We know that,
f(x) = ∑[tex]\frac{f^{k} (x_{0} ) }{k!} (x-x_{0} )^{k}[/tex]
We know that,
f(x) = ∑[tex]\frac{f^{k} (x_{0} ) }{k!} (x-x_{0} )^{k}[/tex]
holds at a point x if and only if
[tex]\lim_{n \to \infty} R_n(x) = 0[/tex]
Let f(x) = sin x. The Taylor series for f(x) is
f(x) = sin x = ∑ [tex](-1)^{k}[/tex] [tex]\frac{x^{2k+1} }{(2k+1)!}[/tex]
According to the above theorem, we must show that [tex]R_{n}[/tex] ⇒0 or all x as n ⇒ infinity. For all x we have
[tex]f^{n+1} (x) =[/tex] ± sin x
or, [tex]f^{n+1} (x)[/tex] = ± cosy
and in all cases we have [tex]| f^{n+1} (x) |\leq 1.[/tex]
We need to conclude that
[tex]0\leq |R_{n}(x) | \frac{|x|^{n+1} }{(n+1)!}[/tex]
Furthermore, it follows that
[tex]\lim_{n \to \infty} \frac{|x|^{n+1} }{(n+1)!} = 0[/tex]
Using this result, it follows that [tex]|R_{n}|[/tex] ⇒0 and hence that [tex]R_{n}[/tex] ⇒0 is n⇒infinity.
Since this, is true for all x, we have proved that the Taylor series for sin x converges to sin x for all x.
Holds at a point x if
[tex]\lim_{n \to \infty} R_n(x) = 0[/tex]
Let f(x) = sin x. The Taylor series for f(x) is
f(x) = sin x = ∑ [tex](-1)^{k}[/tex] [tex]\frac{x^{2k+1} }{(2k+1)!}[/tex]
According to the above theorem, we must show that [tex]R_{n}[/tex] ⇒0 or all x as n ⇒ infinity. For all x we have
[tex]f^{n+1} (x) =[/tex] ± sin x
or, [tex]f^{n+1} (x)[/tex] = ± cosy
and in all cases we have [tex]| f^{n+1} (x) |\leq 1.[/tex]
We need to conclude that
[tex]0\leq |R_{n}(x) | \frac{|x|^{n+1} }{(n+1)!}[/tex]
Furthermore, it follows that
[tex]\lim_{n \to \infty} \frac{|x|^{n+1} }{(n+1)!} = 0[/tex]
Using this result, it follows that [tex]|R_{n}|[/tex] ⇒0 and hence that [tex]R_{n}[/tex] ⇒0 is n⇒infinity.
Since this, is true for all x, we have proved that the Taylor series for sin x converges to sin x for all x.
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What quantity of 60% chlorine solution must be mixed with a 18% chlorine solution to produce 542.5 mL of 42% solution?
Answer: ___mL
The quantity of chlorine that has a 60% concentration is 310 ml.
What is expression?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
Given that, 60% chlorine solution must be mixed with an 18% chlorine solution to produce 542.5 mL of 42% solution
Let "x" be the quantity of chlorine that has a 60% concentration
So, the quantity of chlorine that has 18% concentration = 542.5 - x
thus, As per the given data expression formed as given below:
0.6 * x + 0.18 (542.5 - x) = 542.5 * 0.42
0.6x - 0.18x + 0.18 *542.5 = 542.5 *0.42
0.42x = 130.20
x = 310
therefore, the quantity of chlorine that has a 60% concentration is 310 ml.
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The bill is 19.37 find the amount of a 15% tip
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{15\% of 19.37}}{\left( \cfrac{15}{100} \right)19.37} ~~ \approx ~~ 2.91[/tex]
. MATHEMATICS: 1. Given that x + y =3/4 and x - y =5/2 find the value of 2y + x
Step-by-step explanation:
[tex]x = \frac{13}{8} \\ y = \frac{ - 7}{8} \\ 2y + x = - \frac{14}{8} + \frac{13}{8} = - \frac{1}{8} [/tex]
Answer:
-1/8
Step-by-step explanation:
x + y = 3/4
x -y = 5/2
2x = 13/4 3/4 + 5/2 = 3/4 + 10/4 = 13/4 Next divide both sides by 2
[tex]\frac{2x}{2}[/tex] = [tex]\frac{\frac{13}{4} }{2}[/tex] [tex]\frac{13}{4}[/tex] ÷[tex]\frac{2}{1}[/tex] = [tex]\frac{13}{4}[/tex] x [tex]\frac{1}{2}[/tex] = [tex]\frac{13}{8}[/tex]
x = [tex]\frac{13}{8}[/tex]
plug in 13/8 for x in either of the two original equations.
