. A) Treat Section 3.3 solidly and heavily, then use Section 3.5 as a “capstone section” which pulls it all together; thus, assigning a smaller number of problems. B) Treat Section 3.3 as an introduction to Section 3.5; thus, keeping the homework assignment in 3.3 smaller. Use Section 3.5 to firm up and solidify the concepts for all real numbers. 108 14x book X use Rational Zeros Theorem to find list of possible rational zeros of P (x) 3.3 X find all real zeros of a polynomial P (x) 3.3 X use a graph to estimate integer zeros, check them algebraically and find all real zeros of a polynomial P (x) (exercises 1 or 2) use Upper and Lower Bound Theorems in finding real zeros 3.3 use Descartes Rule of Signs in finding real zeros 3.3 show a polynomial P (x) has no rational zeros 3.3 application problems using calculator graphs 3.3 GCM 3.3 108 14x book Q X add, subtract, multiply, and divide complex numbers, writing result in a + b i form 3.4 simplify in , n > 2 3.4 T X simplify expressions using square roots of negative numbers 3.4 2 T X solve equations with non-real complex solutions, such as 9x + 4 = 0 3.4 T X expand discriminant use for quadratic to indicate 1 distinct real solution (multiplicity 2), 2 distinct real solutions, or 2 non-real complex solutions (exercises 1 or 2) X recognize from a graph the number and type of solutions for a quadratic as well as state the sign of the discriminant (exercises 3 or 4) 108 14x X X X X H Fundamental Theorem of Algebra use Complete Factorization Theorem, and factor f (x) completely use Zeros Theorem, find all zeros of f (x) including the multiplicity of each if > 1 use Conjugate Zeros Theorem - given zeros such as −1(multiplicity 2), 3 − 4 i, find P (x) of degree 4 with integer coefficients read information from calculator graph to help factor/solve P (x) 9 book 3.5 3.5 3.5 3.5 GCM 3.4 3.4 GCM 3.5 108 14x book X find x− and y− intercepts of rational function 3.6 X find vertical asymptotes of rational function 3.6 X find behavior of f (x) near the vertical asymptotes 3.6 − + such as x → 2 , y → ∞ and x → 2 , y → ∞ find horizontal asymptote by comparing degree of numerator and denominator 3.6 X find horizontal asymptote by dividing and finding y−value as x → ∞ 3.6 compare degree of numerator and denominator to determine if a slant asymptote exists 3.6 X determine if function f (x) crosses horizontal asymptote y = c by setting f (x) = c, solving for x, and including the point in the graph 3.6 H use long division and transformations to graph a rational function 3.6 X find intercepts, vertical, horizontal asymptotes, sketch graph of rational funciton 3.6 find slant asymptote, vertical asymptote, intercepts, and graph 3.6 graph rational functions on calculator 3.6 application problems involving rational functions, usually with calculator graphing 3.6 such as, concentration of drug in bloodstream is given by c(t) = t25t+1 , find highest concentration reached, what happens to concentration after long period of time?, how long until concentration is below 0.3 mg/L ? given information such as x−intercepts, vertical asymptote, horizontal asymptote, 3.6 make up a possible function definition for f (x) 10 GCM 3.6 Chapter 4 Exponential and Logarithmic Functions Exponential and logarithmic functions are vital. Concentrate on minimizing the use of the calculator by performing manipulations by hand as long as possible before using a calculator. Emphasize the properties, laws, and use transformations of functions for graphing instead of plotting points. 108 Q Q T T Q Q Q T Q Q T 14x X evaluate exponential functions using a calculator (Math 108 has simple computations, such as e5.2 ) sketch graph of exponential function from table of values X sketch graph of exponential function using transformations of y = ax for a > 1 H e = (1 + n1 )n as n → ∞ X state domain, range, asymptote of exponential function using graph X given the graph and a point (2, 9) on f (x) = ax , find a (we read the y−intercept (0, 1) from the graph) X given two points on f (x) = Cax , find f (x) H find difference quotient of f (x) = 10x sketch graph of hyperbolic cosine function, cosh(x) = 21 (ex + e−x ) by graphing y = 21 ex and y = 21 e−x and using graphical addition use definition of cosh(x) and prove identities such as cosh(−x) = cosh(x) H applications of exponential functions mostly of the type - given the function, evaluate it for a given value X applications of exponential functions - typical compound interest problems using a calculator graph, on what intervals is a function such as f (x) = x2 (2−x ) increasing, decreasing, find maximum or minimum value X X X X X X X X X Q inverse of exponential function is logarithmic function common logarithms, natural logarithms convert between logarithmic form and √ exponential form evaluate log expression such as log9 3 use log properties loga 1 = 