MSLC – Math 1150

Final Exam Review

Disclaimer: This should NOT be used as your only guide for what to study.

1. Use the piece-wise defined function

2 2 if 0( )

2 4 if 3

x xf x

x x

to answer the following:

a. Compute f(0), f(3), f(-1), and f(4).

b. Plot the points you found above and sketch a complete graph of y = f(x).

2. Use your graphing calculator to graph each of the functions below over the interval 2, 2 and approximate

any local extrema. Also, determine the intervals where the functions are increasing and decreasing.

Round your answers to three decimal places where appropriate.

a. 2f x x

b. 2 2

1 1g x x x

c. h x x

d. 1 1k x x x x

3. Determine the average rate of change of the functions between the given values of x.

a. 3 17h x x from 1 to 2x x

b. 2

3 1

xf x

x

from 1 to x x t

c. 1

1g x

x

from 0 to x x a

4. Write the equation of the function F x x transformed in the following ways:

a. shifted 2 units to the left, and shifted up 3 units

b. reflected about the x-axis, then shifted down 3 units

c. shifted 1 unit to the right, and vertically stretched by a factor of 3

d. shifted 1 unit to the left, then reflected about the y-axis

5. Given 3 15 and

4 1f x x g x

x

find:

a. f g

b. g f

c. f f

d. g g

6. Algebraically find the maxima and minima of the functions given.

a. 3 2 5f x x x

b. 2

75 3

x xg x

7. Find the inverse of each function, state the domain and range of both the original function and its inverse.

a. 2 1f x x

b. 3 1g x x

c. 1

3k x

x

d. 5 2

xh x

x

8. Find all real zeros and the multiplicity of those zeros for the given polynomials, then find their y-intercepts

and sketch the graph.

a. 2 1 1 3p x x x x

b. 31

41 3p x x x

c. 2 2

1 3p x x x

9. Find all zeros (both real and complex) of the following polynomials.

a. 5 37P x x x

b. 3 6P x x x

c. 5 3 28 8P x x x x

10. Factor the polynomials into linear and irreducible quadratic factors with real coefficients.

a. 4 3 22 2 2 3P x x x x x

b. 4 2 2 2Q x x x x

11. Factor the polynomials into linear factors with complex coefficients.

a. 4 3 22 2 2 3P x x x x x

b. 4 2 2 2Q x x x x

12. Find a polynomial with integer coefficients having the following properties:

a. Degree: 3 Zeros: 0, i

b. Degree: 3 Zeros: –3, 1 i

13. Find all intercepts and asymptotes, then sketch the graph of each rational function.

a. 2

2

4

xR x

x x

b. 2

2

3

6

x xR x

x x

c. 3

2

4

2 1

xR x

x x

14. Solve the following inequalities. Write your answer using interval notation.

a. 5 1 2 9 3x x

b. 2 1 1

3 2 6x x

c. 2

3 1 0x x

d. 31

xx

x

e. 4 1 17x

f. 2 1 5x

15. Solve the equation.

a. 2 1 2 12 4w w

b. 927 81x x

c. 2ln( ) ln(15 34)r r

d. 4log (2 1) 2x

e. 2

3log 24 4x x

f. 2 4xe

g. 2

6log 6 41 2x x

h. 2 13 3 2x x

i. 2 1 34·5 3x x

16. How much needs to be invested now in order to have $1000 in 5 years given a 4.2% interest rate that is

compounded:

a. daily?

b. continuously?

17. Determine (correct to 3 decimal places) how long it will take for $2000 to double if it’s invested in an

account that gives 6% interest compounded:

a) semiannually?

b) quarterly?

c) monthly?

d) daily?

e) continuously?

18. The frog population in a small pond grows exponentially. The current population is 70 frogs, and the

relative growth rate is 15% per year.

a. Find a function that models the population after t years.

b. Use your equation to find the projected population after 3 years.

Round your answer to the nearest frog.

c. Use your equation to find the number of years required for the frog population

to reach 550 frogs. Round your answer to two decimal places.

