houston community college, short answer. write the word or

Houston Community College, MATH 2412 PreCalculus- Review for Test 3 Spring Semester 2017

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Write the first five terms of the geometric sequence.

1) a1 = 3; r = 14 1)

Find the indicated sum.2) Find the sum of the first 40 terms of the arithmetic sequence: 15, 23, 31, 39, . . . 2)

3) Find the sum of the even integers between 35 and 59. 3)

4)5

i = 2(4i - 3) 4)

Use the dot product to determine whether the vectors are parallel, orthogonal, or neither.5) v = 2i + 4j, w = 4i + 8j 5)

6) v = 4i + j, w = i - 4j 6)

Express the repeating decimal as a fraction in lowest terms.

7) 0. 55 =55100

+55

10,000+

551,000,000

+ ... 7)

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.8) v = i + 4j, w = -3i + j 8)

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Find the term indicated in the expansion.9) (x + 2y)12; 5th term 9)

A) 7920x4y8 B) 7920x8y4 C) 3960x4y8 D) 3960x8y5


Find the quotient z1z2

of the complex numbers. Leave answer in polar form.

10) z1 = 5(cos 200° + i sin 200°)z2 = 4(cos 50° + i sin 50°)

10)

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequencewith the given first term, a1, and common difference, d.

11) Find a 24 when a1 = 6 , d = -2 . 11)

1

Solve the problem.12) Jacie is considering a job that offers a monthly starting salary of $4000 and guarantees her

a monthly raise of $130 during her first year on the job. Find the general term of thisarithmetic sequence and her monthly salary at the end of her first year.

12)

13) A deposit of $7000 is made in an account that earns 7% interest compounded quarterly.The balance in the account after n quarters is given by the sequence

an = 7000 1 + 0.074

n n = 1, 2, 3, ...

Find the balance in the account after 8 years.

13)

14) The magnitude and direction of two forces acting on an object are 35 pounds, N45°E, and55 pounds, S30°E, respectively. Find the magnitude, to the nearest hundredth of a pound,and the direction angle, to the nearest tenth of a degree, of the resultant force.

14)


15) An explosion causes debris to rise vertically with an initial velocity of 4 feet per second. Thefunction s(t) = -16t2 + 64t describes the height of the debris above the ground, s(t), in feet, tseconds after the explosion. What is the instantaneous velocity of the debris 3.3 second(s) after theexplosion?

15)

A) -105.6 feet per second B) 105.6 feet per secondC) -41.6 feet per second D) feet per second


16) A basketball player signs a contract with a starting salary of $830,000 per year and anannual increase of 4% beginning in the second year. What will the athlete's salary be, tothe nearest dollar, in the eighth year?

16)

17) As part of her retirement savings plan, Patricia deposited $250 in a bank account duringher first year in the workforce. During each subsequent year, she deposited $30 more thanthe previous year. Find how much she deposited during her twentieth year in theworkforce. Find the total amount deposited in the twenty years.

17)

18) Let vector u have initial point P1 = (0, 2) and terminal point P2 = (-5, 4). Let vector v haveinitial point Q1 = (3, 0) and terminal point Q2 = (-2, 2). u and v have the same direction.Find u and v . Is u = v?

18)


A piecewise function is given. Use the properties of limits to find the indicated limits, or state that the limit does notexist.

19) f(x) = 4x - 3 if x < 1-3x + 4 if x > 1

a. limx 1-

f(x) b. limx 1+

f(x) c. limx 1

f(x)

19)

A) a. -3b. c. does not exist

B) a. b. c.

C) a. b. c. does not exist

D) a. b. -3c. does not exist

2

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit propertiescan be applied.

20) limx 1

3x - 2 20)

A) -1 B) 2 C) does not exist D) 1

21) limx 1

2x - 74x + 5

21)

A) does not exist B) -75 C) -

12 D) -

59

22) limx 0

4 + x - 2x 22)

A) 4 B) 14 C) 1

2 D) 0

23) limx 2

(x3 + 5x2 - 7x + 1) 23)

A) does not exist B) 29 C) 15 D) 0

24) limx 0

x3 + 12x2 - 5x5x 24)

A) -1 B) 0 C) does not exist D) 5


Write the first five terms of the arithmetic sequence.25) a1 = -23; d = 2 25)

26) an = an - 1 -38

; a1 = -18 26)

3


Choose the table which contains the best values of x for finding the requested limit of the given function.27) lim

x 2(x2 + 8x - 2) 27)

A) x 0.9 0.99 0.999 1.001 1.01 1.1f(x)

B) x -1.9 -1.99 -1.999 2.001 2.01 2.1f(x)

C) x -0.9 -0.99 -0.999 1.001 1.01 1.1f(x)

D) x 1.9 1.99 1.999 2.001 2.01 2.1f(x)


Let v be the vector from initial point P1 to terminal point P2. Write v in terms of i and j.28) P1 = (-4, -3); P2 = (5, 3) 28)


Find the derivative of f at x. That is, find f '(x).29) f(x) = x2 + 5x + 20; x = 8 29)

A) 21 B) 7 C) D) 16

4

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit orfunction value does not exist.

