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A practice final exam in Discrete Math. It includes questions on sequences, subsets, permutations, standard form, distance between numbers, determinants, matrix products, and graphing functions. The exam also covers topics such as completing the square, finding roots, and factoring polynomials. The questions are numbered and there is space for the student to write their answers.
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Discrete math
1.) Find
i=
(i
2 − 1)
2.) Find
i=
6 i
3.) What is the 61st term of the sequence 7, 11 , 15 , 19 , ...?
4.) What’s the 57th term of − 3 , 6 , − 12 , 24 , ...?
5.) What’s the sum of the first 60 terms of the sequence 3, 5 , 7 , 9 ,.. .?
2
6.) Suppose a set A contains 243 objects. How many 92 object subsets of
A are there?
7.) How many ways are there to choose and order 49 objects from a collec-
tion of 304 objects?
8.) How many different ways are there to order 93 different objects?
9.) You’re decorating a room by choosing a color to paint the walls with
and a color of carpet to use for the floor. You have 6 different colors of paint
to choose from for the walls, and 11 different colors of carpet to choose from
for the floor. How many different wall and floor color combinations could you
create?
10.) Write
3
as an integer in standard form.
2
16.) Find x where x
3 (
1 2 x^ + 3)
3 = 8.
17.) Find x where 2
e^2 x ex+
18.) Find x where 4 loge(x) + loge(x
3 ) + 8 = 11.
19.) Find g ◦ f (x) if f (x) = x + 2 and g(x) = x^2.
20.) Find the inverse of g(x) = 7 loge(x + 3).
21.) What is the implied domain of f (x) = x
2 − 2 x + loge(3 − 7 x)?
22.) What is the implied domain of g(x) =
x^3 2 −^7
x − 4?
23 .) Find
x^3 − 3 x^2 − 5 x + 14
x^2 − 4
2
28.) |x − y| is the distance between which two numbers?
29.) Solve for x if | 3 x − 2 | < 4.
Linear algebra
30.) What’s the determinant of the matrix below?
( 2 − 3
1 − 5
31.) Find the product ( 1 0
3 1
32.) What’s the inverse of the matrix below?
( 1 4
2 3
33.) Write the following system of three linear equations in three variables
as a matrix equation
2 x−y + z = 2
y +2z = 1
−x +y − z = 0
34.) Solve for x, y, and z if
x
y z
(^) and
− 1
2
41.) Graph p(x). (Label all x-intercepts.)
p(x) = −2(x + 1)(x + 1)(x − 2)(x
2
42.) Graph r(x) (Label all x-intercepts and all vertical asymptotes.)
r(x) =
−3(x − 1)(x − 1)
4(x + 2)(x^2 + 1)
43.) Graph
h(x) =
ex^ if x 6 = 1;
− 3 if x = 1.
44.) Graph
m(x) =
1 if x ∈ (−∞, 1);
3 if x = 1;
x
2 if x ∈ (1, 2].
