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Unformatted text preview: BAvv 45” ‘3 MATH 3 EARLY FINAL 1217/2004 lnstructor: Frank Béiuerle, Ph.D. No books, graphing utilities or notes allowed. Show your
work. Show your work. Show your work. YourName:
Your TA:
Max Your score Problem 1: 10
Problem 2: .10
Problem 3: 15
Problem 4: 10
Problem 5: 10
Problem 6: 10
Problem 7: 10
Problem 8: 20
Problem 9' 10 Problem 10: 20
Problem 11: 15
Problem 12: 20
Problem 13: 15
Problem 14: 10
Problem 15: 10
Extra Cr 16: 10 TOTAL: 205 Relax, good luck, have a great winter break. 1. ( 10 points) Use the reference angle principle to get the exact value of the following: (a) sin 225° on com 3 ) 2. (10 points) Give a table that gives the exact values AND decimal approximations, rounded
to the nearest hundredth, for sin 0,0080! and tana for a z 0°,a = 30°, 0: = 45°,a = 60°
and a = 90°. Also give the radian measures of all the angles. 1 .
3. (15 points) Consider the function f(;1:) = 2 005(22: + Find the followmg: (a) (1 points) The amplitude of f.
(b) (2 points) The period of f.
(c) (2 points) The phase shift of f. (d) (5 points) The graph of f. . 1 . .
(e) (4 pomts) Now graph 9(2) 2 2 sec(2$ + Into the same coordlnate system above. 4. (10 points) Give the exact values of the following by using (and drawing !) the appropriate
triangles: (a) oosIsin*‘(—§)1 (b) sin [arctan(2)] 5. (10 points) Graph f = arctan($) + 1. Make sure to label all asymptotes. (Extra Credit, 2 points) Use your graph to graphically ﬁnd/ estimate the solution(s) of the
equation arctan(x) + 1 == 1. What are the exact values of all solutions '2' d1 6. (10 points) Find all Solutions for the following equations: 1
t = ~
(a) an x 2 (h) sinzz+3inz6=O 7. (10 points) A vertical ﬂag pole is on the ﬂat roof of a building. From a. point on the ground
50 feet away from the building, the angles of elevation to the top and the bottom of the ﬂag
are 15° and 12° respectively. Find the height of the ﬂag. 8. (20 points) Consider the function f(:c) = 111(x + 1) (a) (5 points) Show that the function f(x) is one’toone. (You can do this algebraically or
graphically) (b) (5 points) Find f‘1[:c). (c) (10 points) Graph both ﬁx) and f‘1(x) in the same coordinate system. 9. (10 points) 200kg of a radioactive substance with a halflife of 20 years is held in a storage
container. How long must one wait until only 2 kg of the substance is left ? Hint: First use the halflife T to compute k for which you can use the following halflife l
formula: k 2 ——112. T 10. (20 points) Give all real solutions to the following equations.
(a) logso + 1) = 2 (13) 21113:: 1+ln(x+1) (c) 62” + 23” + 1 = 0 Hint: Rewrite eh appropriately to factor as a. trinornial (d) lnlnlnx = 0 11. (gpoints) a) Two numbers add up to 33. Find the maximal value for the product of these two numbers. b) Two nonnegative integers add up to 33. Find the maximal and the minimal value for
the product of these two numbers. 12. (20 points) Let 0 be an angle such that 7r 5 0 5 33“ and sin!) 2 —§. Find the following: (a) (2 points) tanB =
(b) (2 points) cost? =
(c) (2 points) cot 0 =
(d) (2 points) sect? =
(e) (2 points) cscB = (f) (3 POintS) 605(29) = . 3 . . . .
(g) (3 points) Is cos( posﬂuve or negatwe? Explam. (h) (4 points) 3 =
calculator and your brain here) in degrees and radians (you should use your 10 ‘5 ' :62—3:c——4 13. (23 points) For the rational function = m do the following: (a) Find all mintercepts (if any). (1)) Find the y—intercept. (c) Find all vertical asymptotes (if any).
((1) Find the horizontal asymptote. (e) Where does the graph intersect the horizontal asymptote ? (f) Sketch the graph (label the axes, asymptotes and all intercepts): 11 14. (10 points) Let a = 7, a := 40° and b = 9. Find one solution of this triangle. (Extra Credit, 5 points) Find the other solution. 12 15. (10 points) 3) Find an equation of the sine function with the following graph: 1)) Give a reason Why the following graphs. cannot represent a polynomial function with
highest degree term —:c5.  ‘0
13 N 16. (Extra Credit, 10 points) Let a sequence be deﬁned by an“ =an+2,a1 = l,n=l,2,3,... (a) Find the next three elements «12, a3, (14 of this sequence. 0200 (c) Find an eicplicit formula for an. (d) What is the value of the sum of the ﬁrst n elements of the sequence, namely What is
the sum ' 1" K: r+1+s+,,+(n—r)+ n
Kai 14 IT ...
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