0, loga a = 1, loga ax = x, aloga x = x given the graph and a point (5, 1) on f (x) = loga x, find a sketch the graph of log functions using transformations of y = loga x for a > 1 algebraically find the domain of log functions, such as f (x) = log(x + 3) or f (x) = log(x2 − 1) given a logarithmic or exponential function, algebraically find the inverse function for example, find the inverse function for problems such as page 350(49-56, 77) and page 337(25-38) applications of logarithmic functions mostly of the type - given the function, evaluate it for a given value using a calculator graph, on what intervals is a function such as f (x) = 6x − 2x ln(2x) increasing, decreasing, find maximum or minimum value using a calculator graph, given f (x) = ex − 2 and g(x) = ln(x + 1) for what values of x is f (x) = g(x), f (x) < g(x) 11 book 4.1 GCM 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 4.2 108 14x , loga M k to expand or Q X use Laws of Logarithms, loga (M N ), loga M N condense expressions including problem 66 X evaluate log expressions, such as log3 (100) − log3 (18) − log3 (50) + 2 Q X use change of base formula Q X use change of base formula to evaluate logs on calculator such as log2 5 application problems X X X T X Q T H H X X X book 4.3 4.3 4.3 4.3 4.3 solve a variety of exponential equations similar to problems 17, 19, 27, 31 using exact values using calculator approximations -Math 108 does simple problems such as 1 - 9 solve logarithmic equations such as log9 (x − 5) + log9 (x + 3) = 1 Math 108 does log(4x) − log(x − 7) = 3 and log3 (x) + log3 (2x + 5) = 1 log3 (6x + 5) = 2 graphically (with calculator), solve log and exponential equations and inequalities algebraically solve inequalities such as log(x − 2) + log(9 − x) < 1 or 2 < 10x < 5 applications involving exponential, log functions, mostly with function given 4.4 4.4 4.4 modeling with exponential and logarithmic functions exponential growth - population including creating the function exponential decay, half-life including creating the function Newton’s Law of Cooling pH Scale Richter scale decibel scale 4.5 4.5 4.5 4.5 4.5 4.5 4.5 12 GCM 4.4 4.4 4.4 4.4 Chapter 5 Trigonometric Functions of Real Numbers Chapter 5 is more abstract than Chapter 6. The two chapters can be done in either order, or some sections can be done together (5.1 & 6.1, 5.2 & 6.2). P X X X X X X X X X X X X X X X X X X H X X X X H H determine if a given point is on the unit circle given P (− 35 , y) on the unit circle, and information about the quadrant, P , or the missing coordinate, find y label arclength t and its terminal point on unit circle, for t in increments of π4 , π6 for positive or negative rotation find terminal point on the unit circle for a given arclength t given terminal point for arclength t, find terminal point for π − t, −t, t + 2π, t + 3π, etc. find the reference number t for given values of t given a unit circle diagram marked in tenths, find the terminal point determined by t, such as t = 2.5, −1.1 given a unit circle diagram, label all arclengths t = 0, π6 , π4 , π3 , π2 , · · · , 2π given terminal point (x, y) on unit circle for arclength t, define sine, cosine, and tangent in terms of x and y. Define reciprocal functions as needed know the sine and cosine of the basic values t = 0, π6 , π4 , π3 , π2 , π, 3π 2 and be able to quickly find the other 4 trig function of these real numbers find the six trig functions of a real number t using the terminal point P (x, y) on unit circle where t is a multiple of the basic values given a terminal point P (x, y) for any real number t on unit circle, find the six trig functions of t where t is not a multiple of the basic values find sin(t), cos(t), tan(t) for t = −1.3, 4.1, etc. given a unit circle diagram marked in tenths using a calculator find sign of expressions such as sin(t) cos(t) for t in quadrant II define rest of trig functions using sine and cosine know and use Pythagorean identities find quadrant for terminal point for t, given csc(t) > 0 and sec(t) < 0 write tan(t) in terms of sin(t), t in quadrant IV given tan(t) = − 34 , cos(t) > 0, find other trig functions (using Pythagorean identities) determine if f (x) = sin(x) cos(x) is even, odd, or neither algebraically graphically - from given graph or using calculator graph simplify expressions such as sec(−x) cot(−x) tan(−x) given sec(x) = 2, find cos(−x) applications - mostly evaluate given function for specific value 13 book 5.1 5.1 GCM 5.1 5.1 5.1 5.1 5.1 5.1 5.1 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 5.2 Transformations are vital for the graphs in sections 5.3 and 5.4. Do not plot points from a table of values. X state amplitude, period, phase shift, reflection over x, vertical shifts for sine, cosine graphs X graph y = A sin k(x − B) + C for sine or cosine functions X given sine or cosine graph with