2
Last Name: First Name:
11
2
36.) f (x) 37.) g(x) 38.) − 2 e −x 13
28.) f (x) 29.) g(x) 30.) − loge(x + 2) 10 2 3 It 5 C 7 8 —l 2 3 It 5 C 7 8 —l 2 3 It 5 —l 2 3 It 5 C 7 8 3 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES OF COARSE MANIFOLDS IN SEMISIMPLE AND SOLVABLE ARITHMETIC GROUPS MLADEN BESTVINA, ALEX ESKIN, & KEVIN WORTMAN Abstract. We provide partial results towards a conjectural gen- eralization of a theorem of Lubotzky-Mozes-Raghunathan for arith- metic groups (over number fields or function fields) that implies, in low dimensions, both polynomial upper bounds for isoperimetric inequalities and finiteness properties. As a tool in our proof, we also provide polynomial upper bounds for isoperimetric inequalities and finiteness results for certain solv- able groups that appear as subgroups of parabolic groups in semisim- ple groups, thus generalizing a theorem of Bux. Our main result is Theorem 5 below. Before stating it, we provide some background. 1 3 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES OF COARSE MANIFOLDS IN SEMISIMPLE AND SOLVABLE ARITHMETIC GROUPS MLADEN BESTVINA, ALEX ESKIN, & KEVIN WORTMAN Abstract. We provide partial results towards a conjectural gen- eralization of a theorem of Lubotzky-Mozes-Raghunathan for arith- metic groups (over number fields or function fields) that implies, in low dimensions, both polynomial upper bounds for isoperimetric inequalities and finiteness properties. As a tool in our proof, we also provide polynomial upper bounds for isoperimetric inequalities and finiteness results for certain solv- able groups that appear as subgroups of parabolic groups in semisim- ple groups, thus generalizing a theorem of Bux. Our main result is Theorem 5 below. Before stating it, we provide some background. 1 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES OF COARSE MANIFO SEMISIMPLE AND SOLVABLE ARITHMETIC MLADEN BESTVINA, ALEX ESKIN, & KEVIN WORTM Abstract. We provide partial results towards a conjectura eralization of a theorem of Lubotzky-Mozes-Raghunathan for metic groups (over number fields or function fields) that impl low dimensions, both polynomial upper bounds for isoperim inequalities and finiteness properties. As a tool in our proof, we also provide polynomial upper b for isoperimetric inequalities and finiteness results for certain able groups that appear as subgroups of parabolic groups in se ple groups, thus generalizing a theorem of Bux. Our main result is Theorem 5 below. Before stating it, some background. 1 x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES SEMISIMPLE AND SOLV MLADEN BESTVINA, A Abstract. We provide pa eralization of a theorem of L metic groups (over number low dimensions, both polyn inequalities and finiteness p As a tool in our proof, we for isoperimetric inequalitie able groups that appear as s ple groups, thus generalizin Our main result is Theorem some background. 3 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES OF COARSE MANIFOLDS IN SEMISIMPLE AND SOLVABLE ARITHMETIC GROUPS MLADEN BESTVINA, ALEX ESKIN, & KEVIN WORTMAN Abstract. We provide partial results towards a conjectural gen- eralization of a theorem of Lubotzky-Mozes-Raghunathan for arith- metic groups (over number fields or function fields) that implies, in low dimensions, both polynomial upper bounds for isoperimetric inequalities and finiteness properties. As a tool in our proof, we also provide polynomial upper bounds for isoperimetric inequalities and finiteness results for certain solv- able groups that appear as subgroups of parabolic groups in semisim- ple groups, thus generalizing a theorem of Bux. Our main result is Theorem 5 below. Before stating it, we provide some background. 1 3 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES OF COARSE MANIFOLDS IN SEMISIMPLE AND SOLVABLE ARITHMETIC GROUPS MLADEN BESTVINA, ALEX ESKIN, & KEVIN WORTMAN Abstract. We provide partial results towards a conjectural gen- eralization of a theorem of Lubotzky-Mozes-Raghunathan for arith- metic groups (over number fields or function fields) that implies, in low dimensions, both polynomial upper bounds for isoperimetric inequalities and finiteness properties. As a tool in our proof, we also provide polynomial upper bounds for isoperimetric inequalities and finiteness results for certain solv- able groups that appear as subgroups of parabolic groups in semisim- ple groups, thus generalizing a theorem of Bux. Our main result is Theorem 5 below. Before stating it, we provide some background. 1 3 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES OF COARSE MANIF SEMISIMPLE AND SOLVABLE ARITHMETIC MLADEN BESTVINA, ALEX ESKIN, & KEVIN WORTM Abstract. We provide partial results towards a conjectura eralization of a theorem of Lubotzky-Mozes-Raghunathan for metic groups (over number fields or function fields) that impl low dimensions, both polynomial upper bounds for isoperim inequalities and finiteness properties. As a tool in our proof, we also provide polynomial upper b for isoperimetric inequalities and finiteness results for certain able groups that appear as subgroups of parabolic groups in se ple groups, thus generalizing a theorem of Bux. Our main result is Theorem 5 below. Before stating it some background. 1 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES SEMISIMPLE AND SOLV MLADEN BESTVINA, A Abstract. We provide pa eralization of a theorem of L metic groups (over number low dimensions, both polyn inequalities and finiteness p As a tool in our proof, we for isoperimetric inequalitie able groups that appear as s ple groups, thus generalizin Our main result is Theorem some background.
