Prove

Problem 101

4) Soit ff une fonction définie sur R\mathbb{R} par: f(x)=x˙3+x˙+1\boldsymbol{f}(\boldsymbol{x})=\dot{x}^{3}+\dot{x}+1 a) Montrer que l'équation f(x)=0\boldsymbol{f}(\boldsymbol{x})=\mathbf{0} admet une solution unique α\alpha dans R\mathbb{R} et que 1<α<0-1<\alpha<0 b) Trouver un encadrement de α\alpha d'amplitude 0,25. (1pt) c) Déduire le signe de ff sur R\mathbb{R} (1pt) d) Montrer que α=α+13\alpha=-\sqrt[3]{\alpha+1} (0,5pt) 5) Soit ff la fonction continue sur [a;b][a ; b] tel que f(a)<0f(a)<0
Montrer c]a;b[;(bc)f(c)=ac\exists c \in] a ; b[;(b-c) f(c)=a-c (1pt)

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Problem 102

BXERCTCR 63 La figure en-dessous est la représentation de la restriction d'une fonction ff sur [2;2][-2 ; 2]. ésou 1) Montrer que pour tout x[2;2]x \in[-2 ; 2] : f(x)=x+1+x1f(x)=|x+1|+|x-1| 2) On suppose que la fonction ff est périodiqued période 4 et on considère l'intervalle : Ik=[4k;4(k+1)[ ouˋ kZI_{k}=[4 k ; 4(k+1)[\text { où } k \in \mathbb{Z}

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Problem 103

c. Justify the statement: 3(n4)2+5n2O(n2)3\left(\frac{n}{4}\right)^{2}+5 n^{2} \in O\left(n^{2}\right).

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Problem 104

Can you conclude that these triangles are congruent? yes no

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Problem 105

Can you conclude that these triangles are congruent? yes no

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Problem 106

f(x)=1(x5)2f(x)=\frac{1}{(x-5)^{2}} is always positive. True False
Question 5 (1 point) All rational functions have at least one vertical asymptote. True False

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Problem 107

Given: ABBC\overline{A B} \cong \overline{B C} and BC\overline{B C} bisects ACD\angle A C D. Prove: ABCD\angle A \cong \angle B C D.
Note: quadrilateral properties are not permitted in this proof.
Step
1 try Type of Statement

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Problem 108

Establish the identity. (cscθ+1)(cscθ1)=cot2θ(\csc \theta+1)(\csc \theta-1)=\cot ^{2} \theta
Multiply and write the left side expression as the difference of two squar \square

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Problem 109

College tuition: The mean annual tuition and fees in the 2013-2014 academic year for a sample of 13 private colleges in California was $33,000\$ 33,000 with a standard deviation of $7300\$ 7300. A dotplot shows that it is reasonable to assume that the population is approximately normal. Can you conclude that the mean tuition and fees for private institutions in California is less than $35,000\$ 35,000 ? Use the α=0.01\alpha=0.01 level of significance and the PP-value method and Excel.
Part 1 of 5 (a) State the appropriate null and alternate hypotheses. H0:μ=35,000H1:μ<35,000\begin{array}{l} H_{0}: \mu=35,000 \\ H_{1}: \mu<35,000 \end{array}
This hypothesis test is a left-tailed \quad \mathbf{} test. Part 2 of 5 Skip Part Check Save For Later Submit Assi Q 2024 McGraw Hill LLC. All Rights Reserved. Terms of Use I Privacy Center

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Problem 110

Use the given information and theorems and postulates you have learned to show that cdc \| d.
18. m4=58,m6=58\mathrm{m} \angle 4=58^{\circ}, \mathrm{m} \angle 6=58^{\circ}
19. m1=(23x+38),m5=(17x+56),x=3\mathrm{m} \angle 1=(23 x+38)^{\circ}, \mathrm{m} \angle 5=(17 x+56)^{\circ}, x=3
20. m6=(12x+6),m3=(21x+9),x=5\mathrm{m} \angle 6=(12 x+6)^{\circ}, \mathrm{m} \angle 3=(21 x+9)^{\circ}, x=5
21. m1=99,m7=(13x+8),x=7\mathrm{m} \angle 1=99^{\circ}, \mathrm{m} \angle 7=(13 x+8)^{\circ}, x=7

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Problem 111

20. Thermodynamics texts 4{ }^{4} use the relationship (yx)(zy)(xz)=1.\left(\frac{\partial y}{\partial x}\right)\left(\frac{\partial z}{\partial y}\right)\left(\frac{\partial x}{\partial z}\right)=-1 .
Explain the meaning of this equation and prove that it is true. [HinT: Start with a relationship F(x,y,z)=0F(x, y, z)=0 that defines x=f(y,z),y=g(x,z)x=f(y, z), y=g(x, z), and z=h(x,y)z=h(x, y) and differentiate implicitly.]

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Problem 112

Explain why the triangles are similar. Then find the missing length, x .
Choose the reason that the triangles are similar. A. All right triangles are similar. B. The Pythagorean theorem states that a2+b2=c2\mathrm{a}^{2}+\mathrm{b}^{2}=\mathrm{c}^{2}. Thus, the corresponding sides are proportional. C. Both triangles are right and scalene. D. One angle pair is given to have the same measure (right triangles). Another angle pair consists of vertical angles with the same measure. Thus, two angles of the large triangl measure to two angles of the small triangle.

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Problem 113

Ex: show that the set {(cos(ncos1x)}n=1\left\{\left(\cos \left(n \cos ^{-1} x\right)\right\}_{n=1}^{\infty}\right. is and orthogonal set write the weight function 11x2\frac{1}{\sqrt{1-x^{2}}}, over the interval [1,1][-1,1]

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Problem 114

1) For cvery 2 red candies there are 3 green candies. How many red if there are 12 green? Dram a picture or diagram to prove your answer.

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Problem 115

(XˉACˉ)AXˉCˉ=Cˉ(\bar{X} \cup A \cup \bar{C}) \cap \overline{A \cap \bar{X} \cap \bar{C}} = \bar{C}

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Problem 116

Use the Two-column Proof for Questions 3-4:
GIVEN: ABDE\quad \overline{A B} \| \overline{D E} PROVE: ABCEDC\triangle A B C \cong \triangle E D C \begin{tabular}{|c|l|} \hline \multicolumn{2}{|c|}{ Statements } \\ \multicolumn{2}{|c|}{ Reasons } \\ \hline 1. ABDE\overline{A B} \| \overline{D E} & 1. Given \\ \hline 2. BACDEC\angle B A C \cong \angle D E C & 2. \\ \hline 3.BD3 . \quad \overline{B D} bisects AE\overline{A E} & 3. Given \\ \hline 4. & 4. Definition of segment bisector \\ \hline 5. & 5. Vertical Angles Thrm. \\ \hline 6. & \\ \hline \end{tabular}
3. What is the correct \#2 reason? a. ASA b. CPCTC c. AAS d. Definition of Congruent s\nvdash^{\prime} s e.) Alternate Interior \Varangle^{\prime} s Thrm.
4. What is the correct \#4 statement? a. ABCEDC\angle A B C \cong \angle E D C b. ACBECD\angle A C B \cong \angle E C D c. BCDC\overline{B C} \cong \overline{D C} d. ACEC\overline{A C} \cong \overline{E C} e. ABDE\overline{A B} \| \overline{D E}

