Semester 2 Final Exam REVIEW Name:_______________________________________

Pre-Calculus

Simplify the expression.

1.

2.

3.

4. 5. ( ) 6.

Verify the identity.

7.

8.

9.

10.

Chapter 7 – Analytic Trigonometry

Use an addition or subtraction formula to find the EXACT value of the expression.

11. 12. 13.

Find and from the given information.

14.

; 15. x in quadrant II

Use a half-angle formula to find the EXACT value of the expression.

16. 17. 18.

Find the EXACT value of each expression, if it is defined.

19.

20. (

√

) 21.

22. ( √

) 23. (

) 24. (

)

Find all solutions of the equation. Work and answers must be in radians.

25. √ 26.

27. 28.

Find all solutions of the equation in the interval ).

29. 30.

0

Graph each point and label them accordingly. Then find the rectangular coordinates of each point.

1. (

) 2. (

)

3. (

) 4. (

)

A point P(r, θ) is given in polar coordinates. Give two other polar representations of the point, one with

r < 0 and one with r > 0.

5. (

) 6. ( )

Convert the rectangular coordinates to polar coordinates with r > 0 and .

7. ( √ ) 8. ( √ √ )

Chapter 8 – Polar Coordinates & Vectors

Match the equation with its graph. Name each shape.

9.

10. 11.

12. 13. 14.

A. B. C.

D. E. F.

A complex number is given. Find the modulus and then write the complex number in polar form.

15. 16. 17. √

Find the product and the quotient

. Express your answer in polar form.

18. (

) (

)

Find the indicated power using DeMoivre’s Theorem. Write your answer in complex number form.

19. ( √ ) 20. ( √ )

Express the vector with initial point P and terminal point Q in component form.

21. ( ) ( ) 22. ( ) ( )

Find u + v, -3u + 5v, | |, and | |.

23. ⟨ ⟩ ⟨ ⟩ 24.

25. Find the vector with | | and .

26. Find the magnitude and direction of the vector .

Find (a) (dot product) and (b) the angle between u and v to the nearest degree.

27. ⟨ ⟩ ⟨ ⟩ 28. √ √

29. Determine whether and are orthogonal.

30. Given , find ( ).

31. Find the work done by the force in moving an object from P(0, 10) to Q(5, 25).

32. A constant force ⟨ ⟩ moves an object along a straight line from point (2, 5) to the point (11,

13). Find the work done if the distance is measured in feet and the force is measured in pounds.

Graph the ellipse and identify the center, vertices, and foci.

1.

2.

Center:___________ Center:___________

Vert:__________ Vert:__________

Foci:________ Foci:________

3. ( )

( )

4. ( ) ( )

Center:___________ Center:___________

Vert:__________ Vert:__________

Foci:________ Foci:________

Find the standard form of the equation of each ellipse.

5. Foci (0, 3), vertices (0, 4) 6. Major axis vertical with length 20;

length of minor axis 10; center: (2, -3)

7. Foci ( 5, 0), length of major axis 12 8. Endpoints of major axis: (7, 9) & (7, 3)

Endpoints of minor axis: (5, 6) & (9, 6)

9. 10.

Chapter 10 - Conics

Convert the equation to standard form by completing the square.

11.

Graph the hyperbola and identify the center, vertices, asymptotes, and foci.

12.

13.

Center: ___________ Center: ___________

Vertices:___________ Vertices:___________

Foci:__________ Foci:__________

Asymptotes:________ Asymptotes:________

14. ( ) ( ) 15. ( )

( )

Center:____________ Center: ____________

Vertices:___________ Vertices:___________

Foci:__________ Foci:__________

Asymptotes:________ Asymptotes:________

Find the standard form of the equation of each hyperbola.

16. Foci (0, ), vertices (0, ) 17. Vertices ( 4, 0), Asymptotes:

18. Endpoints of transverse axis: (0, ) 19. Foci (0, 1), length of transverse axis 1

Asymptotes: y = x

Convert the equation to standard form by completing the square.

20.

Graph the parabola and identify the vertex, directrix, and focus.

21. 22.

Vertex: _______ Vertex: _______

Dir: _______ Dir: ____________

Focus:___________ Focus:_________

23. ( ) ( ) 24. ( )

Vertex: _______ Vertex: _______

Dir: ___________ Dir: __________

Focus: ________ Focus:________

Write an equation in standard form for the parabola satisfying the given conditions.

25. Focus: (8, 0); Directrix: x = -8 26. Vertex: (2, -3); Focus (2, -5)

Find the equation for the parabola whose graph is shown.

27. 28.

Convert the equation to standard form by completing the square.

29.

Find the first five terms of the recursively defined sequence.

1.

; 2. ;

3. Find the sum:

4

1

2

k

k 4. Write the sum using sigma notation:

Determine whether the sequence is arithmetic or geometric. Identify the common difference or the

common ratio.

5.

6. 2, 4, 6, 8, …

7. Determine the common difference, the fifth term, the nth term, and the 100th term of the

arithmetic sequence -12, -8, -4, 0, …

8. The 12th term of an arithmetic sequence is 32, and the fifth term is 18. Find the 20th term.

Chapter 11 – Sequences & Series

9. Which term of the arithmetic sequence 1, 4, 7, … is 88?

10. Find the partial sum of the arithmetic sequence that has .

A partial sum of an arithmetic sequence is given. Find the sum.

11. (

) 12.

20

0

21n

n

13. An arithmetic sequence has first term and common difference . How many terms of

this sequence must be added to get 2700?

14. Determine the common ratio, the fifth term, and the nth term of the geometric sequence

15. The first term of a geometric sequence is 3, and the third term is

. Find the fifth term.

16. Which term of the geometric sequence 2, 6, 18, … is 118,098?

17. Find the partial sum of the geometric sequence 1 + 3 + 9 + + 2187.

18. Find the sum of the infinite geometric series

19. Express as a fraction.

Use the Binomial Theorem to expand.

21. ( )

22. ( )

23. Find the eleventh term in the expansion of ( ) .

24. Find the term containing in the expansion of (√ )

.

1. For the function g whose graph is given, state the value of the given quantity, if it exists.

a) ( ) b) ( ) c) ( )

d) ( ) e) ( ) f) ( )

g) g(2) h) ( ) i) g(0)

Find the limit algebraically.

2. 3. 4.

5. 6. 7.

8. Evaluate the limits using the function below.

( ) {

a) ( ) b) ( ) c) ( ) d) ( ) e) ( )

Chapter 12 – Limits

𝑥

(𝑥 )(𝑥 𝑥) 𝑢

𝑢 𝑢 𝑥

𝑥 𝑥

𝑥 𝑥

ℎ

√

𝑥

𝑥

𝑥

𝑥

(𝑥 )

𝑥

Find an equation of the tangent line to the curve at the given point. Use:

9. ( ) at (1, 1) 10. ( )

at (-1, 1)

Find the derivative of the function at the given number. Use

11. ( ) at -1 12. ( )

at 3

Use the power rule to find the derivative of the function ( ). Then find ( ).

13. ( ) 14. ( )

15. ( ) √

𝑥 𝑎

𝑓(𝑥) 𝑓(𝑎)

𝑥 𝑎

ℎ

𝑓(𝑎 ) 𝑓(𝑎)