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MATH 12
FINAL EXAM PROBLEMS
SPRING 2009
- Rationalize the denominator and simplify:
8 5 6
x x z
- Rationalize the denominator and simplify:
4 2 3
a b c
- Rationalize the denominator and simplify: 10 5 โ 2
- Rationalize the numerator and simplify: 4 h h
- Factor: x^3 โ 3 x^2 โ 4 x + 12
- Factor: x^4 + 27 x
- Factor:
3 2 12 1 3 x โ 9 x + 6 x โ^2
- Factor: 2 ( x x โ 3)1/ 3 + 5( x โ3)4 / 3
9. Factor: ( 4 x + 3 y ) โ^2 โ ( 4 x + 3 y ) โ^3 ( 5 x โ y )
- Simplify:
2 2
x x x x
โ^ +
- Simplify: (^1 )
y x x y
y x
- Solve (^3) t^3 + 2 t^2 โ 48 t โ 32 = 0
- Solve x^3 + 2 x^2 = 16 x + 32
- Solve: 2 2 1
x x 1 x x
=^ โ
- Solve: 3 x โ 4 = 10
- Solve 2 x โ 3 โ 4 > 7
- Solve 6 โ 4 x + 3 โค โ 12
- Solve for x : 3 + 5 x โ 1 = 18
- Solve for x and graph the solution on a number line: 5 x โ 9 > 7
- Solve for x and graph the solution on a number line: 3 + 2 5 + 3 x โค 9
- Solve (^3 2 ) 2
x x
- Solve 2 1 3 2
x x 2
- Solve the inequality 2 2 0 4 5
x x x
24. Solve log 4 x โ log4 ( x โ 6 )= 2
25. Solve log 4 ( x + 1 ) = 1 +log 4 x
- Solve for x : 8 + 4log ( 4 x + 1) = 24
- Solve for x : 5 x^ +^1 = 25 x โ^1
- Solve 73 x +^2 โ 11 = 38
- Solve 2 ex โ^1 = 8
- Solve 3 x + 1 โ x + 1 = 2
- Solve 5 โ x + 3 = 3 x + 4
- Solve for x : x โ 3 + x โ 8 = 5
33. Solve for x: ( )
(^12) 2 x 4 โ x^ โ โ 3 4 โ x = 0
- Solve for x: 22 1 5 x 1 x 1
- Solve for x : (^3) x^3 โ 5 x^2 โ 3 x + 5 = 0. It will be helpful to factor by grouping.
- Solve for x :
2 2 7 4 1 1
x x x x
โโ โ โโ โโ โ โโ =^0
- Find an equation for the line that passes through the point (2, -5) and a. has slope - b. is parallel to the x-axis c. is parallel to the y-axis d. is parallel to the line 2 x โ 4 y =
- Use synthetic division and the Remainder Theorem to show that is a solution to.
x = โ 3 P x ( ) = 0
- Find the remaining zeros of P x ( ). Use the back, if necessary.
Problems 46- 48 use P x ( ) = 6 x^3 โ 5 x^2 โ 22 x + 24.
- Use the Rational Zeros Theorem to list all possible rational zeros.
- Use synthetic division and the Remainder Theorem to show that x = โ 2 is a solution to P x ( ) = 0.
- Find the remaining zeros of and use your result to completely factor .
P x ( ) P x ( )
- Let f x ( ) = x^3 โ 19 x โ 30. a. List the possible rational zeros for f(x). b. Show that x = 2 is a zero and use this information to completely factor f(x). c. Use the result from (b) to graph f(x). Label the y-intercept and all x-intercepts.
- Find the domain of the function ( ) 23 7 2 9 f x x x x 5
- Graph the rational function
2 2
f x x^ x x
. Find and label all asymptotes and places where the graph intersects the coordinate axes.
- Graph the rational function
2 2
f x x x
= +^2
. Find and label all asymptotes and places where the graph intersects the coordinate axes.
- a. Find the vertical and horizontal asymptotes for ( ) 1 2 f x x x
b. Sketch the graph of f(x) using the information from part (a). Label the x-intercept and the asymptotes.
- If f x ( ) = x^3 + 7 , find f โ^1 ( x ).
- Let ( ) 1
f x x x
and let g x ( )^32 x
a. Find ( f D g )( x ), simplify, and state the domain.
b. Find ( g D f )( x )and state the domain.
- Let f x ( ) = 2 x^2 + x + 5. Find and simplify the difference quotient
f ( x h ) f x ( )
h
Problems 57 โ 59 use the functions f x ( ) = x^2 โ 16 and g x ( ) = 3 x โ 5.
- Find (^ ) ( )
f x g x
and the domain of this function.
58. Find ( f D g )( x ).
- Calculate f(x^ h)^ f(x) h
- Let h(x) 2x^12 5
. Find h โ^1 (x).
61 Graph the conic section whose equation is x^2 โ 8 x + 8 y + 8 =. Label all important points for this conic.
62. Graph the conic whose equation is (^ )^ (^ )
x + y โ
- =. Label all important points for this conic.
63. Sketch the graph of (^ )^ (^ )
y โ (^) โ x โ =. Label the center, vertices, and
foci. Write the equations of the asymptotes.
- Graph the conic section whose equation is y^2 โ 8 y + 8 x + 8 = 0. Label all important points for this conic.
65. Graph and identify the conic whose equation is (^ )^ (^ )
x y . Label all important points for this conic.
- Find the center and radius of the circle x^2^ + 2 x + y^2 โ 4 y = 11.
- Find the standard form for the equation of a parabola with focus at (-3, 0) and directrix: x = 3
- Let and
A = โกโข
B
. Find the product AB , if possible.
- Given the matrices A =
โก โ^ โ โค
and B =
, find the
matrix product AB, if possible.
- Given the matrices A = โฃ
and B =
find the matrix product AB.
- Find the inverse of the matrix
A
= โข^ โ โ โฅ
- Use the result from the previous problem to solve the system
4 3 2 7 6 5 7 2 8
x y z x y z x y z