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Problem 117

P Flag question
If f(x)={2x+1 if x0x+2 if x>0f(x)=\left\{\begin{array}{cc}2 x+1 & \text { if } x \leq 0 \\ x+2 & \text { if } x>0\end{array}\right. then f1(x)=f^{-1}(x)=
Select one: a. {12(x1) if x<1x2 if x2\left\{\begin{array}{ccc}\frac{1}{2}(x-1) & \text { if } & x<1 \\ x-2 & \text { if } & x \geq 2\end{array}\right. b. {12(x1) if x0x2 if x>0\left\{\begin{array}{rll}\frac{1}{2}(x-1) & \text { if } & x \leq 0 \\ x-2 & \text { if } & x>0\end{array}\right. c. {12(x1) if x<0x2 if x0\left\{\begin{array}{cl}\frac{1}{2}(x-1) & \text { if } x<0 \\ x-2 & \text { if } x \geq 0\end{array}\right. d. {12(x1) if x1x2 if x>2\left\{\begin{array}{ccc}\frac{1}{2}(x-1) & \text { if } & x \leq 1 \\ x-2 & \text { if } & x>2\end{array}\right.
Clear my choice Next page

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Problem 118

If x+zyx+z \propto y and y+zxy+z \propto x, prove x+yzx+y \propto z

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Problem 119

BONUS: Write a Proof
Prove: EAFC\overline{E A} \cong \overline{F C}

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Problem 120

For the two triangles given, CF\angle C \cong \angle F and BCEF\overline{B C} \cong \overline{E F}. Which of the following is needed as one additional piece of information to prove ABCDEF\triangle A B C \cong \triangle D E F using the Side-Angle-Side (SAS) congruence theorem? (A) AD\angle A \cong \angle D
B BE\angle B \cong \angle E
C ABDE\overline{A B} \cong \overline{D E} (D) ACDF\overline{A C} \cong \overline{D F}

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Problem 121

The xx coordinate of the vertex is: x=b2ax=\frac{-b}{2 a} True False

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Problem 122

The axis of symmetry is the horizontal line at the vertex. True False

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Problem 123

10. For n1n \geq 1, establish that the integer n(7n2+5)n\left(7 n^{2}+5\right) is of the form 6k6 k.
11. If nn is an odd integer, show that n4+4n2+11n^{4}+4 n^{2}+11 is of the form 16k16 k.

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Problem 124

Prove the triangles below are congruent.
Given: BD,ACCE\angle B \cong \angle D, A C \cong C E Prove: ACBECD\triangle \mathrm{ACB} \cong \triangle \mathrm{ECD} 1) BD,ACCE\angle B \cong \angle D, A C \cong C E Given 2) ACBECD\angle A C B \cong \angle E C D [Choose] 3) ACBECD\triangle \mathrm{ACB} \cong \triangle E C D [Choose ]

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Problem 125

Exercice 1 .(04 Pts)
1. Show by induction : nN:k=0n2k=2n+11;(02Pts)\forall n \in \mathbb{N}: \sum_{k=0}^{n} 2^{k}=2^{n+1}-1 ;(02 \mathrm{Pts})
2. Show by contrapositive : nn prime n=2\Rightarrow n=2 or nn odd.(02 Pts)

Exercice 2 .(04 Pts) Let \Re the relation in R\mathbb{R} defined by : aba3b3=aba \Re b \Leftrightarrow a^{3}-b^{3}=a-b
1. Show that \Re is an equivalence relation. ( 02 Pts )
2. Discuss according to mm the number of elements in the class of m.(02Pts)m .(02 \mathrm{Pts})

Exercice 3 .(12 Pts) Let S,A,BP(S)S \neq \emptyset, A, B \in \mathcal{P}(S). We define a function ff as: f:P(S)P(A)×P(B)X(XA;XB)\begin{aligned} f: \mathcal{P}(S) & \rightarrow \mathcal{P}(A) \times \mathcal{P}(B) \\ X & \mapsto(X \cap A ; X \cap B) \end{aligned}
1. Let A={e,{f,g}},B={d}A=\{e,\{f, g\}\}, B=\{d\}. Calculate the power set of AA, and the Cartesian product of P(A)\mathcal{P}(A) and P(B)\mathcal{P}(B). (03 Pts)
2. Show by contradiction that if ff is injective then SABS \subset A \cup B.(02 Pts)
3. Show by contradiction that if ff is surjective then AB0.(02Pts)A \cap B \subset 0 .(02 \mathrm{Pts})
4. Show quickly that ABSA \cup B \subset S and AB.(01Pts)\emptyset \subset A \cap B .(01 \mathrm{Pts})
5. Using the direct proof, show that if AB=SA \cup B=S then ff is injective.( 01 Pts )
6. Using the direct proof, show that if AB=A \cap B=\emptyset then ff is surjective.(01 Pts)
7. Using the previous questions, give the necessary condition to ff to be a bijective function, then calculate f1.(02Pts)f^{-1} .(02 \mathrm{Pts})

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Problem 126

Exercice 1 .(04 Pts)
1. Show by induction : nN:k=0n2k=2n+11;(02Pts)\forall n \in \mathbb{N}: \sum_{k=0}^{n} 2^{k}=2^{n+1}-1 ;(02 \mathrm{Pts})
2. Show by contrapositive : nn prime n=2\Rightarrow n=2 or nn odd.(02 Pts)

Exercice 2 .(04 Pts) Let \Re the relation in R\mathbb{R} defined by : aba3b3=aba \Re b \Leftrightarrow a^{3}-b^{3}=a-b
1. Show that \Re is an equivalence relation. ( 02 Pts )
2. Discuss according to mm the number of elements in the class of m.(02Pts)m .(02 \mathrm{Pts})

Exercice 3 .(12 Pts) Let S,A,BP(S)S \neq \emptyset, A, B \in \mathcal{P}(S). We define a function ff as: f:P(S)P(A)×P(B)X(XA;XB)\begin{aligned} f: \mathcal{P}(S) & \rightarrow \mathcal{P}(A) \times \mathcal{P}(B) \\ X & \mapsto(X \cap A ; X \cap B) \end{aligned}
1. Let A={e,{f,g}},B={d}A=\{e,\{f, g\}\}, B=\{d\}. Calculate the power set of AA, and the Cartesian product of P(A)\mathcal{P}(A) and P(B)\mathcal{P}(B). (03 Pts)
2. Show by contradiction that if ff is injective then SABS \subset A \cup B.(02 Pts)
3. Show by contradiction that if ff is surjective then AB0.(02Pts)A \cap B \subset 0 .(02 \mathrm{Pts})
4. Show quickly that ABSA \cup B \subset S and AB.(01Pts)\emptyset \subset A \cap B .(01 \mathrm{Pts})
5. Using the direct proof, show that if AB=SA \cup B=S then ff is injective.( 01 Pts )
6. Using the direct proof, show that if AB=A \cap B=\emptyset then ff is surjective.(01 Pts)
7. Using the previous questions, give the necessary condition to ff to be a bijective function, then calculate f1.(02Pts)f^{-1} .(02 \mathrm{Pts})

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Problem 127

Exercice 1 .(04 Pts)
1. Show by induction : nN:k=0n2k=2n+11;(02Pts)\forall n \in \mathbb{N}: \sum_{k=0}^{n} 2^{k}=2^{n+1}-1 ;(02 \mathrm{Pts})
2. Show by contrapositive : nn prime n=2\Rightarrow n=2 or nn odd.(02 Pts)

Exercice 2 .(04 Pts) Let \Re the relation in R\mathbb{R} defined by : aba3b3=aba \Re b \Leftrightarrow a^{3}-b^{3}=a-b
1. Show that \Re is an equivalence relation. ( 02 Pts )
2. Discuss according to mm the number of elements in the class of m.(02Pts)m .(02 \mathrm{Pts})

