(Exercises for Chapter 0: Preliminary Topics) E.0.1 CHAPTER 0: Preliminary Topics (A) means refer to Part A, (B) means refer to Part B, etc. (Calculator) means use a calculator. Otherwise, do not use a calculator. SECTION 0.1: SETS OF NUMBERS 1) Which of the following numbers are integers? Rational numbers? Real numbers? (B) π 1 5 7 5 7.13 ) Rewrite 1 in plain English. Is this statement true or false? (B, C) 3) Rewrite π in plain English. Is this statement true or false? (B, C) 4) What do the symbols,, and denote? (C) 5) Fill out the table below so that each row describes a particular set of real numbers. (E) Set-builder Form Graphical Form Interval Form 1, ( { x x < 0} 6) The set \ { } is written in set-difference form. Write the set in set-builder form, graphical form, and interval form. (E) 7) Simplify: a) (, 4 (, ). b) [ 3, 7) ( 4, 9]. (E) 8) ADDITIONAL PROBLEM. (Russell s Paradox). Let T = sets S S S { }. We say that T is not a well-defined set. Explain why the definition of T is paradoxical.
(Exercises for Chapter 0: Preliminary Topics) E.0. SECTION 0.: LOGIC 1) Consider the following (true) if-then statement: If Arnold is a Californian, then Arnold is an American. (B, D) a) Write the converse of this statement. Is it true or false? If it is false, give a counterexample. b) Write the inverse of this statement. Is it true or false? If it is false, give a counterexample. c) Write the contrapositive of this statement. Is it true or false? If it is false, give a counterexample. d) Which of the following is logically equivalent to the given statement? Its converse, its inverse, or its contrapositive? ) Yes or No: If Arnold is a Californian, is that a sufficient condition for Arnold to be an American? (E) 3) Yes or No: If Arnold is a Californian, is that a necessary condition for Arnold to be an American? (E) 4) True or False: x = 9 x = 3. (C) 1) (B). Consider the number: 4.39716 SECTION 0.3: ROUNDING a) Round this number off to the nearest tenth. b) Round this number off to the nearest hundredth. c) Round this number off to three decimal places. d) Round this number off to six significant digits (or figures). ) (B). Fill out the table below. Decimal 7.583 0.00510 600.000 Number of Decimal Places Number of Significant Digits (or Figures)
(Exercises for Chapter 0: Preliminary Topics) E.0.3 SECTION 0.4: ABSOLUTE VALUE AND DISTANCE 1) (B). a) Find π. b) Find 0. c) Find 7.5. ) Write the definition of a. (B) 3) Express the distance between x and a on the real number line using absolute value notation. (D) SECTION 0.5: EXPONENTS AND RADICALS: LAWS AND FORMS 1) Simplify the following expressions. Write not real if the expression is not real-valued. (B, C) 3 a) 49 ; b) 64 ; c) 7 ; d) 64 ; e) ) Simplify the following expressions. (C) a) ( x + 4) 3 b) ( x + 4) 3 9 3 + 3) Disprove the crossed statement x + y = x + y by finding a counterexample. Hint: Use x = 1 and y = 4. (Section 0.: B)
(Exercises for Chapter 0: Preliminary Topics) E.0.4 4) Rewrite the following expressions in power form. (B, C, D) a) 3 y, where y 0 b) 4 ( ), where x 0 3 5 x 5) Simplify the following expressions. You may not have nonpositive exponents in your final expression. (B-E) 5a a 4 a) a 0 a 9 b) ( 4 x ) 3 3 x x ( ) 3 radical forms. ( c) 54n 5 ) n 3 5 n, where a 0, where x > 0. Write your final expression in both power and SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS 1) 7x 4 1 3 x + x π is a polynomial in x. (B) a) What is its degree? b) What is its leading coefficient? c) What is its leading term? d) Identify a 0, a 1, a, a 3, and a 4. ) Write down a quadratic binomial in x with leading coefficient 6 and a 1 = 1. (B, C, D)
