MATH 115 FINAL EXAM, FALL 1992

Part I. NO CALCULATORS!  Name:  Score
1. Find the equations of the line satisfying the given conditions  1  11

a) y-int 4, x-int 3 b) Through P(-1,5) and (3,-2)

2

12

3  13
4  14
5  15
6  16
7  17
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2. a) Find the domain of f(x) = \sqrt{4 − x^{2}}. b) Find the range of f(x) = 2 + \sqrt{x}.

9

19

10 20

Tot

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3. a) Find the center and radius of the circle x^{2} + y^{2} + 8x − 10y + 37 = 0.

b) Find the maximum value of f(x) = −3x^{2} − 6x + 8.

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4. a) Let f(x) = \sqrt{x − 3}, g(x) = x^{2} − 6. Find (f ∘ g)(x).

b) let f(x) = 2x^{3} − 1. Find f^{−1}(x).

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5. a) Let f(x) = x^{3} − x^{2} − 10x − 8. List all the possible rational zeros.

b) given that x = −1 is a zero, find the other zeros.

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Part I (cont) NO CALCUALTORS!  Name:

6. a) Simplify: log_{2}(\sqrt{2}/4). b) Simplify: e^{2 ln x}.

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7. a) Solve for x: 4^{x} = 2^{2x+1} b) Solve for x: log_{9} x = −1/2.

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8. a) Compute cos 5π/3 b) Compute tan^{−1}(−\sqrt{3}).

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9. a) Compute sin 15^{o}. b) Compute cos[sin^{−1}(4/5) + tan^{−1}(5/12)].

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10. a) Solve: tan^{2} x = 3, x ∈ [0, 2π). b) Solve: sin 2x = sin x, x ∈ [0, 2π).

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11.

a) Graph y = 2^{−x}.

b) Graph one cycle of y = 2 sin(2x + π/2).

Part II. Calculators allowed  Name:

12. a) Solve for t: Q = Q_{0}e^{−Rt/C}. b) Solve: log(5x + 1) = 2 + log(2x − 3).

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13. a) Compute to 3 decimals: log_{3} 6.

b) Solve: e^{2x} = 9.

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14. a) Compute to 3 decimals: sec 56^{o}12′. b) Compute to 3 decimals: \frac{sin(π/5)}{1 + ln 2}.

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15. An investment of $2000 earns 4.3% annual interest compounded continously.

a) How much is the investment worth after 4

years b) How long does it take for the investment to double

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16. In a triangle ABC, a = 5, b = 7, c = 8

a) Find angle α. b) Find the area

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17. In a triangle ABC, α = 43^{o}, γ = 75^{o}, a = 8.24

a) Find a

b) Find the area

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Extra space

Name:

18. a) Verify that 2 sin^{2} 2t + cos 4t = 1. b) Write a formula for tan(θ/2) in terms of tan θ.

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19. Express in trigonometric form: 1 − \sqrt{3}i. b) Find all solutions to z^{3} = 1

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20. Solve by Gaussian elimination:

5x + 2y − z = −7

3y + z = 17

x − 2y + 2z = −7

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Extra space