CSET Mathematics 110, 111, 112 (XAM CSET)

Study Tips......Page 13
Testing Tips......Page 16
1.1a. Know why the real and complex numbers are each a field, andthat particular rings are not fields (e.g., integers, polynomialrings, matrix rings)......Page 17
1.1b. Apply basic properties of real and complex numbers inconstructing mathematical arguments (e.g., if a < b and c < 0,then ac > bc)......Page 19
1.1c. Know that the rational numbers and real numbers can beordered and that the complex numbers cannot be ordered, butthat any polynomial equation with real coefficients can besolved in the complex field......Page 22
1.2a. Know why graphs of linear inequalities are half planes and beable to apply this fact (e.g., linear programming)......Page 24
1.2b. Prove and use the following: the Rational Root Theorem forpolynomials with integer coefficients; the Factor Theorem; theConjugate Roots Theorem for polynomial equations with realcoefficients; the Quadratic Formula for real and complexquadratic polynomials; the Binomial Theorem......Page 31
1.2c. Analyze and solve polynomial equations with real coefficientsusing the Fundamental Theorem of Algebra......Page 38
1.3a. Analyze and prove general properties of functions (i.e., domainand range, one-to-one, onto, inverses, composition, anddifferences between relations and functions)......Page 43
1.3b. Analyze properties of polynomial, rational, radical, andabsolute value functions in a variety of ways (e.g., graphing,solving problems)......Page 46
1.3c. Analyze properties of exponential and logarithmic functions ina variety of ways (e.g., graphing, solving problems)......Page 57
1.4a. Understand and apply the geometric interpretation and basicoperations of vectors in two and three dimensions, includingtheir scalar multiples and scalar (dot) and cross products......Page 62
1.4b. Prove the basic properties of vectors (e.g., perpendicularvectors have zero dot product)......Page 67
1.4c. Understand and apply the basic properties and operations ofmatrices and determinants (e.g., to determine the solvability oflinear systems of equations)......Page 70
2.1a. Know the Parallel Postulate and its implications, and justify itsequivalents (e.g., the Alternate Interior Angle Theorem, theangle sum of every triangle is 180 degrees)......Page 77
2.1b. Know that variants of the Parallel Postulate produce non-Euclidean geometries (e.g., spherical, hyperbolic)......Page 81
2.2a. Prove theorems and solve problems involving similarity andcongruen......Page 83
2.2b. Understand, apply, and justify properties of triangles (e.g., theExterior Angle Theorem, concurrence theorems, trigonometricratios, Triangle Inequality, Law of Sines, Law of Cosines, thePythagorean Theorem and its converse)......Page 94
2.2c. Understand, apply, and justify properties of polygons andcircles from an advanced standpoint (e.g., derive the areaformulas for regular polygons and circles from the area of atriangle)......Page 107
2.2d. Justify and perform the classical constructions (e.g., anglebisector, perpendicular bisector, replicating shapes, regular ngonsfor n equal to 3, 4, 5, 6, and 8)......Page 122
2.2e. Use techniques in coordinate geometry to prove geometrictheorems......Page 129
2.3a. Demonstrate an understanding of parallelism andperpendicularity of lines and planes in three dimensions......Page 133
2.3b. Understand, apply, and justify properties of three-dimensionalobjects from an advanced standpoint (e.g., derive the volumeand surface area formulas for prisms, pyramids, cones,cylinders, and spheres)......Page 136
2.4a. Demonstrate an understanding of the basic properties ofisometries in two- and th......Page 140
2.4b. Understand and prove the basic properties of dilations (e.g.,similarity transformations or change of scale)......Page 146
3.1a. Prove and use basic properties of natural numbers (e.g.,properties of divisibility)......Page 149
3.1b. Use the Principle of Mathematical Induction to prove results innumber theory......Page 152
3.1c. Know and apply the Euclidean Algorithm......Page 154
3.1d. Apply the Fundamental Theorem of Arithmetic (e.g., find thegreatest common factor and the least common multiple, showthat every fraction is equivalent to a unique fraction where thenumerator and denominator are relatively prime, prove that thesquare root of any number, not a perfect square number, isirrational)......Page 155