x + y = 3/4
13/8 + y = 3/4 Subtract 13/8 from both sides
y = -7/8 3/4 - 13/8 = 6/8 - 13/8 = -7/8
Check:
substitute x = 13/8 and y = -7/8 back into the original two equations to see if the equations ae true
x + y = 34
13/8 - 7/8 = 3/4
6/8 = 3/4
3/4 = 3/4 Checks
x - y = 5/2
13/8 - (-7/8) = 5/2
13/8 + 7/8 = 5/2
20/8 = 5/2
5/2 = 5/2 Checks
Find the value of 2y + x
2(-7/8) + 13/8
-14/8 + 13/8
-1/8
khalil is working two summer jobs, making $18 per hour tutoring and $9 per hour landscaping. last week khalil worked a total of 14 hours and earned a total of $180. graphically solve a system of equations in order to determine the number of hours khalil worked tutoring last week, x,x, and the number of hours khalil worked landscaping last week, yy.
10 number of hours khalil worked tutoring last week and 3 the number of hours khalil worked landscaping last week
given that
An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way. [1] The majority of the time, size comparisons between two numbers on the number line are made. Different types of inequalities are represented by a variety of notations, including:
The symbol a b indicates that an is smaller than b.
When a > b is used, it indicates that an is bigger than b.
A is not equivalent to b in any scenario. Since an is strictly less than or strictly greater than b, these relationships are known as stringent inequalities. Equivalence is not considered.
khalil is working two summer jobs, making $18 per hour tutoring
$9 per hour landscaping
last week khalil worked a total of 14 hours and earned a total of $180.
let x be number of hours khalil worked tutoring last week
y be the number of hours khalil worked landscaping last week
we can form equation by given information
x + y [tex]\leq[/tex] 13 - (1)
18x + 9y > 180 - (2)
the numbers that satisfy the above equations is (10,3)
10 number of hours khalil worked tutoring last week and 3 the number of hours khalil worked landscaping last week
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the general solution to a linear system is given. express this solution as a linear combination of vectors. x1
The linear combination of the general solution of system of equations is.
[tex]X=(7,-3,0)+S1(-2,0,1)\\[/tex].
A system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. A linear system in three variables determines a collection of planes The intersection point is the solution.
A linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results.
Write vector form of the equations, then the vector with variable can expressed as a linear combination of vectors on right sides.
[tex](x1,x2,x3)=(7-2S1,-3,S1)\\=(7,-3,0)+(-2S1,0,S1)\\\\=(7,-3,0)+S1(-2,0,1)[/tex]
So, the linear combination of vectors is.
[tex]X=(7,-3,0)+S1(-2,0,1)\\[/tex]
The complete question is,
the general solution to a linear system is given. express this solution as a linear combination of vectors.
x1=7-2S1
X2=-3
X3=S1.
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4.A) A straight line L1 whose equation is 3y-2x-2 meets the x-axis at R. Determine the co-ordinates of R byAsecond line l2 is perpendicular to LI at R Find the equation of L2 in the form y=mx+c A Third line l3 passes through (-4,1) and is parallel to LI Find, i) the equation of L3 in the form y=mx+c the co-ordinates of points at which l3 intersects l2.
Answer:
69
Step-by-step explanation:
Prove that ROS is an equivalence relation if and only if RoS= So R.
Transitivity: Assume that for some a, b, xRSy and yRSz, or xRaSy and yRbSz, exist. After that, aSRb, abbreviated as aSyRb. In light of the fact that SoR=RoS, we obtain aRSb, or, to use an example, aRcSb.
How to find the calculation?R and S should be two equivalence relations on X.
I want to demonstrate that the relationship RS is equivalent, but I am unable to demonstrate that it is transitive and reflexive.
regarding transitivity
There are random x and y with (x,y)RS, z with (x,z)R and (z,y)S, and what else?
Do I need to use R=S=S=R somewhere else?
