Honors Pre Calculus

FINAL EXAM REVIEW PACKET 2018-19

Chapter 4.7, 4.8, 6.1, 6.2, 5, 6.3, 9.1, 9.2, 9.3, 10.7, 10.8, 12

Continuity, Sideways Parabolas, and Factoring

Derivatives: power rule, trig, word problems

1) A ship leaves a port at 1:00 PM traveling at 13 knots directly north. Another ship leaves the same port at 2:00

PM traveling due east at 15 knots. At 9:00 PM, how far apart are the ships? Along what heading will the northern

ship have to travel to reach the eastern ship which is now stopped?

2) An airplane leaves an airport and travels due east for 255 miles. It then heads due south for 330 miles.

From its current position, along what heading (bearing) should it travel to reach the airport and how far is

it?

3) The bearing of a lighthouse from a ship was found to be 𝑁37°𝐸. After the ship sailed 3.5 miles due

south the new bearing was 𝑁25°𝐸. Find the distance between the ship and lighthouse at each location.

4) Find the exact value of each expression

a) cos1 0

b)

2

1sin 1

c) 3tan 1

d) 1tan ( 1)

e) sin11

f) 1 2 3sec

3

g) 1cot 1

h) 2csc 1

i)

2

3sincos 1

j) 1 2tan cos

2

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k)

2

1cossec 1

l)

3

2cotcot 1

m)

3

4coscos 1

n) 24.0sinsin 1

o)

7

3tantan 1

p) 1 5cos cos

3

5) Use a calculator to find each value:

a) 1.6cot 1

b) 4sec 1

c)

7

3sin 1

d) 1.5tan 1

e) cos1 3

5

6) Simplify

a) 2(1 tan )(cos 2 1)y y

b) 3csc (cos tan sin )x x x x

c) 2 2sin (5 ) cos (5 )x x

d) cos cos4 4

x x

e) tan 5 tan 2

1 tan 5 tan 2

x y

x y

f) sin tan cos 2 secx x x x

g) 2

tan sin cos

sin

x x x

x

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h) (sec 1)(1 cos )

sin

x x

x

i) 1 1

sec tan sec tanx x x x

j)

1sin (sin ),2

x x

k) cot (sec cos )A A A

7) Solve for 0 360 .

l) 22cos 3sin 3 0

m) 8 tan 2 5x

n) 4sin(3 ) 1 0x

o) sec( 35 ) 4x

p) sin 2 cosx x

8) Solve for 0 2x :

a) 23sin 2sin 1 0x x

b) 3 csc 7x

c) cos 2 cos 2x x

9) Find the solutions of each equation in the indicated interval.

d) (2cos 3)(tan 3) 0,x x x

e) tan sin tan 0, 2 0x x x x

f) 3sin 3cos 0,90 450x x x

g) sin 2 cos 2 sec 2 , 90 90x x x x

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10) Evaluate without a calculator.

h) 3

sin( ) cos2

x x

i) 3 5 4 11

sec csc tan cot4 4 3 6

j) 1tan ( 3)

k) cos(165 )

l) 1 3

sin cos5

m) 4 4

sin cos cos sin3 3 3 3

n) 5

tan4

where

1tan

3

11) Given that 3

sin5

and 7

cos25

, where 3

22

, find each of the following.

a) cos

b) sin

c) sin( )

d) cos( )

e) 1

tan2

f) sin 2

g) cos( )

h) tan( )

12) Suppose 3 3

cos ,csc 2,0 , 2 ,11 2 2

x y x and y

find sin( ).x y

13) If A is acute and 4

cos5

A , then find sin 4A . (exact value)

Honors Pre Calculus

14) Verify each identity

a) csc sin sin cos 2 2

b) sectancotsin

c) 1cotcsccotcsc

d) csc csc cot cot4 2 4 2

e) csc cotsin

cosx x

x

x

1

f) 8 3 3 52 2 2csc cot csc

g) 1

1

1

12 2

sec seccot

h) coscos

i) sin( )

cot cotsin sin

j) cos( ) cos( ) 2cos cos

k) 2cossin2sin1

l)

2

2

tan1

tan12cos

m)

2cos1

2sec2

15) Consider the sequence 3, 6, 11, 18, 27, …

a) State whether the sequence is arithmetic, geometric, or neither.

b) Find a6.

c) Find a formula for an.

d) Find a100.