some x− and y−values labeled, find the amplitude, period, phase shift, write equation using y = a sin k(x − b) or y = a cos k(x − b) H determine appropriate viewing window for calculator graph of f (x) = cos(100x) and graph it graph f, g, f + g on common calculator screen for graphical addition graph 3 given functions on same calculator screen to see how they are related using a calculator graph, find maximum/minimum values of f (x) = sin(x) + sin(2x) using calculator graph, solve given trig equation given a function, answer questions such as even/odd, x−intercepts, (using calculator graph) behavior as x → ±∞, etc. applications - given function, find period, graph with calculator, answer questions H given a nontrig graph of f (x), determine if f (x) is periodic, if so, find period X given a sine or cosine graph with no scale labeled and a function definition such as y = 2 cos(2x), label intercepts so graph matches definition X match graph to function definition graph f (x) = sin (|x|) on calculator, determine if periodic, if so, find period X match graph to function definition X state period, phase shift, reflection over x, for tangent, cotangent, secant, cosecant X graph y = A tan k(x − B) for tangent, cotangent, secant, cosecant determine appropriate viewing window for calculator graph of y = tan(10πx), and graph it applications - mostly given function, evaluate at various values X given a trigonometric graph with no scale labeled and a function definition such as y = − csc(πx), label intercepts, asymptotes so graph matches given definition graph f (x) = cot (|x|) on calculator, determine if periodic, if so, find period 14 book 5.3 GCM 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.3 5.4 5.4 5.4 5.4 5.4 5.4 5.4 5.4 Chapter 6 Trigonometric Functions of Angles Radian measure is vital for calculus. Do not concentrate on degree measure to the exclusion of radian measure. X X X H H X convert from degree measure to radian measure convert from radian measure to degree measure know the basic angles in either degrees or radians find positive and negative angles that are coterminal with the given angle given two angles, determine if they are coterminal given two of these three values for a circle (radius r, central angle θ, arclength s), find the third value using s = rθ X given two of the three (radius r, central angle θ, area A of a sector of the circle), find the third value using A = 12 r2 θ H applications - using arclength s, radius r, central angle θ applications - find the linear speed (v = st ) or angular speed (ω = θt ) of an object X know the definitions of sine, cosine and tangent using ratios of sides of a right triangle and be able to quickly find ratios for reciprocal functions when needed X know sine, cosine of 0, 30◦ , 45◦ , 60◦ , 90◦ , 180◦ , 270◦ , and be able to quickly find other 4 trig functions of these angles X find the six trig functions of an acute angle of a right triangle, given 2 or 3 sides X find the sides and second acute angle of a right triangle, given 1 side and one acute angle X given sec θ = 72 , find other trig functions of acute angle θ (using right triangles) X express the length x of a side of a right triangle in terms of the trig ratios of an acute angle θ, no specific values given, such as problems 43, 44 X use a calculator to find the six trig functions of any angle (degrees or radians) X given the trig function of an angle, use a calculator to find the angle in degrees or radians X right triangle applications, find a side or an acute angle X given some trig functions, use cofunctions to find others 15 book 6.1 6.1 6.1 6.1 6.1 6.1 GCM 6.1 6.1 6.1 6.1 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 6.2 Sections 6.3, 6.4 & 6.5 need some story problems where the picture has not been drawn – the students must learn to draw the pictures. X find reference angle for any angle in radians or degrees X find exact value of trig functions of any angle with reference angle of 30◦ , 45◦ , 60◦ , or in radians, or angle terminating on x− or y−axes X given θ in standard position, P (x, y) on terminal side of θ, r (distance from origin to P ), define the sine, cosine, and tangent of θ in terms of x, y, and r. Define reciprocal functions as needed. X find quadrant for θ given information such as sin θ < 0 and cos θ < 0 X write first trig function in terms of second, given tan θ, cos θ, θ in quadrant III X given sec θ = 5, sin θ < 0, find other trig functions using right triangle or Pythagorean Identities X find area of triangle using A = 12 ab sin θ for two sides a, b and the included angle θ X find area of shaded region using area of triangle and area of sector applications H based on a clock face, find the requested angle in degrees or radians X X X X book 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 6.3 know Law of Sines use Law of Sines to find