As opposed to (x,y)R, I use the notation xRy. For some a, the term "xRSy" specifically refers to "xRaSy." Additionally, writing (xRy and yRz) is more convenient than writing xRySz. I would generally suggest drawing these relationships as graphs, with the vertices representing the points and the edges denoting the relationship.
assumption R=S=R.
Transitivity: Assume that for some a, b, xRSy and yRSz, or xRaSy and yRbSz, exist. After that, aSRb, abbreviated as aSyRb. In light of the fact that SoR=RoS, we obtain aRSb, or, to use an example, aRcSb. Then, as R,S are equivalence relations, xRcSz follows from xRaRcSbSz, and xRSz follows.
I'm sure you can now independently demonstrate reflexivity.
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scientist do not accept a nutrition or scientific hypothesis until it has been thoroughly tested using the
The scientific method is a systematic set of method that scientists follow to gain information. we will explore how the scientific method is used in the study of nutrition.
To make the things simple, you might want to think of the scientific method as a pro way to ask questions and find answers. With this in mind, it makes sense that the stride of the scientific method starts when researchers ask questions.
For example, a research scientist might awe whether people who eat an apple a day visit their family doctor hardly times a year than those who do not eat apples.
The scientific method is a nice tool for analyzing nutrition. It has been used to set nutrition advice, understand how nutrients support health and learn how nutrition supports athletic account.
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The multiplicative inverse of -7/9 is
Answer:
9/-7
Step-by-step explanation:
the reason is all you have to do is reverse the fraction
Answer:
-9/7
Step-by-step explanation:
The multiplicative inverse of -7/9 is_____?
The multiplicative inverse of a number is defined as a number that when multiplied by the original number gives the product as 1
-7/9 × (- 9/7) = 1
Prove divisability
will mark the brainliest!
Step-by-step explanation:
We can prove without use of calculator.
Use exponent product rule:
[tex]a^{n+m} = a^{n}*a^{m}[/tex]
Rewrite with exponent power rule:
[tex](9*9*9*9)-(3^{2}*3^{2}*3^{2}*3)[/tex]
Since [tex]3^{2}=9[/tex], we can eliminate 3 pairs via subtraction, leaving us with:
[tex]9-3=6[/tex]
6 is divisible by 6.
Easy without calculator :)
Hope this helps.
consider the following set of requirements for a university database that is used to keep track of students' transcripts. the university keeps track of each student's name, student number, social security number, current address and phone, permanent address and phone, birthdate, sex, class (freshman, sophomore, ..., graduate), major department, minor department (if any), and degree program (b.a., b.s., ..., ph.d.). some user applications need to refer to the city, state, and zip of the student's permanent address, and to the student's last name. both social security number and student number have unique values for each student. each department is described by a name, department code, office number, office phone, and college. both name and code have unique values for each department. each course has a course name, description, course number, number of semester hours, level, and offering department. the value of the course number is unique for each course. each section has an instructor, semester, year, course, and section number. the section number distinguishes different sections of the same course that are taught during the same semester/year; its values are 1, 2, 3, ..., up to the number of sections taught during each semester. a grade report has a student, section, letter grade, and numeric grade (0, 1, 2, 3, 4 for f, d, c, b, a, respectively). using a modeling tool, complete the following. design a conceptual schema for this company (see chapter 1, page 13 for an example) draw an er diagram for that schema. specify key attributes of each entity type and structural constraints on each relationship type. note any unspecified requirements and make appropriate assumptions to make the specification complete. what to submit submit a word or pdf document containing both the conceptual schema and the er diagram. additional resources summary of the notation for er diagrams try one of the following er modeling tools to complete this assignment. gitmind - free online mind mapping diagrams.net dbdiagram.io start date jan 9, 2023 7:00 pm due date jan 23, 2023 11:59 pm
The conceptual schema and the E-R diagram are attached.
(i) Before making the E-R diagram we discuss the entity relational model for the given problem. So the E-R diagram is the graphical representation of different entities and attributes.
Assumptions for the given problem:
(i) University: a university has its name, address, email id, and mobile number
the university gives admission to students and also university and different departments.
(ii) Student: a student has attributes name, contact number, and social security number as the primary number, address, student number, birth date has its name, department, sex, class, department, and degree program.