30) a. limx 1

f(x) b. f(1) 30)

A) a. limx 1

f(x) = -3

b. f(1) = 1

B) a. limx 1

f(x) = 1

b. f(1) = -3C) a. lim

x 1f(x) = -3

b. f(1) does not exist

D) a. limx 1

f(x) = -3

b. (1) = -3

31) a. limx 1

f(x) b. f(1) 31)

A) a. limx 1

f(x) = 2

b. f(1) = 2

B) a. limx 1

f(x) = 1

b. f(1) = 0C) a. lim

x 1f(x) does not exist

b. f(1) = 2

D) a. limx 1

f(x) = 1

b. f(1) = 2


Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.32) 3 + 12 + 27 + . . . + 75 32)

5

Write the vector v in terms of i and j whose magnitude v and direction angle are given.33) v = 7, = 225° 33)

Use mathematical induction to prove that the statement is true for every positive integer n.34) 10 + 20 + 30 + . . . + 10n = 5n(n + 1) 34)


Find the slope-intercept equation of the tangent line to the graph of f at the given point.35) f(x) = 5x2 + x at (-4, 76) 35)

A) y = 10x B) y = 10x + 1 C) y = -39x - 232 D) y = -39x - 80

Translate the given limit notation into a sentence.36) lim

x 4( x - 2) = 0 36)

A) The limit of x - 4 as x approaches 2 equals the number 0.B) The limit of x - 4 as x approaches 0 equals the number 2.C) The limit of x - 2 as x approaches 4 equals the number 0.D) The limit of x - 2 as x approaches 0 equals the number 4.


Find the unit vector that has the same direction as the vector v.37) v = 5i + 12j 37)

Write the first four terms of the sequence whose general term is given.

38) an =3n

(n + 3)!38)

39) an = 2n - 3 39)

40) an = (-1)n + 1(n + 5) 40)


Find the slope of the tangent line to the graph of f at the given point.41) f(x) = x2 + 5x at (4, 36) 41)

A) 21 B) 13 C) 3 D) 9

Determine for what numbers, if any, the given function is discontinuous.

42) f(x) = x- 4 if x 4x2 -8 if x > 4

42)

A) None B) 0 C) 4 D) -4, 4

6

43) f(x) = -3x2 + 5x 43)

A) None B) 56 C) -

56 D) -3


Find the sum of the infinite geometric series, if it exists.

44) -15 - 5 -53

-59

- . . . 44)

Write a formula for the general term (the nth term) of the arithmetic sequence. Then use the formula for an to find a20,the 20th term of the sequence.

45) 6 , 14 , 22 , 30 , 38 , . . . 45)

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequencewith the given first term, a1, and common ratio, r.

46) Find a10 when a1 = 5, r = -3. 46)

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form.

47) 2 2 (cos 74

+ i sin 74

)5

47)


Evaluate the given binomial coefficient.

48) 117 48)

A) 1 B) 330 C) 7920 D) 165

Write the first three terms in the binomial expansion, expressing the result in simplified form.49) (x + 2) 15 49)

A) x 15 + 30 x 14 + 840 x 13 B) x 15 + 28 x 14 + 420 x 13

C) x 15 + 30 x 14 + 420 x 13 D) x 15 + 28 x 14 + 840 x 13

Complete the table for the function and find the indicated limit.

50) limx 0

x3 - 6x + 8x - 2

x -0.1 -0.01 -0.001 0.001 0.01 0.1f(x)

50)

A) 4.09476; 4.00995; 4.00100; 3.99900; 3.98995; 3.89526limit = 4.0

B) -2.18529; -2.10895; -2.10090; -2.09910; -2.09096; -2.00574limit = -2.10

C) -4.09476; -4.00995; -4.00100; -3.99900; -3.98995; -3.89526limit = -4.0

D) -1.22843; -1.20298; -1.20030; -1.19970; -1.19699; -1.16858limit = -1.20

7


Use the given vectors to find the specified scalar.51) u = 9i - 6j, v = -8i - 9j, w = -10i + 5j; Find u · (v + w). 51)


Use the Binomial Theorem to expand the binomial and express the result in simplified form.52) (x + 5)4 52)

A) x4 + 20x3 + 250x2 + 500x + 625 B) x4 + 5x3 + 150x2 + 250x + 625C) x4 + 20x3 + 150x2 + 20x + 625 D) x4 + 20x3 + 150x2 + 500x + 625

Use the definition of continuity to determine whether f is continuous at a.

