SCHOOL OF UNIVERSITY ARTS AND SCIENCES MATH 00 REVIEW PACKAGE Gearing up for calculus and preparing for the Assessment Test that everybody writes on at. You are strongly encouraged not to use a calculator for any of the following questions, since calculators will not be allowed during the Assessment Test. The test is closed book. A. ARITHMETIC SKILLS Eample : Simplify each a. b. c. ( ) + ( 5) 6 5 5 4 9 ( ) ( ) + 5 7 5 6 5 6 5 5 5 + + 4 9 4 9 5 + + 6 9 9 5 5 + 6 9 5 0 + 8 8 5 8 7-f09
Math 00 Review Package Page. Simplify: a. ( ) 4+ 6 5 b. 4 6 c. 75 5 Note: All answers are given at the end of this package. B. ALGEBRA Eample : Simplify each epression a. b. 6 k k 4 + Solution ( k ) ( )( ) 6 k k 4 k+ k k + + + ( ) + ( + ) ( + ) + ( + ) ( + ) + 7-f09
Math 00 Review Package Page c. d. y + y + y ( ) + y + y + y y + y + y + y y + y y ( ) ( ) ( ) + + + ( ) ( ) ( ). Simplify each epression: a. + h h 7-f09
Math 00 Review Package Page 4 b. p q ( pq) c. 4 + + + 4 + 4+ + 4 7-f09
Math 00 Review Package Page 5 C. SOLVING EQUATIONS Eample : Solve each equation 5 a. 4 + 5, Note:, 4 + 5, LCD + ( ) + ( )( ) ( )( ) ( ) ( + ) ( ) ( + ) + 5 + + 0 4, Linear + 9 0+ 0 4+ 7 7 ( )( ) b. 6 7 6 7 0 ( )( ) + 0 6 7, Quadratic + 0, or + 0, 7-f09
Math 00 Review Package Page 6 c. + 6 9. Solve each equation 0 8 0 6 + 9 8 6 + 9 8 8 6 + 9, cross-multiply ( 4) ( + 9) ( ) 6 9 9 + 6 9 + 8 7 8 9 9 ±, but does not satisfy the 7 7 original equation, therefore, one solution 9 7 a. 5 4 5 + 9 6 b. ( + ) 5 7-f09
Math 00 Review Package Page 7 c. + + d. 0 7-f09
Math 00 Review Package Page 8 D. INEQUALITIES AND ABSOLUTE VALUE Solving Inequations: typically the solution is an interval (or combination of intervals) that can be shown on a number line. Eample 4: Solve the inequality a. + > 7 5 + > 7 5 5 > 8 8 < 5 8 or,, 5 open interval ( ) b. + 4< + 4< separate : + 4, but, + 4 < 7 > < 7 or, (, 7] closed interval ( ) c. 4 + > + > 4 + 4 > 0 ( ) + 4 > 0 ( )( ) + 4 > 0 zeros / 4 regions to"test" show : 4 < < 0, or, > ( ) ( ) or, 4, 0, ABSOLUTE VALUE: The absolute value of any number is always positive. So when solving the equation, there are two solutions: ±. Solving Inequations with absolute value: two cases case (i) if < a,then a< < a case (ii) if > a, then > a, or, < a 7-f09
Math 00 Review Package Page 9 Eample 5: Solve the inequality a. < b. > c. + 5 8 4. Solve the inequality a. 5 9 (case (i)), then < < or,, ( ) (case (ii)), then >, or, < or,,, ( ) ( ) (case(i)) then 8 + 5 8 or,, b. + < 4 c. + 7-f09
Math 00 Review Package Page 0 E. CARTESIAN (XY) PLANE/STRAIGHT LINES Three Forms of Equations of a Line: Slope/Intercept Form: y m + b Point/Slope Form: y y m( ) General Form: a + by + c 0 Eample 6: Find the equation of a line that goes through 4 8y., and is perpendicular to the line 5. Find the equation of a line that goes through (, ) and ( 4, ). 4 8y, 8y 4, y 8 so, m y m+ b y + b,, + b b So,answer : y +, or, 6+ y 0 7-f09
Math 00 Review Package Page F. TRIGONOMETRY Radian Measure: ratio of arc length, s, divided by radius, r of a circle. s r θ, radian ( s r) ; θ rad 57. r r π r for full circle, s c π r: θ π r o that is, π rad 60 π conversion ratios: from degrees to radians. 80 80 from radians to degrees. π o E: π π 80 π : rad 45, that is : 45 4 4 π 4 o Some Special Angles: Isosceles Triangle: π For 45 π 4 sin 45 sin 4 o 45 o π cos 45 cos 4 o π tan 45 tan 4 45 o π o π For 0 and 60 : (Slice) Equilateral Triangle 6 o π sin 0 sin 6 0 o π cos0 cos 6 60 o π tan 0 tan 6 60 60 60 o π sin 60 sin o π cos 60 cos o π tan 60 tan 7-f09