(Exercises for Chapter 0: Preliminary Topics) E.0.5 3) Rewrite ( 3x 4) in general form. That is, perform the square and write the result in descending powers of x. (E) 4) Is 4x a polynomial in x? Why or why not? Is it a rational expression in x? Why or why not? (F) 3 5) Is x a rational expression in x? Is it an algebraic expression in x? (F) SECTION 0.7: FACTORING POLYNOMIALS 1) Factor each polynomial below over, the set of integers. If it is prime (over ), write Prime. (B, C, D) a) 15x 10 + 5x 6 b) x 7 3x 5 c) x + 9 d) a 4 1 e) 4x 3x 60 f) 4x 1x + 9. Hint: Find the discriminant first. g) 6x + 7x 3 h) 3r r +. Hint: Find the discriminant first. i) x 3 + 4x 3x 1 j) 8x 3 + y 3 k) 8x 3 y 3 l) a 4 + 18a ) Yes or No: Is ( x + x) equivalent to x( x + 1)? Why or why not? (B) 3) Factor x 4 + x 6 over the integers ( ). (E)
(Exercises for Chapter 0: Preliminary Topics) E.0.6 SECTION 0.8: FACTORING RATIONAL AND ALGEBRAIC EXPRESSIONS 1) Factor each expression below over, the set of integers. (B) a) x 5 + 4x 3 + x 1 b) x 1/ x 5/ c) ( x + 3) 3/4 + ( x + 3) 5/4 SECTION 0.9: SIMPLIFYING ALGEBRAIC EXPRESSIONS 1) Simplify each expression below. a) 6a 6 9a 8 + 1a 7. (B) b) t3 + 7 t + 3. (B) c) d) e) f) g) 16 x. (B, C) x x 1 ( 5x + 1) ( ) ( x) ( 5x + 1) 10x ( 5x + 1) ( ). (B, D) ( 4r 3) ( 15r ) 5r 3 1 ( 4r 3 ( ) 1/ 4). (B, D) ( 4r 3) ( x 3 + 5) /3 ( 8x) ( 4x 3) ( 3 x3 + 5) 1/3 ( 6x ) ( ) /3 x 3 + 5. (B, D) x 3, if x < 3. (Section 0.4: B; Section 0.9: B, C) x 5x + 6
(Exercises for Chapter 0: Preliminary Topics) E.0.7 ) Rationalize the denominator: 1 x + a. Ignore restrictions. (E) 3) Rationalize the numerator: 4 9 + x x 7. (B, C, E) 4) For each pair of expressions below, write Yes if they are equivalent, and write No if they are not. (F) a) Is ( x + 3) equivalent to x + 3? b) Is ( a 4) equivalent to a 4? c) Is 3 x 3 equivalent to x? d) Is x + y equivalent to x + y? e) Is r + 5 equivalent to r + 5? f) Is ( a + b) equivalent to a + b? g) Is 3x + 3y equivalent to 3 x + y? h) Is ( 3x + 3y) 1/ equivalent to 3( x + y) 1/? i) Is j) Is x x 3 + x equivalent to 1 x + x? x 3 x 3 + x equivalent to 1 1+ x? k) Is 1 x + 1 y equivalent to 1 x + y? l) Is x y + z m) Is n) Is equivalent to x y + z 1 x + 1 equivalent to x 1 + 1? equivalent to 6x 1 ( x 0)? 3x?
(Exercises for Chapter 0: Preliminary Topics) E.0.8 SECTION 0.10: MORE ALGEBRAIC MANIPULATIONS 1) Fill in the boxes below with appropriate, simplified numbers: 7 x x + 4x 5 + 1x 9 = x x + + x 6x 5 ) Fill in the boxes below with appropriate, simplified numbers: 4x 9 16y = x y 3) Fill in the box below with an appropriate, simplified expression: x x + 9 = 1 4) Fill in the box below with an appropriate, simplified number: ( 1x 7) 3 = ( 1x 7) 3 ( 1)
(Exercises for Chapter 0: Preliminary Topics) E.0.9 SECTION 0.11: SOLVING EQUATIONS 1) Solve the following equations for x by giving the appropriate solution set. (B) a) 3x 7 = 8 b) 5x 3x = x c) 6x + 3 = 4x + x ) Solve the quadratic equation x 4x = 1 using (C) a) the Quadratic Formula. b) the Factoring Method. c) the Completing the Square (CTS) Method. 3) Solve the quadratic equation 3x 11 = 70 using the Square Root Method. (C) 4) Solve the quadratic equation 3x + 6x 5 = 0 using the Quadratic Formula. (C) 5) Solve the rational equation x 3 + 4 x = 0. (D) 6) Solve the rational equation x x x +1 = x +1. (D) 7) Solve the radical equation x + 3 = x 5. (E) 8) Solve the absolute value equation x + 4 = 3, and interpret your solution(s) in terms of distances along the real number line. (F) 9) Solve the absolute value equation x 4x + 3 =. Hint: The solution is short. (F)