53) f(x) = x2 - 5, if x < 03, if x 0

a = -4

53)

A) Continuous B) Not continuous


Find all the complex roots. Write the answer in the indicated form.54) The complex square roots of 144(cos 210° + i sin 210°) (polar form) 54)

55) The complex cube roots of 8 (rectangular form) 55)

Use the formula for the sum of the first n terms of a geometric sequence to solve.56) Find the sum of the first 11 terms of the geometric sequence: -7, -14, -28, -56, -112, . . . . 56)

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.

57)5

i = 14(-2)i 57)

Evaluate the factorial expression.

58) n(n + 2 )!(n + 3 )! 58)

Write the first four terms of the sequence defined by the recursion formula.59) a1 = 6 and an = 3an-1 for n 2 59)

Write a formula for the general term (the nth term) of the geometric sequence.60) 4, 12, 36, 108, 324, . . . 60)

Find projwv.61) v = i - 3j; w = 5i + 12j 61)

Find the specified vector or scalar.62) u = i - 2j, v = -9i + 7j; Find u - v. 62)

8

63) u = -8i + 1j and v = 10i + 1j; Find u + v . 63)

Find the product of the complex numbers. Leave answer in polar form.

64) z1 = 2 cos 3

+ i sin 3

z2 = 5 cos 2

+ i sin 2

64)

Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index ofsummation.

65) 67

+78

+89

+910

+ . . . + 1819 65)

Find the angle between the given vectors. Round to the nearest tenth of a degree.66) u = i - j, v = 4i + 6j 66)

9

Answer KeyTestname: MATH 2412 REVIEW FOR TEST 3 MAR 2017-2ND

1) 3, 34

, 316

, 364

, 3256

2) 68403) 5644) 445) parallel6) orthogonal

7) 59

8) v1 =110

(-3i + j), v2 =1310

i +3910

j

9) B

10) 54

(cos 150° + i sin 150°)

11) -4012) an = 3870 + 130n ; $543013) $12,195.4914) F = 57.04; = -23.6°15) C16) $1,092,22317) $820; $10,70018) u = 29, v = 29; yes19) B20) D21) D22) B23) C24) A25) -23, -21, -19, -17, -15

26) -18

, - 12

, - 78

, - 54

, - 138

27) D28) v = 9i + 6j29) A30) A31) C

32)5

i = 13i2

33) v = -7 2

2i -

7 22

j

10


34) S1: 10?= (5 · 1)(1 + 1)

10?= 5 · 2

10 = 10Sk: 10 + 20 + 30 + . . . + 10k = 5k(k + 1)Sk+1: 10 + 20 + 30 + . . . + 10(k + 1) = 5(k + 1)(k + 2)We work with Sk. Because we assume that Sk is true, we add the next multiple of 10, namely10(k+1), to both sides.10 + 20 + 30 + . . . + 10k + 10(k + 1) = 5k(k + 1) + 10(k + 1)10 + 20 + 30 + . . . + 10(k + 1) = (k + 1)(5k + 10)10 + 20 + 30 + . . . + 10(k + 1) = 5(k + 1)(k + 2)We have shown that if we assume that Sk is true, and we add (10(k+1) to both sides of Sk, then Sk+1 is also true. Bythe principle of mathematical induction, the statement Sn is true for every positive integer n.

35) D36) C

37) u =513

i +1213

j

38) 18

, 340

, 380

, 9560

39) -1, 1, 3, 540) 6, -7, 8, -941) B42) C43) A

44) -452

45) an = 8n - 2; a20 = 15846) -98,41547) -128 + 128i48) B49) C50) C51) -13852) D53) A54) 12(cos 105° + i sin 105°), 12(cos 285° + i sin 285°)55) 2, -1 + 3i, -1 - 3i56) -14,32957) -88

58) nn + 3

59) 6, 18, 54, 16260) an = 4(3)n - 1

61) -155169

i -372169

j

11


62) i - 9j63) 2 2

64) 10 cos 56

+ i sin 56

65)18

k = 6

kk + 1

66) 101.3°

12

houston community college, short answer. write the word or

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