19. In the circle pictured below, r is the radius of the circle, is the central angle of the sector,

A is the area of the sector, and s is the length of the arc subtended by the central angle.

Find:

a. A and s if 3 inches, and radians3

r

b. r and s if 2.25 square miles, and 36A

c. A and if 4 meters, and 12.57 metersr s

d. r and A if 13 meters, and radians4

s

20. Use the given information to find the exact value of the six trigonometric functions of the angle in the

picture.

a) 8a , 15b

b) 12a , 13c

c) 2

sin7

d) cot 3

e) 3

cos5

f) csc 5

21. For each part, determine the quadrant lies in and the values of the five remaining trigonometric

functions.

a) csc 65 and cot 0

b) 13

sec and sin 04 3

c) 35

tan and cos 012

22. Find the exact value of each of the following.

a. 1 99

101tan sin

b. 1 4cos cos

3

23. Given triangle ABC with the following properties:

14, 17, 44b c B and C is an obtuse angle

Find the measure of angle C.

Round your answer to 2 decimal places.

24. A team of surveyors have been hired to measure the distance across a canyon. Using a tree at point T on the

opposite site of the canyon as a reference point, they established points A, B, and C and found the following

distances:

AB = 12.25 ft

BC = 6.5 ft

AC = 15 ft

a. Find the measure of angle ABC.

Round your answer to 2 decimal places.

b. Find the distance TC across the

canyon to the nearest foot.

25. Establish the identity:

a. 21 sin

sec tan1 sin

xx x

x

b. sec csc

sin costan cot

c. tan sin 1 cos

tan sin sin

v v v

v v v

d. cot cot 1

cot( )cot cot

x yx y

x y

e. sin sin2 2

x x

f. 1 sin 2 1

1 sec cscsin 2 2

xx x

x

26. Find the exact solution of:

a. 11

sin12

b. tan(165 )

c. 7

cos8

d. tan12

27. Given 2

cot and sin 03

x x find:

a. sin 2x

b. cos2x

c. tan 2x

28. Find all solutions of:

a. 2 cos2 1 0x on the interval 0, 2

b. cos sin 2cos 0x x x

c. 2sin 3 02

x on the interval 0,8

29. Convert to polar coordinates:

a. 2 2

2 2i

b. 3

2 2

33i

30. Use DeMoivre’s Theorem to find 7z in the standard a bi form:

a. 1z i

b. 3 cos sin6 6

z i

31. Given 2 and 3 2 u i j v i j find:

a. u v

b. 3 4u v

c. u v

d. 3 4u v

e. u v

f. the angle (in degrees) between u and v. Round your answer to 3 decimal places.

32. Find the complete solution for each of the following systems of linear equations.

a. 3 2 0

2 8

x y

x y

b.

2 0

3 5 8

2 2 7

x y z

x y z

x y z

33. A boat on a river travels downstream for 20 miles in one hour. The return trip back upstream takes 2.5

hours. How fast does the boat travel in still water, and how fast is the current?

34. A fruit stand sells a box of Strawberries for $7 and a box of Kiwi fruit for $10. If they sold a total of 135

boxes of fruit and had revenue of $1110, how many boxes of each fruit did they sell?

35. From the top of a 250 foot tall lighthouse, the angle of depression to a ship on the water is 27°. How far is

the ship from the lighthouse? Round your answer to two decimal places.

36. Find the equation of the parabola with vertex at the origin and a directrix of 6x .

37. Find the equation of an ellipse with eccentricity of 1

9 and foci at 0, 2

38. Find the transverse axis, vertices, foci, and the equations of the asymptotes of the hyperbola described by

the equation 2 29 4 36x y .

