(Revision 15) Assignment 1 Complete this assignment after you have nished Unit 1, and submit your work to your tutor for grading. This assignment has one bonus question. There is no penalty if you do not attempt it but you may be rewarded if you do. The maximum grade you can obtain in this assignment is 100%. Total points: 100 Weight: 5% (4 points) 1. Use the prime decomposition of integers to express the listed radical in its minimal expression. Do not use decimals. a. m b. m c. Wm d. M+M (8 points) 2. Fill in the table below. Note that you should refer to the section titled "Intervals" on pages 337338 of the textbook and to Table 1 on page 338. Inequality Representation 0n the Real Line (6 points) 3. Give the intervals that correspond to the inequalities listed below. a. All real numbers x so that x 2 2. b. All real numbers x so that 0 3 or 6 S x. Mathematics 265: Introduction to Calculus I l (Revision 15) (9 points) 4. In each of the following exercises, rewrite and simplify the given expression. Give your answer using positive exponents only. 3/2 [ 5x3y22 ] a. 2 xy 2 b. (uvzw3 Buzw)2 (v72) 0. (I x3 + x2y_zz)(xyzz3 (9 points) 5. In each of the following exercises, expand and simplify. a. Jim Jixz +1) J1_8(1 4x)3 b. (r u)2 + 5(3; u + 4u2)(l + u} c. (1 3x + x2)3(2 2x2) (9 points) 6. In each of the following exercises, factor the given expressions. a. 2y3 +6y2 +y+3 b. 3x2 ley + 24y2 c. 50x3 + 20x2 + 2x (12 points) 7. In each of the following exercises, factor and simplify the terms, and then do the indicated operations. 1 12x2 10xy+2y2 a. 93c2 y2 93c2 639/ + y2 Vx2+5x+4 x23x4 b. x2+8x+16 x216 9x3+6x2+x 6xl c. 3 2 27x +1 3x +JC (12 points) 8. Determine which of the equations given below have real solutions and give the solutions of those that do. 2x2 + 3x = 6 b. x4 +6x2 3= 0 Hint. Set a : x2 and express the equation in terms of a. 0. 3x :12x2 5x d. x2+7x=_: 3 2 3x2+3 e. =JC 2 Mathematics 265: Introduction to Calculus I 2 (Revision 15) (9 points) J5 6 Setwg b. Rationalize the denominator of L [2 + x , J6x . . . J8 3 + 5 e. Rationalize the denomlnator of IcyJ; 2yJ}_/ 9. a. Rationalize the numerator of (6 points) 10. Convert from radians to degrees the numbers given below. Note that you should refer to the section titled "Angles" on pages 358359 ofthe textbook. it 371 a. + 5 8 b 773 271 4 (6 points) 11. Convert from degrees to radians for the numbers given below. 3- Fail 0 (10 points) 12. Give the exact value of 771' a. tan [6] b. cos S sin 5 8 8 0. cos2 i (8] Mathematics 265: Introduction to Calculus I 3 (Revision 15) Bonus. (8 points) 13. In Unit 1 of the study guide we defined the trigonometric functions using a right triangle with hypotenuse 1. Use similar triangles to define in any right-angle triangle with hypotenuse z the trigonometric functions as cos 0 = _ sin 0 = 2 tan 0 = _ Hint. The circle below has radius 1. 2 X Mathematics 265: Introduction to Calculus I 4