# MATHEMATICS

Assignment 1

Complete this assignment after you have finished Unit 1, and submit it for grading. Use the assignment drop box to submit your assignment as a single PDF. Do not submit your assignment by email. If you are unable to use the online drop box, make alternative arrangements with your tutor.

This exercise is worth 5 per cent of your final grade.

1. List the following numbers in increasing order:4 pts

√ 13−

√ 5; 4 √ 38− (

√ 3)−1);

√ 2− 61

152 ; 3π −

√ 5

32 ; π.

2. Fill in the table below. Note that you should refer to the section titled “Intervals,”12 pts on pages 337-338 of the textbook, and to Table 1 on page 338.

Interval Inequality Representation on the real line

x < −3

[−3,∞)

1

4 ≤ x < 12

−13

/

(−25, 7]

3. Identify a real number that belongs to the intersection of each of the sets of2 pts intervals given below.

a. [−2.6,−1) and (−4,−7/4]

b. (π 4 , π

2

] , [ −π 3 , π

3

) and

[π 5 , π

3

]

Mathematics 265: Introduction to Calculus I Assignment 1 1

4. In each of the following exercises, rewrite and simplify the given expression.9 pts Give your answer using positive exponents only.

a. (4x2y4)3/2

b. ( x3

y2

)−4 c. (√

x−5 + x−5y7z−2 ) (x9y−2z9)

5. In each of the following exercises, expand and simplify.12 pts

a. 5(3x− 1) + 4(x2 − 3x+ 3)(x+ 6)

b. (t+ 5)2 − (6t+ 8)(7− t)

c. (1 + 2z − 5x)(z + 5x− 7)

d. (1− a+ 2×2)2

6. In each of the following exercises, perform the indicated operations. Give your6 pts answer as a fraction in lowest terms.

a. 2

x+ 1 − 4 x− 1

b. 3

u+ 3 + u− 2

c. 5

x+ 3 +

1

x2 − 9

d. 2√

x2 + 2 − 5 x2 + 2

7. In each of the following exercises, factor the given expression.8 pts

a. 4×2 − 16t2

b. −10y2 + 31y − 15

c. x3 − 4×2 + 5x− 2

d. 27a3 − 64b3

2 Assignment 1 Mathematics 265: Introduction to Calculus I

8. In each of the following exercises, factor and simplify the given expression.15 pts

a. 9a2 + 24ab+ 16b2

9a2 − 16b2

b. x3 − 8

x2 + 2x− 8

c. x2 + 2×2 − 3x 2×3 + 2×2 − 4x

d. x2y − x2

x3 − x3y

e. x2 + 5x+ 4

x2 − 4x− 5

9. Solve each of the quadratic equations below.12 pts

a. a2 − 6a+ 2 = 0

b. 2×2 + 3x = 2

c. 3×2 = x+ 4

d. 25 = 9×2 − 30x

10. a. Rationalize the denominator of √ 5x− 6√ 5x+ 3

.9 pts

b. Rationalize the numerator of √ 2 + y +

√ 2− y

y .

c. Rationalize the denominator of 2 √ 3 + 1√

6− √ 3

.

11. Convert from radians to degrees the numbers given below. Note that you should3 pts refer to the section titled “Angles” on pages 358-359 of the textbook.

a. 5π

6

b. 3π

8

c. −6π 45

Mathematics 265: Introduction to Calculus I Assignment 1 3

12. Convert from degrees to radians the numbers given below.3 pts

a. −270◦

b. 345◦

c. 38◦

13. Give the exact value of6 pts

a. cos ( 11π

4

) .

b. sin ( 7π

6

) .

c. tan ( 5π

3

) .

Bonus Question In Unit 1 of the Study Guide, we defined the trigonometric functions8 pts using a right triangle with hypotenuse 1. Use similar triangles to define, in any right triangle with hypotenuse z, the trigonometric functions as

cos θ = x

z sin θ =

y

z tan θ =

y

x .

Hint: The circle below has radius 1.

4 Assignment 1 Mathematics 265: Introduction to Calculus I