Precalculus Final Exam Review

Precalculus Final Exam Review: NON-CALCULATOR section

1 1 cos2 x 1.) Simplify:  2.) Simplify: csc sin 1 csc2 x

3.) Find all solutions in the interval [0,2 ) : csc x + 2 = 0

4.) Find all solutions in the interval [0,2 ) : 2cos3 x  cos 2 x  0

5.) Evaluate: cos 165° (Hint: 165° = 210° - 45° ) 6.) Simplify: cos 6x cos 3x – sin 6x sin 3x

12 3 4  7.) Given sin u  ,   u  and cscv  ,  v   , find cos (u + v) 13 2 7 2

4 8.) Given cos  and sin  0 , find tan 2 5

9.) Find an equation of a line that passes through (5, 1) and is perpendicular to the line 4x – 2y = 5

 x  3, x  3 10.) Given f (x)   find f(-5)  2x  8, x  3

11.) Find the vertex of the parabola: y = x2 – 2x + 8

12.) Find the x- and y-intercepts of: y = 2x2 – 5x – 3

1 13.) Find the vertical asymptote(s): f (x)  (x  2)(2x  3)

2x 2  9 14.) Find the horizontal asymptote(s): f (x)  3x 2 1

15.) The domain of f (x)  5  e x

16.) Convert from rectangular to polar coordinates: x2 + y2 + 3x – 2y = 0

1 17.) Evaluate: 3log 18.) Solve for x: 27x = 243 b b

19.) Find a formula for the nth term of the sequence. (Assume n begins with 1) 2 3 4 5 , , , ,... 1 4 9 16

20.) Find an for the arithmetic sequence with a1 = 3, d = -7, and n = 54 21.) Find the sum of the infinite geometric sequence: 2, 1, 0.5, 0.25, …

22.) Eliminate the parameter and find the corresponding rectangular coordinates. x  4cos , y  3sin

 5  23.) Write the point 3,  in polar coordinates using three different representations.  3 

 7  24.) Convert from polar to rectangular coordinates:  2,   6 

25.) Convert from polar to rectangular coordinates: r cos 2   2sin

3, x  2   3x 2  5  f (x)  lim f (x) lim   26.) If  find x2 27.) Find x  2  5, x  2  2x  3x 1

 2x 2  3x  2  lim   2 28.) Find x2   29.) Find lim x4 x  2  x  2 

 4 3 30.) Find an angle coterminal to   31.) Find the angle supplementary to   3 7

6 32.) Convert to degrees: 33.) Convert to radians: 40° 5

  7  2 3 34.) Give the exact value: csc  35.) Find  if sec   6  3

2 36.) A right triangle has an acute angle  , such that tan  . Find sin 3

  37.) Given u  2i  3 j and v  4i  2 j , find u  2v and calculate its magnitude and direction

38.) Find the quadrant in which  lies if tan  0 and cos  0

   39.) Determine the period of f(x) if f (x)  2cos3x    2 

40.) Determine the amplitude of f(x) if f (x)  2sin4x   

   41.) Describe the horizontal shift to the graph of g(x), given g(x)  3sin2x    4 

42.) Determine the period of the function: f(x) = 4 tan(5x)    3  43.) Evaluate: sinarctan    5  (x  h)2  3(x  h) (x 2  3x) 1 1 44.) Find lim 45.) Simplify:  h0 h 1 sin x 1 sin x

46.) Solve for x: log(5 – x) – log(2x – 6) = 1

(x  3) 2 (y 1) 2 47.) Find the vertices of the hyperbola:  4 16

48.) Find the center of the ellipse: 4x2 + 5y2 + 16x – 10y + 1 = 0

49.) Given u  2i  3 j and w  i  j and v  3u  5w, find the component form of v .

50.) A vector has a magnitude of 3 and a direction of   240 °. Find the vector.

51.) A vector w has initial point (4, 6) and terminal point (2, -5). Find the component form of the vector.

52.) Determine the magnitude of v : v   3,6

 3  53.) Solve x  5  10 54.) Plot the point whose polar coordinates are  4,   4 

x 2 55.) Graph the rational function f (x)  x  3

56.) Graph f(x) = 4 + log x 57.) Graph f(x) = log (x + 4)

58.) Sketch the graph: f(x) = 2 + sec 4x 60.) Sketch the graph: f(x) = -3 sin (2x)

   61.) Graph and write the equation for the vertical asymptotes of y  tan2x    4 

(x 1) 2 (y  3) 62.) Sketch a graph of f(x) = 3 – ex 63.) Sketch a graph of   1 9 4 Precalculus Review: Calculator Active

1.) Given a triangle with a = 42, b = 10, and A = 94°, find C.

2.) Find the number of years required for a $3500 investment to triple at a 7% interest rate compounded continuously.

3.) The sun is 23° above the horizon. Find the length of a shadow cast by a flagpole 17 feet tall.

4.) Find the direction of v if v   3,6 .

5.) A triangle has b = 20, c = 28 and C = 50°. Find the area of the triangle.

6.) A triangle has a = 50.2 cm, b = 29.7 cm, and c = 63 cm. Find the area.

7.) Ship A is 60 miles from a lighthouse on shore. Its bearing from the lighthouse is S 17° W. Ship B is 74 miles from the same lighthouse with a bearing of S 48° W. Find the number of miles between the ships.

8.) Solve 2x2 – 3x – 7 < 0

9.) Find all exact, real solutions of 4x3 – 38x – 6 = 0.