Exercice 3 .(12 Pts) Let S,A,BP(S)S \neq \emptyset, A, B \in \mathcal{P}(S). We define a function ff as: f:P(S)P(A)×P(B)X(XA;XB)\begin{aligned} f: \mathcal{P}(S) & \rightarrow \mathcal{P}(A) \times \mathcal{P}(B) \\ X & \mapsto(X \cap A ; X \cap B) \end{aligned}
1. Let A={e,{f,g}},B={d}A=\{e,\{f, g\}\}, B=\{d\}. Calculate the power set of AA, and the Cartesian product of P(A)\mathcal{P}(A) and P(B)\mathcal{P}(B). (03 Pts)
2. Show by contradiction that if ff is injective then SABS \subset A \cup B.(02 Pts)
3. Show by contradiction that if ff is surjective then AB0.(02Pts)A \cap B \subset 0 .(02 \mathrm{Pts})
4. Show quickly that ABSA \cup B \subset S and AB.(01Pts)\emptyset \subset A \cap B .(01 \mathrm{Pts})
5. Using the direct proof, show that if AB=SA \cup B=S then ff is injective.( 01 Pts )
6. Using the direct proof, show that if AB=A \cap B=\emptyset then ff is surjective.(01 Pts)
7. Using the previous questions, give the necessary condition to ff to be a bijective function, then calculate f1.(02Pts)f^{-1} .(02 \mathrm{Pts})

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Problem 128

33. cos6xsin6x=cos2x(114sin22x)\cos ^{6} x-\sin ^{6} x=\cos 2 x\left(1-\frac{1}{4} \sin ^{2} 2 x\right)

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Problem 129

k) 1sin2xcos2xcos4x=tan4x+tan2x+1\frac{1-\sin ^{2} x \cos ^{2} x}{\cos ^{4} x}=\tan ^{4} x+\tan ^{2} x+1

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Problem 130

Mb Apbs (1015) YouTube bal 00 m/math/00 \mathrm{~m} / \mathrm{math} / geometry/proving-triangles-congruent-by-sss-sas-asa-and-aas Complete the proof that PRTPSQ\triangle P R T \cong \triangle P S Q. \begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & SPTQPR\angle S P T \cong \angle Q P R & Given \\ \hline 2 & PSPR\overline{P S} \cong \overline{P R} & Given \\ \hline 3 & PSQPRT\angle P S Q \cong \angle P R T & Given \\ \hline 4 & mRPT=mRPS+mSPTm \angle R P T=m \angle R P S+m \angle S P T & \\ \hline 5 & mQPS=mQPR+mRPSm \angle Q P S=m \angle Q P R+m \angle R P S & Additive Property of Angle Measure \\ \hline 6 & mRPT=mRPS+mQPRm \angle R P T=m \angle R P S+m \angle Q P R & Substitution \\ \hline 7 & mQPS=mRPTm \angle Q P S=m \angle R P T & \\ \hline 8 & PRTPSQ\triangle P R T \cong \triangle P S Q & \\ \hline \end{tabular} Sign out Nev 15

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Problem 131

If k=1pk\sum_{k=1}^{\infty} p^{k} diverges, then k=1(p+0.0001)k\sum_{k=1}^{\infty}(p+0.0001)^{k} diverges for a fixed real number pp. True False

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Problem 132

Establish the identity. cos(π2+θ)=sinθ\cos \left(\frac{\pi}{2}+\theta\right)=-\sin \theta
Choose the sequence of steps below that verifies the identity. A. cos(π2+θ)=cosπ2cosθ+sinπ2sinθ=(0)cosθ+(1)sinθ=sinθ\cos \left(\frac{\pi}{2}+\theta\right)=\cos \frac{\pi}{2} \cos \theta+\sin \frac{\pi}{2} \sin \theta=(0) \cos \theta+(1) \sin \theta=-\sin \theta B. cos(π2+θ)=sinπ2cosθcosπ2sinθ=(1)cosθ+(0)sinθ=sinθ\cos \left(\frac{\pi}{2}+\theta\right)=\sin \frac{\pi}{2} \cos \theta-\cos \frac{\pi}{2} \sin \theta=(1) \cos \theta+(0) \sin \theta=-\sin \theta C. cos(π2+θ)=sinπ2cosθ+cosπ2sinθ=(0)cosθ(0)sinθ=sinθ\cos \left(\frac{\pi}{2}+\theta\right)=\sin \frac{\pi}{2} \cos \theta+\cos \frac{\pi}{2} \sin \theta=(0) \cos \theta-(0) \sin \theta=-\sin \theta D. cos(π2+θ)=cosπ2cosθsinπ2sinθ=(0)cosθ(1)sinθ=sinθ\cos \left(\frac{\pi}{2}+\theta\right)=\cos \frac{\pi}{2} \cos \theta-\sin \frac{\pi}{2} \sin \theta=(0) \cos \theta-(1) \sin \theta=-\sin \theta

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Problem 133

The False Claims Act prohibits knowingly submitting false or inaccurate claims for payment from government programs such as Medicare or Medicaid.
True
False

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Problem 134

Which quadrilateral shown could be proved to be a parallelogram by Theorem 6.2C (Quad with opp. s0\angle \mathrm{s} \cong \rightarrow 0 )? IJKL EFGH QRST ABCDA B C D

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Problem 135

Which quadrilateral shown could be proved to be a parallelogram by theorem 6.2B (Quad with opp. sides \cong \rightarrow \square )? QRST EFGH ABCDA B C D MNOP

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Problem 136

The function f(x)=x33f(x)=x^{3}-3 is one-to-one. a. Find an equation for f1\mathrm{f}^{-1}, the inverse function. b. Verify that your equation is correct by showing that f(f1(x))=xf\left(f^{-1}(x)\right)=x and f1(f(x))=xf^{-1}(f(x))=x. a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. f1(x)=f^{-1}(x)= \square , for xx \geq \square B. f1(x)=x+33f^{-1}(x)=\sqrt[3]{x+3}, for all xx C. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \leq \square D. f1(x)=f^{-1}(x)= \square , for xx \neq \square b. Verify that the equation is correct. f(f1(x))=ff\left(f^{-1}(x)\right)=f \square f1(f(x))=f1()=\begin{aligned} f^{-1}(f(x)) & =f^{-1}(\square) \\ & =\square \end{aligned} and f1(f(x))==\quad \begin{aligned} f^{-1}(f(x)) & = \\ & =\end{aligned}
Substitute. Simplify.

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Problem 137

The function f(x)=x3+5f(x)=x^{3}+5 is one-to-one. a. Find an equation for f1f^{-1}, the inverse function. b. Verify that your equation is correct by showing that f(f1(x))=xf\left(f^{-1}(x)\right)=x and f1(f(x))=xf^{-1}(f(x))=x. a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \geq \square B. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \leq \square C. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \neq \square D. f1(x)=x53f^{-1}(x)=\sqrt[3]{x-5}, for all xx b. Verify that the equation is correct. f(f1(x))=f() and f1(f(x))=f1:() Substitute. =\begin{array}{rlrlrl} f\left(f^{-1}(x)\right) & =f(\square) & \text { and } & f^{-1}(f(x)) & =f^{-1}:(\square) & \\ & & & \text { Substitute. } \\ & =\square & & & & \end{array}

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Problem 138

The function f(x)=x3+2f(x)=x^{3}+2 is one-to-one. a. Find an equation for f1\mathrm{f}^{-1}, the inverse function. b. Verify that your equation is correct by showing that f(f1(x))=xf\left(f^{-1}(x)\right)=x and f1(f(x))=xf^{-1}(f(x))=x. a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \neq \square B. f1(x)=x23f^{-1}(x)=\sqrt[3]{x-2}, for all xx C. f1(x)=f^{-1}(x)= \square , for xx \leq \square D. f1(x)=f^{-1}(x)= \square , for xx \geq \square b. Verify that the equation is correct. f(f1(x))=f() and f1(f(x))=f1()= Substitute.  Simplify. \begin{array}{rlrlrl} f\left(f^{-1}(x)\right) & =f(\square) & \text { and } & f^{-1}(f(x)) & =f^{-1}(\square) & \\ & =\square & & \text { Substitute. } \\ & & & & \text { Simplify. } \end{array} Simplify.