a side or an angle of a given triangle use Law of Sines to solve a unique triangle by finding all sides and angles not given given two sides and one angle of a triangle, when the ambiguous case of Law of Sines applies, find remaining sides, angles for two triangles, or one triangle, or realize there is no triangle X application problems for Law of Sines Need to include some problems for which pictures have not been drawn 6.4 6.4 6.4 6.4 X know Law of Cosines X use Law of Cosines to find a side or an angle of a given triangle X use Law of Cosines or Law of Sines, as needed, to solve a triangle by finding all sides and angles not given given 3 sides of a triangle, find the area using Heron’s formula find area of given figures, problems 31 - 35 H find perimeter, area of given figures X application problems using Law of Cosines, Law of Sines, right triangles, or a combination 6.5 6.5 6.5 6.5 6.5 6.5 16 GCM 6.4 6.5 6.5 Chapter 7 Analytic Trigonometry Algebra techniques reappear in full force in Chapter 7. X X X H H H X X X X H X H X X X H use cofunction identities, such as sin 60◦ = cos 30◦ use factoring, finding common denominator, Pythagorean identities, definitions using sine, cosine, even/odd identities, reciprocal identities, etc. - to simplify trig expressions - to prove trig identities verify identities by starting on one side and ending with the other side This is the only valid proof writing technique. verify identities by working separately on LHS and RHS and showing the two sides are the same expression. (A valid verification technique, but not valid for writing proofs in later classes.) Never perform the same operation on both sides at the same time. See Warning on page 530. make a trig substitution in an expression and simplify, √ such as let x = 3 sin θ in 9 − x2 show an equation is not an identity, such as show sin 2x = 2 sin x is not an identity (problem 99) Does a calculator graph suggest (sin x + cos x)2 = 1 ? If so, verify it. ◦ 17π , 12 use addition/subtraction formulas to find the exact value of sin 75 , cos etc. use addition/subtraction formulas to rewrite an expression, then simplify using exact values prove identities using addition/subtraction formulas write an expression, such as sin x +hcos x, in terms of sine only i with a calculator, graph f (x) = − 21 cos(x + π) + cos(x − π) for two periods describe the graph using a simpler function g(x), prove f (x) = g(x) algebraically applications x , cos x2 , tan x2 2 given sec x = 2, x in quadrant IV, find sin 2x, cos 2x, tan 2x, sin use lowering powers formula use half angle formulas to find cos 165◦ , sin 9π , etc. 8 simplify expressions using double or half-angle formulas use product-to-sum formulas to rewrite/evaluate expressions use sum-to-product formulas to rewrite/evaluate expressions prove identities using double angle, half-angle formulas prove identities using lowering powers formulas (problem 70) prove identities using product-to-sum formulas prove identities using sum-to-product formulas (problems 71-75, 77-81, 85) with a calculator, graph f (x) = sin(2x) cos(2x), make a conjecture, and prove the conjecture (problems 83, 84 in text, several problems in GCM) applications 17 book 7.1 7.1 GCM 7.1 7.1 7.1 7.1 7.1 7.1 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 7.3 −1 X X X X −1 −1 know the domain and range of sin (x),cos (x), tan (x) √ −1 1 , cos−1 (1), tan−1 − 33 , using exact values evaluate inverse functions such as sin 2 use calculator to evaluate inverse sine, inverse cosine, or inverse tangent functions use inverse function property relationship for sine, cosine, and tangent to evaluate, such as cos(cos−1 23 ) X evaluate composition of trig and inverse trig functions functions √ −1 3 π −1 such as tan 2 sin 3 or cos sin 2 X evaluate expressions such as sin cos−1 53 , or cos sin−1 35 − cos−1 53 , using exact values X rewrite a given expression such as cos(2 tan−1 x) as an algebraic expression in x (using right triangle info) H with calculator, graph f (x) = sin−1 (x) + cos−1 (x), make a conjecture, prove it solve trig equation with calculator graph and with exact values X application problems with inverse functions H work with calculator graph of inverse functions using transformations book 7.4 7.4 7.4 7.4 GCM 7.4 7.4 7.4 7.4 7.4 7.4 7.4 Section 7.5 problems 1 - 40 show up in Math 170. X X X X X X X H X book algebraically find all solutions of a trig equation in x, by using identities, factoring 7.5 √ 3 algebraically find all solutions of a trig equation such as sin(3x) = 2 7.5 algebraically find all solutions in [0, 2π) of trig equation 7.5 graph two given trig functions by hand and find their points of intersection 7.5 simplify equation then find all solutions in [0, 2π) 7.5 use addition, subtraction formulas such as cos x cos 3x − sin x sin 3x = 0 7.5 use double or half-angle formulas such as