16) Find an for the sequence -2, 6, -18, 54, …

17) Express 0.131313131313… as an infinite series and as a rational number.

18) Find S6 as an exact value for the series: 27 -9 +3 -1 +...

19) Find an explicit formula for the sequence that is made up of all 2 digit numbers that are divisible by 3.

20) Find the sum of the first 50 odd integers.

21) Find the sum of all numbers less than 1000 that are divisible by 3.

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22) Find the sum:

∑1

𝑖2 + 1

6

𝑖=2

23) Find the third partial sum and the sum of the series: 1

5

10ii

24) Use sigma notation to write the sum:

a) 1 1 1 1

...3(5) 3(6) 3(7) 3( )n

b) 1 1 1 1 1 1 1

12 6 24 120 720 5040 40320

25) Simplify:

a)

3 !

3 2 !

n

n b)

! 1 !

1 ! 1 !

n n

n n

26) Evaluate each limit or state that it does not exist.

a) 3

20

2limx

x x

x x

b) 2

2lim 3 5x

x

c)

3

3 2

2 5lim

5 7 1x

x x

x x

d) 24

4lim

12x

x

x x

e) 1

1lim

1x x

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f) 0

4 2limx

x

x

g) 3

10lim

3x x

h) 5

5lim

5x

x

x

i) 2

2

3lim

9x

x

x

j) 4 3 2lim 4 5 2 4x

x x x x

k) 2

6lim

2x x

Honors Pre Calculus

27)

Honors Pre Calculus

28) Justin Bieber goes skydiving. He jumps from a plane at an altitude of 20,000 feet.

a) Find his velocity after 6 seconds.

b) If Justin is supposed to release the chute 10 seconds before ‘would be impact’ when should he

release the chute?

c) How fast is he falling when the chute is released?

29) A marshmallow was thrown upward from the top of a 300 foot building with an initial velocity of 30

ft/ sec.

a) Find the velocity of the marshmallow at 2 seconds.

Selena Gomez lives on the 4th floor of the building (60 feet above ground) and just as she was sticking

her head out to wave to Taylor Swift, the marshmallow hit her.

b) How long was the marshmallow in motion before it hit Selena?

c) What was the velocity of the marshmallow when it hit Selena?

30) Use the definition of the derivative to find the derivative of:

a) 2( ) 3 4 7f x x x b)

1( )

5f x

x

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31) Find the derivative:

a) 𝑦 = 2(cos 𝑥) + 𝑡𝑎𝑛𝑥

b) 𝑔(𝑥) =√𝑥

4

c) 𝑠(𝑡) = 2𝑡5 − 6𝑡3 + 5𝑡2 − 10

32) Find the equation of the tangent line (in slope-intercept form) to the curve 3 2( ) 2 4 5 9f x x x x when 1x .

33) Find the area between the curve 2( ) 2 11f x x x and the x-axis on the interval [-1,3] using:

a) 4 left endpoint rectangles

b) 4 right endpoint rectangles

c) 4 midpoint rectangles

d) the limit process

34) Use the formal definition of continuity to determine if ( )f x is continuous.

2

2 4 2

( ) 2 0

sin 0

x x

f x x x

x x

Honors Pre Calculus

35) Sketch a curve the satisfies the following:

36) Plot each point.

Find two additional polar representations of the point. Convert each point to rectangular coordinates.

a) 5,3 b) 3

2,4

36) Covert each point to polar coordinates.

a) (-1, -1) b) 2, 2 3

37) Name four different polar coordinates for point A. Convert to radians!

38) Convert to polar form:

𝑥2 + 𝑦2 − 2𝑎𝑥 = 0

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39) Convert to rectangular form:

𝑟 =2

1 + 𝑠𝑖𝑛𝜃

40) Graph the polar equation.

a) 3r b) 3

41) Graph each parabola. Use three points and identify the axis of symmetry.

42) Factor and simplify:

a)

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b)