Download sin 75o = π π π π 6 10 2 xxyy + +

Math 130 Final Exam
Spring 2014
Show all work to receive full credit. Each problem is worth 4 points.
1. The terminal point P ( x, y ) determined by a real number t is (
9 40
, ) . Find
41 41
a) sin(t ) =
b) cos(t ) =
c) cot(t ) =
d) sec(t ) =
2. a) Write the equation of the function that takes f ( x) = x stretches it vertically by a
factor of 2, shifts to the left 5 units and shifts up 3 units.
b) Draw your function from part a), labeling 2 points:
⎛ ( x + 5) 2 ⎞
3. Use the Laws of Logarithms to expand completely: ln ⎜ 3
⎟
⎝ x x −1 ⎠
4. a) Use an addition or subtraction formula to find the exact value of the expression:
sin ( 75o ) =
b) Use an addition or subtraction formula to write the expression as a trigonometric function
of one number, and then find its exact value.
⎛π ⎞
⎛π ⎞
tan ⎜ ⎟ + tan ⎜ ⎟
⎝ 18 ⎠
⎝ 9 ⎠ = ⎛ π ⎞ ⎛π ⎞
1 − tan ⎜ ⎟ tan ⎜ ⎟
⎝ 18 ⎠ ⎝ 9 ⎠
5. Solve the equation 3x3 − 6 x 2 − x = 0
6. Solve the inequality:
2x −1
≤ 1 Include your sign chart.
x+3
7. Find the equation of the circle in standard form, the center of the circle and the radius
for x2 + 6 x + y 2 − 10 y = 2
8. Find the quotient and remainder using either long division or synthetic division for
x3 + 2 x 2 − 10
x+3
⎧2 x + 4 if x ≤ −3
9. Sketch the graph of the piecewise defined function: f ( x) = ⎨
⎩ 5 − x if x > −3
10. Given f ( x) = 3x − 1 and g ( x) =
2
, compute each of the following and simplify your
x +8
answers.
a)
f o g ( x) =
b)
g o f ( x) =
11. For the function f ( x) =
4x + 3
, find the exact values of:
x −8
a) The equation of any horizontal asymptote:
b) The equation of any vertical asymptote:
c) The values of any x-intercepts:
d) The value of any y-intercept:
12. Solve the equation: 3e2 x − 18 = 0
13. Write the domain of each expression in interval notation:
a)
x
x −4
2
b) ln ( 2 x − 5)
c) e 2 x
14. Sketch a graph of one complete period for the function y = 5sin(π ( x + 1)) , labeling both
the x-axis and the y-axis carefully.
a) Amplitude:
b) Period:
c) Phase shift:
15. Find the equation of the line in slope-intercept form, that goes through the point (3, −11)
and has slope
1
6
16. Simplify completely:
x2
3x
÷ 2
2
x − 49 3x + 20 x − 7
17. A 600 ft. guy wire is attached to the top of a communications tower. If the wire makes an
angle of 45 degrees with the ground, how tall is the communications tower? Compute the
exact answer.
18. Find the exact value of each:
a) ln(1) =
b) e2ln5 =
c) sin −1(1) =
1
d) sin(cos −1 ( )) =
2
19. Simplify the trigonometric expression completely:
csc( x) − sin( x)
cot( x)
20. Solve the equation 2sin( x) cos( x) − cos( x) = 0 on the interval [0, 2π ] .
21. Solve the equation: log3 (2 x − 5) = 2
22. Compute the following. You do not need to rationalize denominators. Given an angle θ
1
with terminal side in quadrant I and sin(θ ) = ,
6
a) Draw a triangle including θ , labeling the sides.
b) tan(θ ) =
1
c) sin( θ ) =
2
d) cos(2θ ) =
23. Find the inverse function for f ( x) =
24. Form the difference quotient
2x − 5
3
f ( a + h) − f (a )
for the function f ( x) = 3x 2 − 7 x + 11 and
h
simplify your answer completely.
x
5
25. Write the expression tan(cos −1 ( )) as an algebraic expression in the variable. Show your
work by drawing a triangle.