Math 12 (Spring 2001) Kouba Final Exam

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Your HW/Exam ID Number

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3. YOU WILL NEED A CALCULATOR ON THIS EXAM.

4. No notes, books, or classmates may be used as resources for this exam.

5. Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

6. Put units on answers where units are appropriate.

7. You have until 6:00 p.m. to finish the exam.

8. Make sure that you have 7 pages including the cover page.

9. There is candy at the front table for your pleasure.

1.) (5 pts. each) Assume that 3π/2 < α < 2π and sin α = −3/5. Find exact values for each by drawing an appropriate circle and triangle.

a.) cos α

b.) tan α

c.) tan(2α)

d.) sin(2α)

e.) cos(α/2)

2.) (12 pts.) Determine an equation of the line which passes through the point (−1, 2) and which is parallel to the line y = 3x + 5.

3.) (12 pts.) Use the fact that 75^{o} = 45^{o} + 30^{o} to find the exact value of sin 75^{o}.

4.) (12 pts.) Solve for x : 3^{2x} + 3^{x} = 6 5.) (15 pts.) Find the intercepts, vertex (complete the square), domain, range, and graph for the parabola y = (x − 4)(x + 2).

6.) (12 pts.) Use a sign chart to find the domain for y = ln(9 − x^{2}).

7.) (12 pts.) A rectangle is inscribed in the first quadrant below the line y = 6 − 3x in the diagram. Determine the dimensions x and y and the area A of the rectangle of maximum area .

8.) (12 pts.) Derive the inverse function y = f^{−1}(x) for f(x) = \frac{2 − x}{x + 5}.

9.) (12 pts.) Use a calculator to solve (in radians) 7 tan θ = 30 for 0 ≤ θ ≤ 2π. Round answers to two decimal places.

10.) (12 pts.) Prove that the following equation is an identity.

\frac{cos θ}{sin θ} + \frac{sin θ}{cos θ} = 2 csc 2θ

11.) A wheel of radius 8 feet is spinning at the rate of 5 revolutions per second.

a.) (12 pts.) Find the angular velocity of the wheel in radians per second.

b.) (12 pts.) Find the linear velocity of a point on the circumference of the wheel in feet per second. Round your answer to two decimal places.

12.) (12 pts.) Find the center and radius for the circle x^{2}−6x+y^{2}+8y = 0.

13.) (16 pts.) Find the exact solutions (in degrees) to the following equation for 0^{o} ≤ θ ≤ 360^{o} : 2 sin^{2} θ − 3 sin θ + 1 = 0

14.) (12 pts.) Let f(x) = 3x^{2}−2x+7. Simplify the expression \frac{f(x + h) − f(x)}{h}. (The answer represents the average rate of change of function f from x + h to x .)

The following EXTRA CREDIT PROBLEM is worth 15 points. This problem is OPTIONAL.

1.) The figure shows a circular sector with radius 6 in. and central angle π/6 radians. Compute the exact area of the shaded region.

TRIGONOMETRY IDENTITIES

1. sin(α + β) = sin α cos β + cos α sin β

2. sin(α − β) = sin α cos β − cos α sin β

3. cos(α + β) = cos α cos β − sin α sin β

4. cos(α − β) = cos α cos β + sin α sin β

5. tan(α + β) = \frac{tan α + tan β}{1 − tan α tan β}

6. tan(α − β) = \frac{tan α − tan β}{1 + tan α tan β}

7. sin(2α) = 2 sin α cos α

8. cos(2α) = cos^{2} α − sin^{2} α = 2 cos^{2} α − 1 = 1 − 2 sin^{2} α 9. tan(2α) = \frac{2 tan α}{1 − tan^{2} α}

10. sin(α/2) = ±\sqrt{\frac{1−cos α}{2}}

11. cos(α/2) = ±\sqrt{\frac{1+cos α}{2}}

12. tan(α/2) = \frac{sin α}{1 + cos α}