The purpose of this review sheet is to give an overview of what we did this quarter, and to remind you of what you are supposed to know for the final. It’s not enough to just study from this sheet! Go over your notes, the book, the homework, the worksheets and the chapter tests.
These problems are intended to help you study for the final. However, you shouldn’t assume that each problem on this handout corresponds to a problem on the final. Nor should you assume that if a topic doesn’t appear here, it won’t appear on the final.
The final exam is on Monday (Mar 19) from 10:30 – 12:30 in this room. You are
allowed one 8 ½ inch by 11 inch piece of paper on which you can put any notes you want to.
1. in quadrant III; find the exact value of each of the remaining trig functions
2. Graph the function:
3. Graph the function:
4. Find the amplitude, period, and phase shift of
5. A race car is driven around a circular track at a constant speed of 180 miles per hour. If the diameter of the track is mile, what is the angular speed of the car? Express your answer in revolutions per hour.
6. Find the exact value of
7. Find the inverse function of the function on the interval
8. Establish the identity:
9. Find the exact value of:
10. Find if
11. Solve on the interval 12. Solve on the interval Solve the following triangles:
13.
14.
15.
17. Two observers simultaneously measure the angle of elevation of a helicopter. One angle is measured as 25o, the other as 40o. If the observers are 100 feet apart and the helicopter lies over the line joining them, how high is the helicopter? 18. Write the polar equation in rectangular coordinates
(x, y). Identify the type of figure represented by the equation and graph it. 19. Sketch a graph of the polar equation
20. Write the complex number in the standard form
21. Find and for
22. Write the expression in the standard form
23. Find all the complex cube roots of 27.
24. The vector v is represented by the directed line segment . Write v in the form ai + bj.
25. Find the distance from to 26. Find the dot product and the angle between v and w
27. Decompose v into two vectors, one parallel to w and the other perpendicular to w
28. Find the direction angles of the vector
29. A swimmer can maintain a constant speed of 5 miles per hour. If the swimmer heads directly across a river that has a current moving at the rate of 2 miles per hour, what is the actual speed of the swimmer. If the river is 1 mile wide, how far downstream will the swimmer end up from the point directly across the river from the starting point?
30. For the equation , identify the conic section without completing the square and without applying a rotation of axes.
31. For the equation , rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.
32. Identify the conic that the polar equation represents and graph it. 33. Graph the curve whose parametric equations are .
Find the rectangular equation of the curve.
1. in quadrant III;
; must be negative because is in Quardant
III. ;
2. . The graph of is shifted units to the right, stretched vertically by a factor of 3 and shifted up by 2 units.
3. The graph of is stretched horizontally by a factor of
2, shifted left units, and stretched vertically be a factor of 4.
4. Amplitude: ; Period:
5. mile; mile
rad/hr rev/hr
6. follows the form of the equation but
we cannot use the formula directly since is not in the interval . We
need to find an angle in the interval for which . The angle
is in quadrant II so tangent is negative. The reference angle of is and we want to be in quadrant IV so tangent will still be negative. Thus, we
have . Since is in the interval , we can apply
the equation above and get .
7.
The domain of equals the range of and is . To find the domain of , note that the argument of the inverse cosine function is and that it must lie in the interval .
That is, or , this is the domain
of the inverse function and the range of .
9.
10. (82)
11.
k any integer. On the interval the solution set is
12.
or on the
interval the solution set is
Solve the following triangles:
13. ;
14.
or
For both values , therefore there are two triangles.
Two triangles: or
;
16.
17.
H x
A B
Let h = height of the helicopter, x = the distance from observer A to the helicopter, and the angle
feet high
18.
Graph is a circle with center (4, -2) and radius 5.
19.
Check for symmetry:
Polar axis: Replace by . The result is The test fails.
. The graph is symmetric with respect to the line
The pole: Replace r by , . The test fails
Due to symmetry with respect to the line assign values of from
0
3 2 1 0 -1
20.
21.
23. 27
The three complex cube roots of are:
24.
25. to
26.
The decomposition of v into 2 vectors and so that is parallel to w and
is perpendicular to w is given by: and
28.
29. Let the positive x-axis point downstream, so that the velocity of the current is Let the velocity of the swimmer and the velocity of the swimmer relative to the land. Then The speed of the swimmer is
, and the heading is directly across the river, so . Then
mi/hr
The angle downstream that the swimmer goes is
Since the river is 1 mile wide, the distance the swimmer travels is miles
The time it takes to swim this distance is hours The swimmer will end up miles downstream
30. ,
hyperbola
Hyperbola: center at (0, 0), transverse axis is the axis, Vertices at foci at
Asymptotes:
32.
Ellipse; directrix is perpendicular to the polar axis 1 unit to the right of the pole;
