Chapter 7 Analytic Trigonometry€¦ · Chapter 7 – Analytic Trigonometry . Use an addition or...
Transcript of Chapter 7 Analytic Trigonometry€¦ · Chapter 7 – Analytic Trigonometry . Use an addition or...
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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
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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. (
)
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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.
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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
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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. (
) (
)
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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 .
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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.
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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
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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
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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:________
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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.
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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
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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
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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.
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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 (√ )
.
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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
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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)