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Problem 139

Click on the graph to plot a point. Click a point to delete it.
Use the dropdown menus and answer blanks below to prove the triangle is right.
Answer Attempt 1 out of 2
I will prove that triangle IJK is right by demonstrating that two of its sides are perpendicular to one another slope of \square slope of \square \square ==
The slopes of these two sides are \square .That being the case, the two sides are \square Therefore the triangle is \square

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Problem 140

Verify the identity using the fundamental trigonometric identities. tan(θ)+cot(θ)=sec(θ)csc(θ)\tan (\theta)+\cot (\theta)=\sec (\theta) \csc (\theta)
Use Reciprocal Identities to rewrite the expression in terms of sine and cosin tan(θ)+cot(θ)=sin(θ)cos(θ)+sin(θ)=cos(θ)sin(θ)\begin{aligned} \tan (\theta)+\cot (\theta) & =\frac{\sin (\theta)}{\cos (\theta)}+\frac{\square}{\sin (\theta)} \\ & =\frac{\square}{\cos (\theta) \sin (\theta)} \end{aligned}
Use a Pythagorean Identity to simplify the numerator of the expression. =cos(θ)sin(θ)=\frac{\square}{\cos (\theta) \sin (\theta)}
Use Reciprocal Identities again to simplify. ==\square Submit Answer
15. [1/1 Points] DETAILS MY NOTES SPRECALC8 7.1.042.

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Problem 141

The function f(x)=x311f(x)=x^{3}-11 is one-to-one. a. Find an equation for f1f^{-1}, the inverse function. b. Verify that your equation is correct by showing that f(f1(x))=xf\left(f^{-1}(x)\right)=x and f1(f(x))=xf^{-1}(f(x))=x. a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. f1(x)=f^{-1}(x)= \square , for xx \neq \square B. f1(x)=x+113f^{-1}(x)=\sqrt[3]{x+11}, for all xx C. f1(x)=f^{-1}(x)= \square , for xx \leq \square D. f1(x)=f^{-1}(x)= \square , for xx \geq \square b. Verify that the equation is correct. f(f1(x))=f() and f1(f(x))=f1()= Substitute. = Simplify. \begin{array}{rlrlrl} f\left(f^{-1}(x)\right) & =f(\square) & \text { and } & f^{-1}(f(x)) & =f^{-1}(\square) & \\ & =\square & & \text { Substitute. } \\ & =\square & & \text { Simplify. } \end{array}

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Problem 142

The function f(x)=(x+7)3f(x)=(x+7)^{3} is one-to-one. a. Find an equation for f1(x)f^{-1}(x), the inverse function. b. Verify that your equation is correct by showing that f(f1(x))=xf\left(f^{-1}(x)\right)=x and f1(f(x))=xf^{-1}(f(x))=x. a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. f1(x)=f^{-1}(x)= \square , for xx \geq \square B. f1(x)=f^{-1}(x)= \square , for x\mathrm{x} \leq \square C. f1(x)=x137f^{-1}(x)=x^{\frac{1}{3}}-7, for all xx D. f1(x)=f^{-1}(x)= \square , for xx \neq \square b. Verify that the equation is correct. f(f1(x))=f(x37) and f1(f(x))=f1((x+7)3) Substitute. =x=1x Simplify \begin{array}{rlrlrl} f\left(f^{-1}(x)\right) & =f(\sqrt[3]{x}-7) & \text { and } & f^{-1}(f(x)) & =f^{-1}\left((x+7)^{3}\right) & \text { Substitute. } \\ & =x & & ={ }^{-1} x & \text { Simplify } \end{array}
The equation is \square

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Problem 143

8. (a) Verify that for all n1n \geq 1, 261014(4n2)=(2n)!n!2 \cdot 6 \cdot 10 \cdot 14 \cdots \cdots(4 n-2)=\frac{(2 n)!}{n!}

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Problem 144

10. For all n1n \geq 1, prove the following by mathematical induction: (b) 12+222+323++n2n=2n+22n\frac{1}{2}+\frac{2}{2^{2}}+\frac{3}{2^{3}}+\cdots+\frac{n}{2^{n}}=2-\frac{n+2}{2^{n}}.

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Problem 145

College tuition: The mean annual tuition and fees for a sample of 13 private colleges in California was $33,000\$ 33,000 with a standard deviation of $7300\$ 7300. A dotplot shows that it is reasonable to assume that the population is approximately normal. Can you conclude that the mean tuition and fees for private institutions in California is less than $35,000\$ 35,000 ? Use the α=0.01\alpha=0.01 level of significance and the PP-value method with the TI-84 Plus calculator.
Part: 0/50 / 5 \square
Part 1 of 5 (a) State the appropriate null and alternate hypotheses. H0:H1:\begin{array}{l} H_{0}: \square \\ H_{1}: \square \end{array}
This hypothesis test is a (Choose one) \nabla test. \square

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Problem 146

Soit ff la fonction numérique définie sur R\mathbb{R} par: f(x)=x+12x+x11f(x)=|x+1|-2|x|+|x-1|-1 fundefined\overbrace{f} sa courbe représentative dans un repère orthon 1) Montrer que la fonction ff est paire. 2) Étudier les variations de la fonction ff. 3) Tracer la courbe Cf\mathscr{C}_{f}. 4) Résoudre graphiquement l'inéquation suivante: x+1+x1>2x+2x+45|x+1|+|x-1|>2|x|+\frac{2 x+4}{5}

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Problem 147

Show that the given functions are inverse functions of each other. Then display the graphs of each function and the line y=xy=x on a graphing calculator and note that each is the mirror image of the other across y=xy=x. y=10x/2 and y=2log10xy=10^{x / 2} \text { and } y=2 \log _{10} x
Transform the function y=10x/2y=10^{x / 2} to show that it is the inverse of y=2log10xy=2 \log _{10} x. y=10x/2y=10^{x / 2} \rightarrow \square \square

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Problem 148

Is the statement true or false? 6<86<8
Answer
TRUE FALSE

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Problem 149

III. Consider a rectangle ABCD such that AB=8 cm\mathrm{AB}=8 \mathrm{~cm} and AD=4 cm\mathrm{AD}=4 \mathrm{~cm}. EE and FF are two points of [AB][\mathrm{AB}] and [AD][\mathrm{AD}] respectively Such that BE=DF=x\mathrm{BE}=\mathrm{DF}=\boldsymbol{x}; where 0<x<40<\boldsymbol{x}<4. Let S\mathbf{S} denote the area of the shaded part FECD. 1) Prove that S=x2+8x+322S=\frac{-x^{2}+8 x+32}{2} 2) Calculate x\boldsymbol{x} so that S=18 cm2\mathbf{S}=18 \mathrm{~cm}^{2}. 3) Prove that for all xx in ]0;4[,S>10 cm2] 0 ; 4\left[, \mathbf{S}>10 \mathrm{~cm}^{2}\right.. 4) Determine the set of values of xx so that S>20 cm2S>20 \mathrm{~cm}^{2}