39. Write each of the following sums of a sequence using sigma notation.

a. 2+4+6+…+20

b. 1 1 1 1

1 2 2 3 3 4 999 1000

40. Find the sum of the following:

a. 4

2

1k

k

b. 12

4

10k

c. 20

0

1 2n

n

d. 5

0

37

2

j

j

ANSWERS

1. a) 0 2f ; 3 2f ; b)

1 3f ; 4 4f

2. a. Min: 0,0 ; Increasing: 0, 2 ; Decreasing: 2,0

b. Mins: 1,0 & 1,0 ; Max: 0,1 ; Increasing: 1,0 1, 2 ; Decreasing: 2, 1 0,1

c. Min: (0,0); Increasing: 0, 2 ; Decreasing: 2,0

d. Min: 0.578, 0.385 ; Max: 0.578,0.385 ;

Increasing: 2, 0.578 0.578, 2 ; Decreasing: 0.578,0.578

3. a) 3 b) 1

6 2t c)

1 1

1

a

a a

4. a) 2 3y x b) 3y x c) 3 1y x d) 1y x

5.a) 320 6

4 1

xf g x

x

b)

3

1

4 5 1g f x

x

c) 3 3 5 5f f x x d)

4 1

4 5

xg g x

x

6. a) Min.: 1,2 b) Max.: 5 247

,6 36

7. a) 1 1

2

xf x

Domain ,f x ; Range ,f x

Domain 1 ,f x ; Range 1 ,f x

b) 1 3 1g x x Domain ,g x ; Range ,g x

Domain 1 ,g x ; Range 1 ,g x

c) 1 13k x

x

Domain ,3 3,k x ; Range ,0 0,k x

Domain 1 ,0 0,k x ; Range 1 ,3 3,k x

d) 1 5

5 2

xh x

x

Domain 5 5

, ,2 2

k x

; Range 1 1

, ,2 2

k x

Domain 1 1 1, ,

2 2k x

; Range 1 5 5, ,2 2

k x

8. a) Zeros: 12 (multiplicity:1);

–1 (multiplicity:1);

–3 (multiplicity:1);

y-int: 0, 3

b) Zeros: –1 (multiplicity:3);

3 (multiplicity:1);

y-int: 340,

c) Zeros: –1 (multiplicity:2);

3 (multiplicity:2);

y-int: 0,9

9. a) Zeros: 0; 7i ; 7i b) Zeros: 2; 1 2i ; 1 2i c) Zeros: –2; ; ; 1+ 3; 1 3i i i i

10. a) 21 3 1x x x b) 2 21 2 2x x x

11. a) 1 3x x x i x i b) 2

1 1 1x x i x i

12. a) 3P x x x ; b) 3 2 4 6x x x

13. a) no y-intercept,

x-intercept: 2,0 ;

Horizontal Asymptote: 0y ;

Vertical Asymptotes: 0; 4x x

b) y-intercept: 0,0

x-intercepts: 0,0 & 3,0

Horizontal Asymptote: 1y

Vertical Asymptotes: 2; 3x x

c) y-intercept: 0, 4

x-intercepts: 3 4,0

Slant Asymptote: 1 12 4

y x ;

Vertical Asymptotes: 12

1; x x

14. a) 32

,19

b)

1,3

c) , 3 3, 1 d) 2

, 1 ,03

e)

9, 4

2

f) , 2 3,

15. a. 3

2w b. 27x c. 17r d.

17

2x e. 3x f.

ln 4

2x

g. 5 or 1x x h. ln 2

ln 3x or 0x i.

