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Mathematics 141 final exam
Wednesday, June 17, 2009
please show your work to get full credit for each problem
2. Convert 47◦ 330 to decimal degrees.
1. Convert π/3 radians to degrees.
3. Find the arclength s intercepted by a central angle A = 20◦ in a circle of radius 5
centimeters.
20◦
4. Compute the exact value of
(a) cos 30◦
s =?
(using square roots and fractions when necessary)
(c) tan 60◦
(b) sin π/4
5. Find the exact values of the six trigonometric functions of the angle A given the triangle
drawn below: (using square roots and fractions when necessary)
√
29
A
5
6. Use a calculator to solve for the angle A if sin A = 2/3
2
page two
7. Show that
1
1
+
= 2 csc2 θ
1 − cos θ
1 + cos θ
8. Solve for each indicated side or angle in the triangles below:
z=
?
rs
ete
m
8
y =?
34◦
x =?
70◦
2 cm
2
cm
A =?
7c
m
h =?
page three
9. Convert to exponential form:
10. Convert to logarithmic form:
10p = c
log3 N = x
3
11. Simplify:
(a) logb b
8
2
e3x (e4x )
(c)
e−5x+2
2
(b) cos A+sin A
12. Expand using logarithmic properties and simplify:
13. Solve for H:
100x3
log
w5
!
14. Solve for x:
e.04x = 2
log H = −6/5
15. What is the inverse function of g(x) = − log x ?
16. For the rational function f (x) =
2x2 − 2x
3x2 + 3x − 6
(a) state the domain of the function
(b) find equations for the asymptotes
to the graph of this rational function
17. For the polynomial function p(x) = x4 (x − 5)(x + 49)2
(a) state the degree of this polynomial
(b) list the zeroes of the polynomial and
the associated multiplicities
18. One solution of x3 + x2 − 7x − 15 = 0 is known to be x = 3. Use long or synthetic
division and the quadratic formula to find the other solutions. (which will be complex numbers)
19. Write down an equation for a circle of radius 13 which is centered at the point (5, 12).
20. If f (x) = x2 − 3x, compute & simplify the value of the difference quotient
f (x + h) − f (x)
h
page four
21. The minimum value of the quadratic function F (x) = 3x2 + (x − 2)2 occurs
and this minimum value is equal to
.
at x =
22. Start with the graph of y = x3 . Shift the graph 5 units to the right, and then reflect
the graph over the x-axis. Write down the equation of the new graph.
23. For the linear function f (x) = 3x − 2 and the quadratic function g(x) = x2 − 4 find
simplified values or expressions for:
(a) g(−3)
(b) f g
(c) g(f (x))
continue to next page
page five
3.5 y 3 2.5 2 1.5 1 0.5 x 0 ‐3 ‐2 ‐1 ‐0.5 0 1 2 3 4 5 24. Given the graph of a function f above,
• state the intervals on which the function values f are decreasing:
• state the domain of this function
• state the range of this function
• the value of local maximum value of f is
• compute the function values f (−2) =
which occurs at x =
and f (5) =
• list the zero of the function f
• is the function f one-to-one?
• calculate the average rate of change in the values of f when −2 ≤ x ≤ 1.
• solve for x: f (x) = 1
6 page six
25. Determine whether each of the following statements is true or false:
• the solutions sets of the linear inequality x − 42 < 3 and the absolute value inequality
|x − 42| < 3 are identical.
• the graph of an exponential function, f (x) = bx , where b is positive and not equal to 1,
always has a horizontal asymptote y = 0.
(
26. Graph the function
(including any asymptotes)
F (x) =
3 − (x + 1)2 if −3 ≤ x ≤ 0
− log3 x
if 0 < x ≤ 3
y
2
1
x
−3
−2
−1
1
−1
2
3