Graphs 27.) Graph the following functions: 3, x, x^2 , x^3 , 2
x, 3
x, (^1) x , (^) x^12 , ex, loge(x). 28.) Graph f : (− 1 , 1] → R where f (x) = x^3. 29.) Graph g : { 2 , 3 , 5 } → R where g(x) = 3x − 10. 30.) Graph − loge(x + 2) and label its x-intercept. 31.) Graph 4(x + 1)^2 + 2 and label its vertex. 32.) Graph
−x + 1 and label its y-intercept. 33.) Graph p(x). (Label all x-intercepts.) p(x) = −2(x + 1)(x + 1)(x − 2)(x − 2)(x 2
34.) Graph r(x) (Label all x-intercepts and all vertical asymptotes.) r(x) = −3(x − 1) 4(x + 2)(x + 2) 35.) Graph h(x) =
x^3 if x = −1; 2 if x = −1. 36.) Graph m(x) =
x if x ∈ [− 2 , 1); 3 if x = 1; √ x if x ∈ (1, ∞). 8
10
10
36.) f (x) 37.) g(x) 38.) − 2 e
−x 13
28.) f (x) 29.) g(x) 30.) − loge(x + 2) 10 2 3 It 5 C 7 8
S 2
—l 2 3 It 5 C 7 8 3 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES OF COARSE MANIFOLDS IN SEMISIMPLE AND SOLVABLE ARITHMETIC GROUPS MLADEN BESTVINA, ALEX ESKIN, & KEVIN WORTMAN Abstract. We provide partial results towards a conjectural gen- eralization of a theorem of Lubotzky-Mozes-Raghunathan for arith- metic groups (over number fields or function fields) that implies, in low dimensions, both polynomial upper bounds for isoperimetric inequalities and finiteness properties. As a tool in our proof, we also provide polynomial upper bounds for isoperimetric inequalities and finiteness results for certain solv- able groups that appear as subgroups of parabolic groups in semisim- ple groups, thus generalizing a theorem of Bux. Our main result is Theorem 5 below. Before stating it, we provide some background. 1 3 x x^2 x^3 √ (^2) x √ (^3) x 1 x 1 x^2 ex loge(x) FILLING BOUNDARIES OF COARSE MANIFOLDS IN SEMISIMPLE AND SOLVABLE ARITHMETIC GROUPS MLADEN BESTVINA, ALEX ESKIN, & KEVIN WORTMAN Abstract. We provide partial results towards a conjectural gen- eralization of a theorem of Lubotzky-Mozes-Raghunathan for arith- metic groups (over number fields or function fields) that implies, in low dimensions, both polynomial upper bounds for isoperimetric inequalities and finiteness properties. As a tool in our proof, we also provide polynomial upper bounds for isoperimetric inequalities and finiteness results for certain solv- able groups that appear as subgroups of parabolic groups in semisim- ple groups, thus generalizing a theorem of Bux. Our main result is Theorem 5 below. Before stating it, we provide some background. 1
29.) g(x) 30 10
x − 1 25.) 2x − 3