4.1a. Prove and apply basic principles of permutations andcombinations......Page 158
4.1b. Illustrate finite probability using a variety of examples andmodels (e.g., the fundamental counting principles)......Page 162
4.1c. Use and explain the concept of conditional probability......Page 166
4.1d. Interpret the probability of an outcome......Page 168
4.1e. Use normal, binomial, and exponential distributions to solveand interpret probability problems......Page 169
4.2a. Compute and interpret the mean, median, and mode of bothdiscrete and continuous distributions......Page 173
4.2b. Compute and interpret quartiles, range, variance, and standarddeviation of both discrete and continuous distributions......Page 178
4.2c. Select and evaluate sampling methods appropriate to a task(e.g., random, systematic, cluster, convenience sampling) anddisplay the results......Page 184
4.2d. Know the method of least squares and apply it to linearregression and correlation......Page 189
4.2e. Know and apply the chi-square test......Page 195
5.1a. Prove that the Pythagorean Theorem is equivalent to thetrigonometric identity sin2x + cos2x = 1 and tha......Page 198
5.1b. Prove the sine, cosine, and tangent sum formulas for all realvalues, and derive special applications of the sum formulas(e.g., double angle, half angle)......Page 200
5.1c. Analyze properties of trigonometric functions in a variety ofways (e.g., graphing and solving problems)......Page 207
5.1d. Know and apply the definitions and properties of inversetrigonometric functions (i.e., arcsin, arccos, and arctan)......Page 213
5.1e. Understand and apply polar representations of complexnumbers (e.g., DeMoivre's Theorem)......Page 217
5.2a. Derive basic properties of limits and continuity, including theSum, Difference, Product, Constant Multiple, and QuotientRules, using the formal definition of a limit......Page 221
5.2b. Show that a polynomial function is continuous at a point......Page 227
5.2c. Know and apply the Intermediate Value Theorem, using thegeometric implications of continuity......Page 231
5.3a. Derive the rules of differentiation for polynomial,trigonometric, and logarithmic functions using the formaldefinition of derivative......Page 233
5.3b. Interpret the concept of derivative geometrically, numerically,and analytically (i.e., slope of the tangent, limit of differencequotients, extrema, Newton’s method, and instantaneous rateof change)......Page 241
5.3c. Interpret both continuous and differentiable functionsgeometrically and analytically and apply Rolle’s Theorem, theMean Value Theorem, and L’Hopital’s rule......Page 248
5.3d. Use the derivative to solve rectilinear motion, related rate, andoptimization problems......Page 254
5.3e. Use the derivative to analyze functions and planar curves (e.g.,maxima, minima, inflection points, concavity)......Page 258
5.3f. Solve separable first-order differential equations and applythem to growth and decay problems......Page 265
5.4a. Derive definite integrals of standard algebraic functions usingthe formal definition of integral......Page 271
5.4b. Interpret the concept of a definite integral geometrically,numerically, and analytically (e.g., limit of Riemann sums)......Page 277
5.4c. Prove the Fundamental Theorem of Calculus, and use it tointerpret definite integrals as antiderivatives......Page 279
5.4d. Apply the concept of integrals to compute the length of curvesand the areas and volumes of geometric figures......Page 282
5.5a. Derive and apply the formulas for the sums of finite arithmeticseries and finite and infinite geometric series (e.g., expressrepeating decimals as a rational number)......Page 288
5.5b. Determine convergence of a given sequence or series usingstandard techniques (e.g., Ratio, Comparison, Integral Tests)......Page 295
5.5c. Calculate Taylor series and Taylor polynomials of basicfunctions......Page 299
6.1a. Demonstrate understanding of the development ofmathematics, its cultural connections, and its contributions tosociety......Page 302
6.1b. Demonstrate understanding of the historical development ofmathematics, including the contributions of diversepopulations as determined by race, ethnicity, culture,geography, and gender......Page 303
CONSTRUCTED-RESPONSE EXAMPLES......Page 305
CURRICULUM AND INSTRUCTION......Page 311
SAMPLE TEST......Page 317
ANSWER KEY......Page 330
Rigor Table......Page 331
Rationales with Sample Questions......Page 332