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Problem 150

 Objectif : caracteˊriseˊ Les sous groupe de (R,+)\underline{\text { Objectif : caractérisé Les sous groupe de }(\mathbb{R},+)} \multimap Une partie AA de R\mathbb{R} est dite dense dans R\mathbb{R} si pour tout xx et touty dans R\mathbb{R} tels que x<yx<y il existe aAa \in A tel que : x<a<yx<a<y. \multimap Une partie GG de R\mathbb{R} est un sous groupe de (R,+)(\mathbb{R},+) si {Gx,yG,xyG\left\{\begin{array}{l}G \neq \emptyset \\ \forall x, y \in G, x-y \in G\end{array}\right. 1 Montrer que si H{0}H \neq\left\{0^{\}}\right.est un sous groupes de (Z,+)(\mathbb{Z},+) alors !nN\exists!n \in \mathbb{N}^{*} tel que : H=nZH=n \mathbb{Z}. nn est appelé le générateur de HH.
2 Soit maintenant G{0}G \neq\{0\} un sous groupe de (R,+)(\mathbb{R},+) et notons G+=GR+G^{+}=G \cap \mathbb{R}^{+}. Montrer que G+G^{+}admet une borne inférieure otée aa. 3 On suppose dans cette question que a>0a>0. \multimap Montrer que aGa \in G et déduire que aZGa \mathbb{Z} \subset G \multimap Montrer que G=aZG=a \mathbb{Z}
4 On suppose maintenant que a=0a=0. Soient xx et yy deux réels tels que : x<yx<y - ojustifier l'existence d'un élément g1Gg_{1} \in G tel que 0<g1<yx0<g_{1}<y-x. \multimap En Considérant la partie entière de xg1\frac{x}{g_{1}}. Montrer qu'il existe gGg \in G tel que : x<g<yx<g<y. \rightarrow En déduie que GG est dense dans R\mathbb{R}.  Reˊsultat :\underline{\text { Résultat }}: Tout sous groupe de (R,+)(\mathbb{R},+) est soit dense soit discrêt c-à-d de la forme aZa \mathbb{Z} avec aR+a \in \mathbb{R}^{+}. 5a5 a et bb deux réls strictement positifs . \multimap Montrer que a.Z+b.Za . \mathbb{Z}+b . \mathbb{Z} est un sous groupe de (R,+)(\mathbb{R},+). \rightarrow Montrer que : a.Z+b.Z est dense dans RabQa . \mathbb{Z}+b . \mathbb{Z} \text { est dense dans } \mathbb{R} \Leftrightarrow \frac{a}{b} \notin \mathbb{Q} \multimap Quelle est la nature de sous groupe : G=Z+2ZG=\mathbb{Z}+\sqrt{2} \mathbb{Z}.
6 Soit AA une partie de R\mathbb{R}. \multimap Montrer que AA est dense dans RxR,(an)AN\mathbb{R} \Leftrightarrow \forall x \in \mathbb{R}, \exists\left(a_{n}\right) \in A^{\mathbb{N}} tel que : (an)\left(a_{n}\right) converge vers xx. \multimap Soit αR\alpha \in \mathbb{R} montrer : αQ(pn),(qn)ZN telles que :nNpn.αqn0 et (pn.αqn)n converge vers 0\alpha \in \mathbb{Q} \Leftrightarrow \exists\left(p_{n}\right),\left(q_{n}\right) \in \mathbb{Z}^{\mathbb{N}} \text { telles que }: \forall n \in \mathbb{N} p_{n} . \alpha-q_{n} \neq 0 \text { et }\left(p_{n} . \alpha-q_{n}\right)_{n} \text { converge vers } 0
On pourra considérer le sous groupe αZ+Z\alpha \mathbb{Z}+\mathbb{Z} 7 Soit ff une fonction de R\mathbb{R} dans R\mathbb{R}. On note Ef={TR;xR,f(x+T)=f(x)}E_{f}=\{T \in \mathbb{R} ; \forall x \in \mathbb{R}, f(x+T)=f(x)\} l'ensemble des périodes de ff.

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Problem 151

Exercice 2: Soit xx un réel . 3
1 Montrer que : E(x)+E(x+12)=E(2x)E(x)+E\left(x+\frac{1}{2}\right)=E(2 x)
2 Démontrer que n2\forall n \geqslant 2 on a: k=0n1E(x+kn)=E(nx)\sum_{k=0}^{n-1} E\left(x+\frac{k}{n}\right)=E(n x)
Utiliser le fait que : xk=0n1E(x+kn)E(nx)x \rightarrow \sum_{k=0}^{n-1} E\left(x+\frac{k}{n}\right)-E(n x) est 1n\frac{1}{n} périodique . 3 Calculer : limnE(10nx)10n\lim _{n} \frac{E\left(10^{n} x\right)}{10^{n}}. Déduire en utilusons la qst 6 exercice 1 . Exercice 3: Objectif: Inégalité de Cauchy-Schwarz et application . 11 Si a1,,ana_{1}, \ldots, a_{n} et b1,,bnb_{1}, \ldots, b_{n} sont des nombres réels, montrer : (i=1naibi)2i=1nai2i=1nbi2\left(\sum_{i=1}^{n} a_{i} b_{i}\right)^{2} \leq \sum_{i=1}^{n} a_{i}^{2} \cdot \sum_{i=1}^{n} b_{i}^{2}
2 Montrer que pour nNn \in \mathbb{N}^{*} et a1,,anRa_{1}, \ldots, a_{n} \in \mathbb{R} on a : i=1naini=1nai2\sum_{i=1}^{n} a_{i} \leq \sqrt{n} \sqrt{\sum_{i=1}^{n} a_{i}^{2}}
3 Montrer que pour tout réels x,y,zx, y, z strictement positifs : x+yx+y+z+y+zx+y+z+z+xx+y+z6\sqrt{\frac{x+y}{x+y+z}}+\sqrt{\frac{y+z}{x+y+z}}+\sqrt{\frac{z+x}{x+y+z}} \leq \sqrt{6}

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Problem 152

Université Mustapha Stambouli de Mascara Faculté des sciences de la nature et de la vie Première année T C Biologie 12/11/2024
Fiche de TD2T D_{2} : Fonctions numeriques Exercice 1. Déterminer le domaine de definition des fonctions suivantes: f1(x)=x+1x+1x1;f2(x)=xlax1;f3(x)=1sinxf_{1}(x)=\frac{x+1}{|x+1|-|x-1|} ; \quad f_{2}(x)=\frac{x}{\sqrt{\operatorname{la} x-1}} ; \quad f_{3}(x)=\frac{1}{\sin x}
Exercice 2. On se donne la fonction f(x)=(x[x])2f(x)=(x-[x])^{2}[x][x] est la partie entière de xx définie par [x]=kZ si kx<k+1[x]=k \in Z \quad \text { si } \quad k \leq x<k+1
1. Montrer que : xR:0f(x)<1\forall x \in \mathbb{R}: 0 \leq f(x)<1. Que pews-on dire?
2. Montrer que ff est périodique de période T=1T=1.
3. Étudier la continuité de f sur [1,2][-1,2].

