Download 16 Problems for Final Examination - Elementary Functions | MATH 112 and more Exams Mathematics in PDF only on Docsity!
Name:
Final Examination: Math 112, CRN 32893, 12:00-12:50 MTWF
Instructor: Rob Fisette
Directions: Follow specific directions for each question. SHOW ALL WORK
in the space provided or on the backs of pages. Magic answers (those where
work is not clearly shown) will receive no credit. Work quickly but carefully,
completing the problems you know first before returning to work on the more
difficult problems. Mark your answers clearly. When asked for exact values,
decimal approximations will not be accepted (if x
= −2, then x = ±i
2 is
an exact answer, while x ≈ ± 1. 41 i is not).
Good luck, have fun, and enjoy your summer break.
- True/False: Decide if each statement is True or False.
(a) cos(cos−^1 x) = x for all real numbers x.
(b) If sin x = sin
7 π 4
, then x must be
π 4
(c) If cos x < 0 and tan x > 0, then sin x < 0.
(d) Suppose c is a solution of the equation cos x =
. Then −c is also a solution.
(e) The terminal side of the angle of measure − 13 π 3
lies in the fourth quadrant.
(f) i^4040040004 = i.
- State the domain and range of f (x) = sin−^1 x.
- Find the exact values of the functions below:
(a) cos−^1
cos
2 π 3
(b) sin−^1
sin
2 π 3
(c) tan
[
sin−^1
)]
- (a) Sketch a complete graph of the function f (x) = sin x.
(b) Find the period, amplitude, and phase shift of the function g(x) = −3 sin(8πx + 2π).
(c) Sketch a complete graph of the function g.
- Find all solutions of the equation −5 cos^2 x − 12 sin x + 9 = 0.
- Recall that the functions f (x) = ln x and g(x) = ex^ are inverse functions.
(a) Find the domain and range of h(x) = 2165e−.^65 x.
(b) Algebraically find the inverse h−^1 (x) of this function, and state its domain and range.
- Two ships leave a port at the same time. One travels in a straight course at 31 mph and the other travels a straight course at 24 mph. The angle between the courses is 56◦. How far apart are they after 3 hours? You should use your calculator to approximate your final answer.
- A small plane passes directly over Carl’s head at an altitude of 626 feet. Two seconds later, the angle of elevation from his eyes to the plane is 53◦. If his eyes are 6 feet off the ground, how far did the plane travel in those two seconds? How fast is it travelling?
- Write the complex number − 7 − i 3 − i
in standard form (a + bi, where a and b are real) and in polar form.
- Use Demoivre’s Theorem to calculate the 8th roots of unity (that is, the roots of the equation x^8 = 1) and plot them in the complex plane. (Be clear about where you intend them to be.