sin 2x + cos x = 0 7.5 use sum-to-product formulas such as sin x + sin 3x = 0 7.5 applications 7.5 with calculator graph, find all solutions of trig equation 7.5 with calculator graph, find solutions of trig equation and trig inequalities explain the difference between cos−1 (−0.1234) and solving cos(x) = −0.1234 given a trig function, find x−intercepts algebraically and using calculator graph, determine if function is even, odd, or neither (algebraically and from graph) 18 GCM 7.5 7.5 7.5 Chapter 8 Polar Coordinates Chapter 8 ties together many concepts studied previously. For polar graphing, transfer the information from a rectangular graph to the polar graph (examples 4 and 5). Do not plot points from a table of values (example 3). If you need some polar graph paper, contact Barbara Kenny. X using polar coordinate system, plot 4, 2, − 2π 3 π 4 , 3, − 2π 3 book 8.1 X given − , find two other polar coordinate representations of the point, one with r < 0, one with r > 0 X convert rectangular coordinates to polar coordinates X convert polar coordinates to rectangular coordinates convert rectangular equations to polar form convert polar equations to rectangular form 8.1 match a polar equation with a given graph test for symmetry wrt the polar axis, wrt the pole, wrt the vertical line θ = π2 X sketch the graph of a polar equation r = 2 + 2 cos θ by graphing it on a rectangular system, then transferring the information to a polar graph use symmetry information to help graph polar equation H graph a polar equation with a calculator sketch a graph of a rectangular equation by first converting to a polar equation applications 8.2 8.2 8.2 X graph a complex number √ in the complex plane X find the modulus |z| = a2 + b2 for a complex number a + b i in the complex plane, sketch {z = a + b i | a + b < 2} or {z |z| = 3} X write a given (rectangular form) complex number in polar form X given two complex numbers, z and w, in polar form, find zw, wz , answer in polar form given two complex numbers, z and w, in rectangular form, write them in polar form find zw find wz find z1 X use DeMoivre’s Theorem to find (a + bi)n X find roots of√a complex number, such as the fourth roots of −81i H solve z 3 − 4 3 − 4i = 0 8.3 8.3 8.3 8.3 8.3 8.3 19 GCM 8.1 8.1 8.1 8.1 8.2 8.2 8.2 8.2 8.3 8.3 8.3 8.2 Chapter 9 System of Equations The calculus folks are interested in our students being able to solve linear and nonlinear systems of equations by substitution. Substitution is the ”fallback” method for anything. Other methods of solving systems of equations are nice, but not mandatory. Leave chapter 9 to the end, in case something has to be skipped. Section 9.4 is optional. 108 14x X solve nonlinear systems of equations by substitution X solve nonlinear systems of equations by elimination X solve nonlinear systems of equations from a given graph solve nonlinear systems of equations by graphing on calculator applications of nonlinear systems H solve nonlinear systems involving other functions, such as logarithmic and exponential functions (problem 55) (Math 147 folks could also add in a nonlinear system using trig functions.) X T T T X X T solve systems of 2 linear equations, including 1 solution, no solutions, infinite solutions, expressing the answer as an ordered pair Possible methods: − graphing by hand − graphing on calculator − substitution − elimination for infinite solutions, show a general solution such as (x, 2x + 5) for any real x solve system of 2 equations in terms of a and b, such as problems 39 - 42 Choose some typical application problems, such as transportation problems, mixture problems, investment problems. By graphing on calculator, find the triangular region formed by three given lines, then find the area of the region (problem 58 in text, several in GCM). - Math 108 does mixture problems and investment problems book 9.1 9.1 9.1 9.1 9.1 9.1 9.1 9.2 9.2 9.2 9.2 9.2 9.2 Solving systems of 3 equations is required. The method is left up to the instructor, including the technique mentioned below that is not in the text.You can use problems from sections 9.3 or 9.4, regardless of the technique you choose. Be clear with the students regarding your expectations. 108 14x book X solve systems of 3 linear equations, including 1 solution, no solutions, infinite solutions 9.3/9.4 expressing the answer as an ordered triple Possible methods: − Gaussian elimination 9.3 − (not in the text, but perfectly acceptable) Eliminate a variable, say x, in 2 equations, then eliminate x again in 2 other equations. Now we have 2 equations with two unknowns. Proceed as before. − matrices 9.4 X for infinite solutions, show a general solution such as (2z − 4, −z + 5, z) for any real z 9.3/9.4 X typical applications 9.3/9.4 solve systems of 4 linear equations 9.3 20 GCM 9.2 GCM

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