Exercice 3. Utiliser la règle de l'Hospital, pour calculer les deur limites suivartes: limx0(exex)sinxx2;limx02ln(1+x)2x+x2x3\lim _{x \rightarrow 0} \frac{\left(e^{x}-e^{-x}\right) \sin x}{x^{2}} ; \lim _{x \rightarrow 0} \frac{2 \ln (1+x)-2 x+x^{2}}{x^{3}}
Exercice 4. On considère les fonctions suivantes : f1(x)={cos2(πx) si x11+lnxx si x>1,f2(x)={xsin1x si x00 si x=0,f3(x)={ln(x2+1)x si x00 si x=0,f4(x)={x1+e12 si x00 si x=0\begin{array}{l} f_{1}(x)=\left\{\begin{array}{ll} \cos ^{2}(\pi x) & \text { si } \\ x \leq 1 \\ 1+\frac{\ln x}{x} & \text { si } x>1 \end{array}, \quad f_{2}(x)=\left\{\begin{array}{ll} x \sin \frac{1}{x} & \text { si } x \neq 0 \\ 0 & \text { si } x=0 \end{array},\right.\right. \\ f_{3}(x)=\left\{\begin{array}{ll} \frac{\ln \left(x^{2}+1\right)}{x} & \text { si } x \neq 0 \\ 0 & \text { si } x=0 \end{array}, \quad f_{4}(x)=\left\{\begin{array}{ll} \frac{x}{1+e^{\frac{1}{2}}} & \text { si } x \neq 0 \\ 0 & \text { si } x=0 \end{array}\right.\right. \end{array}
1. Étudier la continuité et la dérivabilté de f1f_{1} au point x=1x=1.
2. Montrer que f2,f3f_{2}, f_{3} et f4f_{4} sont continues au point x=0x=0.
3. f2f_{2} et f4f_{4} sont-elles dérivables au point x=0x=0.

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Problem 154

I'm sorry, I can't assist with that.

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Problem 155

(4×5)×3=4×(5×3)(4 \times 5) \times 3=4 \times(5 \times 3) associative identity distributive commutative

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Problem 156

5. Show that C[a,b]C[a, b], together with the usual scalar multiplication and addition of functions, satisfies the eight axioms of a vector space.

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Problem 157

(a) log53+log54=log512\log _{5} 3+\log _{5} 4=\log _{5} 12 (b) log85log83=log853\log _{8} 5-\log _{8} 3=\log _{8} \frac{5}{3} (c) 2log35=log32 \log _{3} 5=\log _{3}

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Problem 158

(c) (n1)+2(n2)+3(n3)++n(nn)=n2n1\binom{n}{1}+2\binom{n}{2}+3\binom{n}{3}+\cdots+n\binom{n}{n}=n 2^{n-1}. [Hint: After expanding n(1+b)n1n(1+b)^{n-1} by the binomial theorem, let b=1b=1; note also that n(n1k)=(k+1)(nk+1)]\left.n\binom{n-1}{k}=(k+1)\binom{n}{k+1} \cdot\right]

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Problem 159

\begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & VV is the midpoint of RT\overline{R T} & Given \\ 2 & VV is the midpoint of SU\overline{S U} & Given \\ 3 & RUST\overline{R U} \cong \overline{S T} & Given \\ 4 & RVTV\overline{R V} \cong \overline{T V} & \\ 5 & SVUV\overline{S V} \cong \overline{U V} & \\ 6 & STVURV\triangle S T V \cong \triangle U R V & \\ \hline \end{tabular}

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Problem 160

VV is the midpoint of RT\overline{R T} and SU\overline{S U}. Complete the proof that STVURV\triangle S T V \cong \triangle U R V. \begin{tabular}{|l|l|l|} \hline & Statement & Reason \\ \hline 1 & VV is the midpoint of RT\overline{R T} & Given \\ 2 & VV is the midpoint of SU\overline{S U} & Given \\ 3 & RUST\overline{R U} \cong \overline{S T} & Given \\ 4 & RVTV\overline{R V} \cong \overline{T V} & \\ 5 & SVUV\overline{S V} \cong \overline{U V} & \\ 6 & STVURV\triangle S T V \cong \triangle U R V & \\ \hline \end{tabular}

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Problem 161

RR is the midpoint of QS,QTRU\overline{Q S}, \overline{Q T} \cong \overline{R U}, and SURT\overline{S U} \cong \overline{R T}. Complete the proof that QRTRSU\triangle Q R T \cong \triangle R S U. \begin{tabular}{|l|l|l|l|} \hline & Statement & Reason \\ \hline 1 & RR is the midpoint of QS\overline{Q S} & \\ 2 & QTRU\overline{Q T} \cong \overline{R U} & & == \\ 3 & SURT\overline{S U} \cong \overline{R T} & & == \\ 4 & QRRS\overline{Q R} \cong \overline{R S} & & == \\ 5 & QRTRSU\triangle Q R T \cong \triangle R S U & & == \\ \hline \end{tabular}

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Problem 162

Determine whether the statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
When working problems involving probability with permutations, the denominators of the probability fractions consist of the total number of possible permutations.
Choose the correct answer below. A. The statement is true B. The statement is false. When working problems involving probability with permutations, the numerators of the probability fractions consist of the total number of possible permutations. C. The statement is false. When working problems involving probability with combinations, the numerators of the probability fractions consist of the total number of possible permutations. D. The statement is false. When working problems involving probability with combinations, the denominators of the probability fractions consist of the total number of possible permutations.

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Problem 163

Determine whether the following statement is true or false. The odds against E can always be found by reversing the ratio representing the odds in favor of E.
The statement is \square because the odds in favor of EE are found by taking the probability that \square a dividing by the probability that \square and the odds against EE are found by taking the probability that \square and dividing by the probability that \square

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Problem 164

xxercise 06()06(* *)
Let EE be a set and f:EEf: E \rightarrow E such that fff=ff \circ f \circ f=f. Show that ff is injective if and only if gg is surjective.
Exercise 07()07(* *)

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Problem 165

Determine if the triangles can be proved congruent, if possible, by SSS, SAS, ASA, AAS, or HL. Write your answer on the blank. If not congruent, write "not congruent."
1. \qquad SAS
2. \qquad AASA A S
3. \qquad 4. not Congruend.
5. \qquad S.SS
6. \qquad SSS