(iii) Department: the department has its name, department code, office number, phone
(iv) Courses: each course has its name, description, course number, number of semester hours, level, and offering department.
(v) Section: section has an instructor, semester, year, course, section number
(vi) Grade report: a grade report contain a student section, letter grade, and numeric grade
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Steven wants to buy some candy with his birthday money. He can choose from Jelly Belly jelly beans that sell for $5.99 per pound and Brach's variety that sells
for $3.99 per pound. He likes to have twice the amount of Jelly Belly jelly beans as Brach's variety. If he spent a total of $63.88 how many pounds of each type of candy did he buy?
Part 1 of 2
Steven bought (blank) lb of Jelly Belly jelly beans
Part 2 of 2
Steven bought (blank) lb of Brach's variety.
Answer:
Jelly bean is 8 pound
Brach's variety is 4 pound
Step-by-step explanation:
jelly belly = x
brach's variety = y
5.99x + 3.99y = 63.88 _______(1)
x = 2y ______(2)
5.99x + 3.99y = 63.88
5.99(2y) + 3.99y = 63.88
11.98y + 3.99y = 63.88
15.97y = 63.88
15.97y/15.97 = 63.88/15.97
y = 4
Substitute y = 4 into equ 2
x = 2y
x = 2(4)
x = 8
Therefore, Steven bought 8 lb of jellybean and 4 lb of brach variety
the first part of the problem asks them to calculate the work done on the gas when the gas goes from the initial state to the final state shown. here is part of their dialogue: nora: the gas is compressed because the volume decreases. that means that there is work done on the gas, and the work is positive. bob: ok. that's right because we're using the first law with q w. we need to use that area thing to find the work. nora: yeah. we want the area under the line between the dots. bob: that line shows the process, right? nora: yup, that's the one. so the area is one-half times base times height. it's the area of that triangle. bob: but you have to add the area of that skinny rectangle below it, don't you? nora: i don't think so. that bottom dot for the final state tells you where the bottom of the area is. bob: why don't we ever agree? tell you what. we'll each try our own way and let webassign tell us who's right. nora: hmmm... i hope i get the big, green check mark!
To determine how much effort the gas undergoes as it changes from its original condition to its final one, Nora and Bob utilize the area underneath the line separating the two states and the first rule of thermodynamics.
When a gas changes from its initial condition to its final state, Nora and Bob explain how to compute the work that is done. They talk about compressing gas, thus something is being done to the gas.
You suggest using the area under the line separating the final and initial states to calculate the work done as well as the application of the first equation of thermodynamics, q=w.
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What is the result when -3x - 4 is subtracted from -4x - 3?
Cual es la propiedad de 8x0=0?
Find the degree measure of each interior angle of quadrilateral PQRS.
Note: Figure not to scale.
angle s= 2yº
angle p= 3yº
angle q= 4y°
angle r= y°
S=72, P=108, Q=144, and R=36 are the degree measures for each quadrilateral angel.
Calculating the angles:Since the sum of all angles is 360.
∠S+∠P+∠Q+∠R=360
2y+3y+4y+y=360
10y=360
y=36.
Hence, we get ∠S=72, ∠P=108, ∠Q=144 and ∠R=36.
Name some of the properties of the quadrilateral and explain what it is?A polygon with four sides, four vertices, and four angles is called a closed quadrilateral. It is made by joining four non-collinear points together. In all cases, the inner angles of quadrilaterals add up to 360 degrees.
Do the interior angles of all quadrilaterals add up to 360?The fact that the sum of the interior angles is always equal to 360 degrees is a property shared by all of the above four-sided shapes. Each interior angle of a regular quadrilateral, such as a square, will be equal to: 90 degrees is 360/4.
Are there four equal angles in every quadrilateral?They split in two directions. There are both regular and irregular quadrilaterals. The diagonals of a regular quadrilateral must cross each other and have four equal sides and angles. The only quadrilateral that meets all of these requirements is square
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Classify these events. Selecting a king from a deck of cards, replacing it and selecting a queen from the same deck. These events would be considered:
Group of answer choices
A) Subjective
B) Empirical
C) Independent
D) Dependent
Picking a king from a deck of cards, swapping it out for a queen, then continuing. These occurrences would constitute Dependent events.
'What is dependent probability?'