(Exercises for Chapter 0: Preliminary Topics) E.0.10 SECTION 0.1: SOLVING INEQUALITIES 1) Solve the following inequalities. If the solution set is empty, write. If not, write the solution set in set-builder form (or write ), graphical form, and interval form. (B, C) a) Solve x < 5x + 1. b) Solve x + < x +1. c) Solve x 5, and interpret your solution(s) in terms of distances along the real number line. d) Solve x > 4, and interpret your solution(s) in terms of distances along the real number line. e) Solve x > 4. SECTION 0.13: THE CARTESIAN PLANE and CIRCLES We consider the usual Cartesian xy-plane. 1) Consider the points P 5, 1 scaled in meters. (C) ( ) and Q, 5 ( ). Assume that the coordinate axes are a) Find the distance between P and Q. That is, find the length of PQ, the line segment with endpoints P and Q. b) Find the midpoint of PQ. ) Write the standard form of the equation of the circle with center ( 0, 0) and radius 5. Graph this circle in the xy-plane. (D) 3) Write the standard form of the equation of the circle with center ( 4, 3) and radius 7. Graph this circle in the xy-plane. (D) 4) A circle has as its equation: x + y + x 8y 19 = 0. Find the standard form of this equation, and identify the center and the radius of the circle. Graph this circle in the xy-plane. (D) 5) The points P 5, 1 ( ) and Q(, 5) are the endpoints of a diameter of a circle. Find the standard form of the equation of the circle. Hint: Use Exercise 1. (C, D)
We consider the usual Cartesian xy-plane. (Exercises for Chapter 0: Preliminary Topics) E.0.11 SECTION 0.14: LINES 1) A line has slope and passes through the point ( 3,1). Find two other points on this line. Graph this line in the xy-plane. (C) ) Find an equation of the line with slope 0 that passes through the point ( 3,1). (D) 3) Find an equation of the line with undefined slope that passes through the point 3,1 ( ). (D) 4) Find a Point-Slope Form for the equation of the line passing through the points 3,1 ( ) and ( 7, 1). (E) 5) Find the Slope-Intercept Form of the equation of the line in Exercise 4. What is the y-intercept of the line? (E) 6) Find the Slope-Intercept Form of the equation of the line that passes through the points ( 4, ) and ( 5, 4). (E) 7) What is the slope of any line parallel to the line y = 3 x 5? What is the slope of 4 any line perpendicular to the line y = 3 x 5? (F) 4 8) Find a Point-Slope Form for the equation of the line that passes through the point, 3 ( ) and that is perpendicular to the line y = 7x +. (E, F) 9) Consider the line with equation 4x + 3y = 4. (G) a) Find the x-intercept of the line. b) Find the y-intercept of the line. c) Find the Two-Intercept Form of the equation of the line. (E, G)
(Exercises for Chapter 0: Preliminary Topics) E.0.1. SECTION 0.15: PLANE AND SOLID GEOMETRY KNOW ALL OF THE FORMULAS LISTED IN THIS SECTION. 1) Derive the formula for the area of a trapezoid with bases b 1 and b and height h by breaking it up into a pair of triangles. (B) ) A right circular cylinder has base radius 5 inches and height 3 inches. (C) a) What is the volume of the cylinder? b) What is the lateral surface area of the cylinder? c) What is the total surface area of the cylinder? d) What is the volume of the right circular cone that has the same base radius and height? (This cone fits snugly within the cylinder.) 3) The volume of a sphere is 100 cubic meters. (Calculator) (C) a) What is the radius of the sphere? Give an exact answer and an approximate answer rounded off to four significant digits (or figures). b) What is the surface area of the sphere rounded off to four significant digits (or figures)? SECTION 0.16: VARIATION 1) a is jointly proportional to (or varies jointly as ) b and the square root of c. Find the particular mathematical model related to this statement if a is 48 when b is 4 and c is 9. (B) ) w is directly proportional to (or varies directly as ) the square of m and is inversely proportional to (or varies inversely as ) the cube of n. Find the particular mathematical model related to this statement if w is 7 when m is 5 and n is 1. (B) 3) If you give a flat 0% tip whenever you buy food at a restaurant, are your tips directly or inversely proportional to your bills at the restaurant? (Ignore the issue of taxes.) (B)