# 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.

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

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