A dependent variable is precisely what it sounds like, just like an independent variable. It is something on which other elements depend.
A test score, for instance, may be a dependent variable because it could vary based on a number of variables, including how much you studied, how much sleep you got the night before the test, and even how hungry you were. Typically, when attempting to establish a connection between two variables, the goal is to determine what causes the dependant variable to vary in the manner that it does.
Many people struggle to recall which variable is the independent variable and which variable is the dependent variable.
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Can someone help me with this?
Answer:
9
Step-by-step explanation:
parenthesis
exponents
multiply
división
add
Subtract
do it in this order and it makes it 10 times easier
What is an equivalent way to write the expression without grouping symbols?
3(4x−5)
Drag and drop the values into the boxes to create an equivalent expression.
30POINTS! please answer!!!
The equivalent way to write the expression is 12x-15. The solution has been obtained by simplifying the expression.
What is simplified expression?
Simplified expression is that expression which does not contains any like terms. When in an expression, all the like terms are combined and all of the specified brackets are solved, then only unlike terms are left in the simplified expression which cannot be further reduced.
We are given 3(4x-5).
There are no like terms in the given expression.
Therefore, as the next step, we need to solve the bracket for simplifying the expression.
Thus, we obtain
⇒ 3*4x - 5*3
⇒ 12x -15
Hence, the equivalent way to write the given expression is 12x-15.
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In a class there are 42 students. the numbers of girls is 2times the numbers of boys. how many boys and girls are there in the class
Answer: 14 boys & 28 girls
Step-by-step explanation:
the number of girls= 2x
the number of boys= x
now create an algebraic equation:
x + 2x = 42
3x = 42
x = 14 (boys)
multiply 14 by 2 as the number of girls is two times the number of boys.
14 x 2 = 28 girls
Determine whether the function represents exponential growth or exponential decay. Identify the percent rate of change. Y=30(0.95)t
y = 30 ( 0.95 ) t y=30(0.95)^t y=30(0.95)t is an exponential decay function. Because 0.95 < 1 0.95<1 0.95<1. Therefore, 0.95 0.95 0.95 is the decay factor 1 − r 1-r 1−r.
What is meant by function ?
Functional refers to an object's operation or usefulness. It also denotes that it relates to how it performs or operates.A mathematical machine known as a function accepts one or more numbers as inputs and outputs another number.One or more functions may be inputs to a functional, which outputs a number. A Functional is a function of Functions, then.All biological activities, including digestion, elimination, respiration, and others, can be performed by a cell.The functional unit of life is referred to be such for this reason. Each and every living thing is made up of cells, which are the smallest unit of life.To learn more about function refer to
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1. You are the manager of a small store that specializes in hats, sunglasses, and other accessories. You are considering a sales promotion of a new line of hats and sunglasses. You will offer the sunglasses only to those who purchase two or more hats, so you will sell at least twice as many hats as pairs of sunglasses. Moreover, your supplier tells you that, due to seasonal demand, your order of sunglasses cannot exceed 100 pairs. To ensure that the sale items fill out the large display you have set aside, you estimate that you should order at least 210 items in all. Assume that you will lose $3 on every hat and $2 on every pair of sunglasses sold. Given the constraints above, how many hats and pairs of sunglasses should you order to lose the least amount of money in the sales promotion? [Using Graphic method]
Graphical techniques are widely used to assess whether the data support the fractionation of a single source or the mixture of two putative sources.
For 210 items in total there will be 140 number of hats and 70 number of pairs of sunglasses.
What is meant by Graphic method?To determine if the data support the mixing of two potential sources or the fractionation of a single source, graphical approaches are frequently used.
To better interpret their data, researchers now employ a variety of graphical tools, including scatterplots, box plots, and histograms. Graphs are excellent for illustrating trends, patterns, and correlations between variables as well as for displaying and summarizing vast volumes of numerical data.
We generate a graph for each equation in order to solve systems of equations or simultaneous equations graphically. Then, we search for the intersection of the two graphs. The system of equations would have a solution at the coordinates of the point of intersection.
Let x be the number of hats and y be the number of pairs of sunglasses.