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Problem 166

```latex Given: JK=LM,KM=LMK\overline{J K} = \overline{L M}, \angle K M = \angle L M K
Prove: MKLKM\triangle M K \cong \triangle L K M
\begin{tabular}{|c|c|} \hline Statements & Reasons \\ \hline 1. JK=LM\overline{J K} = \overline{L M} & 1. Given \\ \hline 2. KM=LMK\angle K M = \angle L M K & 2. Given \\ \hline 3. & 3. \\ \hline 4. JMKLKM\triangle J M K \cong \triangle L K M & 4. \\ \hline \end{tabular}
- Gina Wisson (Ali Thines Algebral, 20) ```

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Problem 167

a. Use synthetic division to show that 2 is a solution of the polynomial equation below. 13x3+15x210x144=013 x^{3}+15 x^{2}-10 x-144=0 b. Use the solution from part (a) to solve this problem. The number of eggs, f(x)f(x), in a female moth is a function of her abdominal width, in millimeters, modeled by the equation below. f(x)=13x3+15x210x41f(x)=13 x^{3}+15 x^{2}-10 x-41
What is the abdominal width when there are 103 eggs? a. The number 2 is a solution to the equation because the remainder of the division, 13x3+15x210x14413 x^{3}+15 x^{2}-10 x-144 divided by x2x-2, is \square

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Problem 168

Consider the curve given by the equation (2y+1)324x=3.3(2y(2 y+1)^{3}-24 x=-3 . \quad 3(2 y (a) Show that dydx=4(2y+1)2\frac{d y}{d x}=\frac{4}{(2 y+1)^{2}}. (b) Write an equation for the line tangent to the curve at the point (1,2)(-1,-2) (c) Evaluate d2ydz2\frac{d^{2} y}{d z^{2}} at the point (1,2)(-1,-2). (d) The point (16,0)\left(\frac{1}{6}, 0\right) is on the curve. Find the value of (y1)(0)\left(y^{-1}\right)^{\prime}(0).

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Problem 169

Establish the identity. secθ+tanθtanθsecθ+tanθsecθ=cosθcotθ\frac{\sec \theta+\tan \theta}{\tan \theta}-\frac{\sec \theta+\tan \theta}{\sec \theta}=\cos \theta \cot \theta

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Problem 170

b) (12)3=(2)\left(-\frac{1}{2}\right)^{-3}=(-2)

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Problem 171

Verify using the indicated test that the infinite series converges. (Hint: Use partial fractions.) n=11n(n+1)\sum_{n=1}^{\infty} \frac{1}{n(n+1)}
By the telescoping series test, we have n=11n(n+1)=limn\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=\lim _{n \rightarrow \infty} \square ) == \square , and thus the series converges.

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Problem 172

5. Given: ABCD\overline{A B} \cong \overline{C D}
Prove: ACBD\overline{A C} \cong \overline{B D} \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|}{ Statements } & \multicolumn{1}{|}{ Reasons } \\ \hline 1. ABCD\overline{A B} \cong \overline{C D} & 1. \\ \hline 2. AB=CDA B=C D & 2. \\ \hline 3. AC+CD=ADA C+C D=A D & 3. \\ \hline 4. AB+BD=ADA B+B D=A D & 4. \\ \hline 5. CD+BD=ADC D+B D=A D & 5. \\ \hline 6. AC+CD=CD+BDA C+C D=C D+B D & 6. \\ \hline 7. AC=BDA C=B D & 7. \\ \hline 8. ACBD\overline{A C} \cong \overline{B D} & 8. \\ \hline \end{tabular} (c) Gina Wilson (All Things Algebra*, LLC), 20

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Problem 173

2. Let f(x)=x3+5xf(x)=x^{3}+5 x. Is ff even, odd, or neither? Prove your answer algebraically.

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Problem 174

Complete each proof.
4. Glven: XX is the midpoint of WY,WXXZ\overline{W Y}, \overline{W X} \cong \overline{X Z}

Prove: XYXZ\overline{X Y} \cong \overline{X Z} \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|}{ Statements } & \multicolumn{1}{c|}{ Reasons } \\ \hline 1. XX is the midpoint of WY\overline{W Y} & 1. Given \\ \hline 2. WX=XYW X=X Y & 2. Defintion of Midpoint \\ \hline 3. WXXZ\overline{W X} \cong \overline{X Z} & 3. Given \\ \hline 4. WX=XZW X=X Z & 4. Definition of congment segments \\ \hline 5. XY=XZX Y=X Z & 5. Transitive Property \\ \hline 6. XYXZ\overline{X Y} \cong \overline{X Z} & 6. Definition of congruent sceimontp \\ \hline \end{tabular}
5. Given: ABCD\overline{A B} \cong \overline{C D}

Prove: ACBD\overline{A C} \cong \overline{B D} \begin{tabular}{|l|l|} \hline \multicolumn{1}{|c|}{ Statements } & \multicolumn{1}{c|}{ Reasons } \\ \hline 1. ABCD\overline{A B} \cong \overline{C D} & 1. \\ \hline 2. AB=CDA B=C D & 2. \\ \hline 3. AC+CD=ADA C+C D=A D & 3. \\ \hline 4. AB+BD=ADA B+B D=A D & 4. \\ \hline 5. CD+BD=ADC D+B D=A D & 5. \\ \hline 6. AC+CD=CD+BDA C+C D=C D+B D & 6. \\ \hline 7. AC=BDA C=B D & 7. \\ \hline 8. ACBD\overline{A C} \cong \overline{B D} & 8. \\ \hline \end{tabular} (c) Gina Wilson (All Things Algebra ,LLC{ }^{\otimes}, \mathrm{LLC} ).

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Problem 175

(1) ガウスの法則の微分形から、ポアソンの方程式を導出せよ。
なお、電界は時間的に変化しない静電界とする。適宜、以下の等式を利用すること。 (a) ガウスの法則の微分形 divE=ρϵ0E:\operatorname{div} \boldsymbol{E}=\frac{\rho}{\epsilon_{0}} \quad \boldsymbol{E}: 電界、 ρ\rho : 電荷密度 (b) 電位 VV と電界 E\boldsymbol{E} の関係式 E=gradV\boldsymbol{E}=-\operatorname{grad} V (2)位置 xx における電位が V(x)=V0(xd1)2V(x)=-V_{0}\left(\frac{x}{d}-1\right)^{2} 、( dd : 定数)のとき、ポアソンの方程式から電荷密度 ρ\rho を求めよ。ただし、誘電率を ε0\varepsilon_{0} とする。
なお、ポアソンの方程式は、 div(gradV)=(V)=(2Vx2+2Vy2+2Vy2)=ρϵ0\operatorname{div}(\operatorname{grad} V)=\boldsymbol{\nabla} \cdot(\boldsymbol{\nabla} V)=\left(\frac{\partial^{2} V}{\partial x^{2}}+\frac{\partial^{2} V}{\partial y^{2}}+\frac{\partial^{2} V}{\partial y^{2}}\right)=-\frac{\rho}{\epsilon_{0}} である。 また、 xx のみが VV の変数である場合 div(gradV)=(V)=(2Vx2)=ρϵ0\operatorname{div}(\operatorname{grad} V)=\boldsymbol{\nabla} \cdot(\boldsymbol{\nabla} V)=\left(\frac{\partial^{2} V}{\partial x^{2}}\right)=-\frac{\rho}{\epsilon_{0}} である。

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Problem 176

Consider the partial derivatives fx(x,y)=5x4y68x3yfy(x,y)=6x5y52x4\begin{array}{c} f_{x}(x, y)=5 x^{4} y^{6}-8 x^{3} y \\ f_{y}(x, y)=6 x^{5} y^{5}-2 x^{4} \end{array}
Is there a function ff which has these partial derivatives? Yes
If so, what is it? f=f= (Enter none if there is no such function.) Are there any others? Yes Submit answer Next item

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Problem 177

4. [-/1 Points] DETAILS MY NOTES MARSVECTORCALC6 3.1.009.
Can there exist a C2C^{2} function f(x,y)f(x, y) with fx=2x3yf_{x}=2 x-3 y and fy=4x+yf_{y}=4 x+y ? Yes No
Additional Materials eBook Submit Answer
5. [-/2 Points]

DETAILS MY NOTES MARSVECTORCALC6 3.1.025.
A function u=f(x,y)u=f(x, y) with continuous second partial derivatives satisfying Laplace's equation 2ux2+2uy2=0\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 is called a harmonic function. Show that the function u(x,y)=9x327xy2u(x, y)=9 x^{3}-27 x y^{2} is harmonic. Since 2ux2=\frac{\partial^{2} u}{\partial x^{2}}= \square and 2uy2=\frac{\partial^{2} u}{\partial y^{2}}= \square ,2ux2+2uy2=0\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.
Additional Materials \square eBook Submit Answer Home My Assignments

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Problem 178

4) cot2θ=cos2θ1cos2θ\cot ^{2} \theta=\frac{\cos ^{2} \theta}{1-\cos ^{2} \theta}

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Problem 179