−x − 1 26.) −3(x + 2)^2 − 1 36.) −3(x + 2)^2 − 1 3 37.) p(x) 4 38.) q(x)
28.) f (x) 29.) g(x) 30.) − loge(x + 2) 10 x − 3 −3(x + 2)^2 − 1 36.) −3(x + 2)^2 − 1 39.) f (x) 37.) p(x) 40.) g(x)
29.) g(x) 30 10
x − 1 25.) 2x − 3
−x − 1 26.) −3(x + 2)^2 − 1 36.) −3(x + 2)^2 − 1 3 37.) p(x) 4
28.) f (x) 29.) g(x) 30.) − loge(x + 2)
10
√ 3 x − 1 25.) 2x − 3 24.) √ −x − 1 26.) −3(x + 2)
2
36.) −3(x + 2)^2 − 1 39.) f ( 37.) p(x) 40.) g( 27.) 28.) f (x) 29.) g(x) 30.) − loge(x + 2)
10
√ 1 25.) 2x − 3 −x^ + 1 36.) −3(x + 2)
2
24.) ex 26.) ex^ − 1 25.) ex^ 26.) 27.) ex^ − 1 28.) 24.) ex^ 25.) loge(x) 26.) e x − 1 27.) loge(x + 2) 25.) ex^ 26.) loge(x) 27.) ex^ − 1 28.) loge(x + 2) 25.)24.) ex^ ex^ 26.) log25.) loge(x) e(x) 24.) ex^ 25.) loge(x) 26.) e x − 1 27.) loge(x + 2) 25.) ex^ 26.) loge(x) 27.) ex^ − 1 28.) loge(x + 2) 24.) ex^ 25.) loge(x) 26.) e x − 1 27.) loge(x + 2) 25.) ex^ 26.) loge(x) 27.) ex^ − 1 28.) loge(x + 2)
39.) 4(x + 2)^2 + 1 40.) −
x + 2
41.) p(x) 42.) r(x)
43.) h(x) 44.) m(x)
14
31.) 4(x + 1)^2 + 2 32.)
−x + 1
33.) p(x) 34.) r(x)
35.) h(x) 36.) m(x)
11
x − 3
−3(x + 2)^2 − 1
2
36.) −3(x + 2)^2 − 1 39.) f (x)
37.) p(x) 40.) g(x)
38.) q(x)
12
x − 1 25.) 2x − 3
−x − 1 26.) −3(x + 2)^2 − 1
2
36.) −3(x + 2)^2 − 1 39.) f (
37.) p(x) 40.) g(
38.) q(x)
12
2
11
1
23.) p(x) 24.) rQc)
I I I I I I -~ -3 -l -, a i a 1
I I I I I I I I,
1
I I I I I I I I,
1
31.) 4(x + 1)^2 + 2 32.)
−x + 1
33.) p(x) 34.) r(x)
35.) h(x) 36.) m(x)
11
x − 3
−3(x + 2)^2 − 1
36.) −3(x + 2)
2 − 1 39.) f (x)
37.) p(x) 40.) g(x)
3
x − 1 25.) 2x − 3
−x − 1 26.) −3(x + 2)^2 − 1
36.) −3(x + 2)^2 − 1 39.) f (x
37.) p(x) 40.) g(x
2
24.) ex^ 25.) l
26.) ex^ − 1 27.) l
28.) f (x) 29.) g
2
25.) ex^ 26.) loge(
27.) ex^ − 1 28.) loge(
29.) f (x) 30.) g(x)
2
24.) ex^ 25.) loge(x)
26.) e
x − 1 27.) loge(x + 2)
28.) f (x) 29.) g(x)
2
5.) ex^ 26.) loge(x)
7.) ex^ − 1 28.) loge(x + 2)
9.) f (x) 30.) g(x)
2
24.) ex^ 25.) loge(x)
26.) e
x − 1 27.) loge(x + 2)
28.) f (x) 29.) g(x)
25.) ex^ 26.) loge(x)
27.) ex^ − 1 28.) loge(x + 2)
29.) f (x) 30.) g(x)
24.) ex^ 25.) loge(x)
26.) e
x − 1 27.) loge(x + 2)
28.) f (x) 29.) g(x)
2
25.) ex^ 26.) loge(x)
27.) ex^ − 1 28.) loge(x + 2)
29.) f (x) 30.) g(x)
2
24.) ex^ 25.) loge(x)
26.) e
x − 1 27.) loge(x + 2)
28.) f (x) 29.) g(x)
2
25.) ex^ 26.) loge(x)
27.) ex^ − 1 28.) loge(x + 2)
29.) f (x) 30.) g(x)
2
26.) ex^ − 1 27.) loge(x + 2)
28.) f (x) 29.) g(x)
2
27.) ex^ − 1 28.) loge(x + 2)
29.) f (x) 30.) g(x)
2