Math 00 Review Package Page Reciprocal Relations: cscθ sinθ secθ cosθ cotθ tanθ E. π sec 6 π cos 6 Trigonometric Functions and Angles in the XY plane Y r θ y r + y X sinθ cosθ tanθ y r r y (positive angles measured counter clockwise) Eample 7: find the eact value of sin 5 Note: The reference angle for 5 is 45. ( ) o o o sin 5 sin 45 sin 45 Trigonometric Identities (true for all Ө) sinθ tanθ, cosθ sin θ + cos θ + tan θ sec θ + cot θ csc θ sin θ sinθ cosθ, cos cos sin θ θ θ sin θ cos θ 7-f09
Math 00 Review Package Page ( ) ( ) sin A± B sin Acos B± cos Asin B cos A± B cos Acos Bm sin Asin B Eample 8: Show cos( θ ) cosθ cos( θ ) cos( 0 θ) cos 0cosθ + sin 0sinθ cosθ + 0sinθ cosθ Solving Trigonometric Equations (true for some Ө) Since trigonometric functions are periodic, typically there are infinitely many solutions when solving these equations. To simplify the solution, an interval is usually chosen to be 0 θ π, or, [ 0, π ] (i.e., one full circle). Even in one full circle each primary trigonometric function has two solutions due to the nature of their ratios in the 4 quadrants. To help find the second solution, we can apply the following Properties: For 0 π, once is found, can be found as follows: for sin : π for cos : π for tan : π + 7-f09
Math 00 Review Package Page 4 Eample 9: Solve cos sin, in [ 0, π ] cos sin cos 0 cos sin cos sin cos ( ) cos sin 0 cos 0 or cos ( 0)( optional) π π π π sin 0 sin sin π 6 π π 6 5π 6 ( optional) So, there are four solutions in [ 0, π ]: π π 5π π,,, (in order) 6 6 7-f09
Math 00 Review Package Page 5 6. Find the remaining trigonometric ratios if π csc, π < < 7. Find the eact value of sin05 o. 8. Find all values of in the interval [ ] 0, π that satisfy the trigonometric equation: + cos cos. 7-f09
Math 00 Review Package Page 6 G. FUNCTIONS AND FUNCTION NOTATION Eample 0: Graph the function y + 8+ 8 and state: domain, range, increasing vs decreasing, verte, basic function and corresponding shifts. y + 8+ 8 y + 8+ 6 6+ 8 ( 4), parabola with verte ( 4,)( opens up) y + + Y ( ) [ ) ( ] [ ) Domain, D: all, or,, Range, R: y, or,, decreasing :, 4 increasing : 4, ( 4, ) X basic parabola: f ( ) y f ( + 4) + { 44 vertical horizontal shift shift Eample : Sketch a graph of the piecewise function g( ) +, if < 0 +, if 0 and find: a. g ( ) b. g ( 0) c., such that g( ) 7-f09
Math 00 Review Package Page 7 g() a. g ( ) ( ) + b. g ( 0) 0+ c. +,or, + 4,or, 6 Combinations +,,, 4444 and Composition 444 f g ( ) fog g f ( ) go f of functions. Eample : Given f ( ) 5 and ( ) a. ( f + ) g b. ( g f ) c. f o g d. go f e. go g g, find: 7-f09
Math 00 Review Package Page 8 Solutions: a. ( f + g) f ( ) + g( ) 5 + b. ( ) ( ) ( ) g g f, 5 f 5 c. f o g f ( g( ) ) d. 5 ( ( )) ( 5) go f g f ( 5) e. o ( ) ( ) ( ) g g g g 9 9. Find the domain and range of y. Sketch a graph. 0. Find f g. o, if f ( ) and g( ) 7-f09
Math 00 Review Package Page 9. Epress F( ) in the form f o g. In other words, identify f ( ) and ( ) g. F( ) sin 7-f09
Answer Key For MATH 00 Review Package. a. 9 b. c. 7. a. ( + h) b. q p c. + 4. a. 9 5 b. ± 5 c. 8 d. 4. a. [, 4] b. (,0) U (,) c. (, 4] U [, ) 4 5. y 6. 7. 8. sin cos, sec tan, cot + π 5π 0,,, π 9. Y Y X domain, D: [, ) range, R: [, ) (,-) 0.. ( ) sin, ( ) o ( ( )) sin ( ) f g f g f g F 09 05 / JJ:da 7-f09