4.2 In the diagram below QSTV;PQST;QT=TR=9 cmQ S\|T V ; P Q\| S T ; Q T=T R=9 \mathrm{~cm} and PS=15 cmP S=15 \mathrm{~cm}. 4.2.1 Prove VR=712 cmV R=7 \frac{1}{2} \mathrm{~cm}. (4) 4.2.2 a) Calculate PQP Q if PQ=165VRP Q=\frac{16}{5} V R (2) \square Gr 10 November Examination: Paper 2 Page 10

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Problem 180

56. (a) Let f(x)f(x) be a continuous function defined on the interval [0,a][0, a], where aa is a positive constant. Prove that 0af(x)dx=0af(ax)dx\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x. (b) (i) Prove that 0π2ln(1+cosx1+sinx)dx=0π2ln(1+sinx1+cosx)dx\int_{0}^{\frac{\pi}{2}} \ln \left(\frac{1+\cos x}{1+\sin x}\right) d x=\int_{0}^{\frac{\pi}{2}} \ln \left(\frac{1+\sin x}{1+\cos x}\right) d x. (ii) Prove that 0π2ln(1+cosx1+sinx)dx=0\int_{0}^{\frac{\pi}{2}} \ln \left(\frac{1+\cos x}{1+\sin x}\right) d x=0. (c) Find ddx[ln(1+cosx1+sinx)]\frac{d}{d x}\left[\ln \left(\frac{1+\cos x}{1+\sin x}\right)\right]. (d) Using integration by parts, evaluate 0π2x(sinx+cosx+1)(1+cosx)(1+sinx)dx\int_{0}^{\frac{\pi}{2}} \frac{x(\sin x+\cos x+1)}{(1+\cos x)(1+\sin x)} d x.

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Problem 181

3 It is given that, at any point on the graph of y=f(x),dy dx=exsinxyy=\mathrm{f}(x), \frac{\mathrm{d} y}{\mathrm{~d} x}=\mathrm{e}^{x} \sin x-y. (i) Show that d3y dx3=d2y dx22y\frac{\mathrm{d}^{3} y}{\mathrm{~d} x^{3}}=\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 y.
The graph of y=f(x)y=\mathrm{f}(x) passes through the origin OO. (ii) Find the Maclaurin series for yy, up to and including the term in x4x^{4}. [4] (iii) Hence, find the Maclaurin series for e2xsin2x\mathrm{e}^{2 x} \sin 2 x, up to and including the term in x2x^{2}. [2]

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Problem 182

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Problem 183

8. (a) Verify that for all n1n \geq 1, 261014(4n2)=(2n)!n!2 \cdot 6 \cdot 10 \cdot 14 \cdots(4 n-2)=\frac{(2 n)!}{n!} (b) Use part (a) to obtain the inequality 2n(n!)2(2n)2^{n}(n!)^{2} \leq(2 n) ! for all n1n \geq 1.

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Problem 184

8. (a) Verify that for all n1n \geq 1, 261014(4n2)=(2n)!n!2 \cdot 6 \cdot 10 \cdot 14 \cdots \cdots(4 n-2)=\frac{(2 n)!}{n!} (b) Use part (a) to obtain the inequality 2n(n!)2(2n)2^{n}(n!)^{2} \leq(2 n) ! for all n1n \geq 1.

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Problem 185

Задача 3: К35.1 Выяснить, являются ли подпространства соответствующего векторного пространства каждая из следующих совокупностей векторов: (1) векторы плоскости с началом в OO, концы которых не лежат на данной прямой (2) векторы пространства Rn\mathbb{R}^{n}, координаты которых - целые числа

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Problem 186

Задача 4: К35.1 Выяснить, являются ли подпространства соответствующего векторного пространства каждая из следующих совокупностей векторов: (1) многочлены четной степени с коэффициентами из полы FF (давайте считать, что нулевой многочлен является многочленом четной степени) (2) многочлены с коэффициентами из поля FF, не содержащие четных степеней переменной xx
Задача 5: К35.2 Доказать, что следующие совокупности векторов пространства Fn,FF^{n}, F - поле, образуют подпространства. (1) векторы, у которых совпадают первая и последняя координаты (2) векторы, у которых координаты с четными номерами равны 0 (3) векторы вида ( α,β,α,β,\alpha, \beta, \alpha, \beta, \ldots )

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Problem 187

The expression: cos(10x)cos(4x)=cos(40x2)\cos (10 x) \cos (4 x)=\cos \left(40 x^{2}\right) A) False SELECT ALL APPLICABLE C B) True is an identity.

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Problem 188

4. State whether or not each of the following pairs of triangles are congruent. Explain how you know. [P] a. c. b. d.

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Problem 189

Naomi makes a conjecture that the sum of two odd integers is always an even integer. Which choice is the best proof of her conjecture? CLEAR CHECK
Let mm and nn both represent odd numbers, and let m+nm+n Every time you add two odd numbers, the sum is an even be an even number. Therefore m+n=n+mm+n=n+m which number. shows that the sum of two odd numbers is even.
Look at different examples: 7+7=14,11+7=187+7=14,11+7=18, 23+3=2623+3=26. So the sum of two odd numbers must be even.

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Problem 190

if y=ln(ex+e2x+1)\quad y=\ln \left(e^{x}+\sqrt{e^{2 x}+1}\right) then Prove in (ex21)y=y\left(e^{x^{2}}-1\right) y^{\prime \prime}=y^{\prime}

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Problem 191

TVX\triangle T V X is equilateral. Complete the proof that VWXVUT\triangle V W X \cong \triangle V U T. \begin{tabular}{|l|l|l|} \hline \multicolumn{1}{|l|}{ Statement } & Reason \\ \hline 1 & ΔTVX\Delta T V X is equilateral \\ 2 & WU\angle W \cong \angle U & \\ 3 & WVXTVU\angle W V X \cong \angle T V U & \\ 4 & VXTV\overline{V X} \cong \overline{T V} & \\ 5 & ΔVWXΔVUT\Delta V W X \cong \Delta V U T & \\ \hline \end{tabular}

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Problem 192

\text{If } y^{x} = x^{y} \text{ then prove } y' = \frac{y^{2}(1-\ln x)}{x^{2}(1-\ln y)} \\

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Problem 193

SAT/ACT Consider DEF\triangle D E F and PQR\triangle P Q R, where DFPRD F \cong P R and EFQRE F \cong Q R. Which additional pies of information would allow you to conclude that DEFPQR\triangle D E F \cong \triangle P Q R ? (A) DP\angle D \cong \angle P (c) DQ\angle D \cong \angle Q (B) EQ\angle E \cong \angle Q (D) FR\angle F \cong \angle R

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Problem 194

c) sinxcosxtanx=1sin2x\frac{\sin x \cos x}{\tan x}=1-\sin ^{2} x

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Problem 195

11. Determine if the triangles are similar; state the reason; complete the similarity statement RST\triangle R S T \sim \qquad

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Problem 196

6. V is the midpoint of WZ\overline{W Z} and XY\overline{X Y}.
Theorem: \qquad ΔWxvΔZxv\Delta W x v \cong \Delta Z x v

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Problem 197

1998 AP Calculus AB Scoring Guidelines
5. Consider the curve defined by 2y3+6x2y12x2+6y=12 y^{3}+6 x^{2} y-12 x^{2}+6 y=1. (a) Show that dydx=4x2xyx2+y2+1\frac{d y}{d x}=\frac{4 x-2 x y}{x^{2}+y^{2}+1}. (b) Write an equation of each horizontal tangent line to the curve. (c) The line through the origin with slope -1 is tangent to the curve at point PP. Find the xx- and yy-coordinates of point PP.

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Problem 198

Are 13\frac{1}{3} and 25\frac{2}{5} equivalent fractions

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Problem 199

Question 8 (1 point) The graph of f(x)=log(x4)f(x)=\log (x-4) has a vertical asymptote at x=4x=-4. True False

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Problem 200

Prove the Product Rule for Logarithms. a. Find ln(5×7)=\ln (5 \times 7)= \square (Round to the nearest hundredth) b. Find ln5+ln7=\ln 5+\ln 7= \square

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