ln 4 ln 5 3ln 3

ln 3 2ln 5x

16. a. $810.59 b. $810.58

17. a) 11.725 yrs b) 11.639 yrs c) 11.581 yrs d) 11.553 yrs e) 11.552 yrs

18. a. 0.1570 tn t e b. 3 110n c. 13.74t years

19. a. 3.142 inchess 24.712 inA b. 2.676 milesr 1.681 miless

c. (radians) 225.13 mA d. 16.552 metersr 2107.589 mA

20. a. 8

sin17

15

cos17

8

tan15

17

csc8

17

sec15

15

cot8

b. 12

sin13

5

cos13

12

tan5

13

csc12

13

sec5

5

cot12

c. 2

sin7

45

cos7

2

tan45

7

csc2

7

sec45

45

cot2

d. 1

sin10

3

cos10

1

tan3

csc 10 10

sec3

cot 3

e. 4

sin5

3

cos5

4

tan3

5

csc4

5

sec3

3

cot4

f. 1

sin5

2

cos5

1

tan2

csc 5 5

sec2

cot 2

21. a. is in Quadrant III; 1

sin65

8

cos65

1

tan8

csc 65 65

sec8

cot 8

b. is in Quadrant II; 11

sin13

48

cos13

11

tan48

13csc

11

13sec

48

48cot

11

c. is in Quadrant IV; 35

sin37

12

cos37

35

tan12

37csc

35

37sec

12

12cot

35

22. a. 99

20 b.

2

3

23. 122.49C

24. a. 101.84ABC b. 31 feetTC

25. a.

2

2

2

2

2

2

2 2 2

2 2 2

1 sinsec tan

1 sin

1 sin 1 sin·

1 sin 1 sin

1 2sin sin

1 sin

1 2sin sin

cos

1 2sin sin

cos cos cos

sec 2sec tan tan (sec tan )

xx x

x

x xRHS

x x

x xRHS

x

x xRHS

x

x xRHS

x x x

x x x x x x

b.

2 2

2 2

sec cscsin cos

tan cot

1 1

cos sinsin cos

cos sin

sin cos

cos sin

sin cos

cos sin

sin cos cos sin· sin cos

cos sin sin cos

x xx x

x x

x x RHSx x

x x

x x

x x RHSx x

x x

x x x xx x

x x x x

____________________________________________________________________________________

c.

2

2

2

2

2

tan sin 1 cos

tan sin sin

sin

cossin

sincos

sin

cossin sin cos

cos

sin

sin sin cos

sin

1 cos

sin 1 cos

1 cos 1 cos

sin 1 cos

1 cos

sin 1 cos 1 cos

sin sin

v v v

v v v

v

v RHSv

vv

v

v RHSv v v

v

vRHS

v v v

vRHS

v

v vRHS

v v

v vRHS

v

v v v

v v

d.

cot cot 1cot( )

cot cot

cos( )

sin( )

cos cos sin sin

sin cos cos sin

1cos cos sin sin

sin sin

1sin cos cos sin

sin sin

cos cos1

cot cot 1sin sin

cos cos cot cot

sin sin

x yx y

x y

x yRHS

x y

x y x yRHS

x y x y

x y x yy x

RHS

x y x yy x

x y

x yx y

y x x y

y x

____________________________________________________________________________________

e.

sin cos sin cos2 2

1 co

sin

s 0

sin2 2

sin

co

cos2

s

LHS x x

LHS x

x x

x

L

x

HS x

f.

1 sin 2 11 sec csc

sin 2 2

sin 2 1

sin 2 sin 2

11

2sin cos

1 11 csc sec 1 sec csc

2 2

xx x

x

xRHS

x x

RHSx x

x x x x

26. a. 6 2

4

b.

3 3

3 3

c.

2 2

4

d. 2 3

27. a. 12

13 b.

5

13 c.

12

5

28. a. 3 5 11 13

, , ,8 8 8 8

b.

2k

c.

2 4 14 16, , ,

3 3 3 3

29. a. 3

1, or 1,1354

b.

43, or 3,240

3

30. a. 8 8i b. 2187 3 2187

2 2i

31. a. 5 i j b. 6 11 i j c. 26 d. 157 e. 4 f. 60.255

32. a. 4, 6x y b. 17 55 36

, , 23 23 23

x y z

33. Boat’s speed in still water = 14mph; Current’s speed = 6mph

34. 80 boxes of Strawberries; 55 boxes of Kiwi fruit

35. 490.65 feet

36. 2 24y x

37. 2 2

1320 324

x y

38. Transverse axis: the x-axis; Vertices: 2,0 ; Foci: 13,0 ; Asymptotes: 3

2y x

39. a. 10

1

2k

k

b.

999

1

1

1k k k

40. a. 30 b. 90 c. –399 d. 4655

32