The constraints exists:
2y = x (because for every 2 hats sold, 1 pair of sunglasses is sold)
y <= 100 (because the supplier can only provide 100 pairs of sunglasses)
x + y >= 210 (because you want to sell at least 210 items in total)
x, y >= 0 (because the number of items sold cannot be negative)
Lose in every hat is $3 and in Pair of Sunglasses is $ 2
Total loss in promotion or cost of promotion is
3x + 2y
For least purchase,
x + y = 210
⇒ x + x/2 = 210
⇒3x/2 = 210
⇒ x =140
y = 140/2 = 70
So, for 210 items in total there will be 140 number of hats and 70 number of pairs of sunglasses.
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For 210 items in total there will be 140 number of hats and 70 number of pairs of sunglasses.
What is meant by total?
Total refers to an entire or complete amount, and the verb "to total" means to combine two numbers or to destroy anything. In mathematics, you add integers to get the total; the outcome is the total. The result of adding 8 and 8 is 16. A car that has been totalled in an accident is irreparably damaged.the outcome of addition. For instance, adding 3 and 2 balls results in a total of 5 balls. Another illustration: The result of 12 + 4 + 6 = 22 is 22. Math is enjoyable.When you add up or count up all of something's components, you obtain the entire number or cost of that thing.Let x be the number of hats and y be the number of pairs of sunglasses.
The constraints exists:
2y = x (because for every 2 hats sold, 1 pair of sunglasses is sold)
y <= 100 (because the supplier can only provide 100 pairs of sunglasses)
x + y >= 210 (because you want to sell at least 210 items in total)
x, y >= 0 (because the number of items sold cannot be negative)
Lose in every hat is $3 and in Pair of Sunglasses is $ 2
Total loss in promotion or cost of promotion is
3x + 2y
For least purchase,
x + y = 210
⇒ x + x/2 = 210
⇒3x/2 = 210
⇒ x =140
y = 140/2 = 70
So, for 210 items in total there will be 140 number of hats and 70 number of pairs of sunglasses.
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The sum of four consecutive integers is 105. What are the integers?
The depth , in inches, of snow in my yard hours after it started snowing this morning given by =1.4+3. If the depth of snow is 9 inches now, what will the depth be one hour from now?
The depth of snow in the yard one hour after the snow depth was 9 inches, obtained using the linear function used for finding the snow depth, D = 1.4·t + 3 is; 10.4
What is a linear function?A linear function is a function that produces a straight line graph when plotted.
The function that can be used to calculate the depth of the snow D in inches is; D = 1.4·t + 3
Where;
D = The snow depth in inches
t = The time duration (hours)
The present depth of the snow = 9 inches
The depth of the snow one hour from now can be found as follows;
The function for the depth of the snow; D = 1.4·t + 3 is a linear function
Therefore, the slope, which is the coefficient of the input variable, t, when the coefficient of the output variable, D, is 1
Therefore, the slope = 1.4
The slope is the rate of change of the output variable D when the input variable increases by one unit
The slope therefore, indicates that the depth of the snow in the yard, increases by 1,4 inches per hour
If the depth of the snow is presently 9 inches, the depth of the snow 1 hour later is therefore; D = 9 inches + 1.4 inches = 10.4 inches
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what is the width of an rectangle with a length of 3/7 ft and a area of 2ft²? write an equation to show your work.
Width of the rectangle is 14/3 feet.
What is Rectangle?Rectangle is a two dimensional figure which has four sides and four angles and all the angles are right angles.
Area of a rectangle = Length × Width
Length = 3/7 feet
Area = 2 feet²
Substituting in the equation for the area,
2 = 3/7 × Width
Width = (2 × 7) / 3
= 14/3 feet
Hence the measure of width of the rectangle is 14/3 feet.
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Suppose y = c1ekx + c2e-kx where k > 0 is a constant, and c1 and c2 are arbitrary constants. Find the following. Enter c1 as c1 and c2 as c2
The c1 and c2 are arbitrary constants.
What is the arbitrary constant?
y=mx+c is the general equation of a straight line in two dimensions, where m and c are arbitrary constants that represent the gradient of the line and the y-intercept. And ∫2x dx=x 2+c where c is an arbitrary constant whose value can be determined by a boundary condition.
y = c1ekx + c2e-kx is a general solution to the homogeneous linear differential equation y' + ky = 0. This means that any specific solution to this equation can be obtained by plugging in appropriate values for c1 and c2.
The general solution y = c1ekx + c2e-kx is a linear combination of ekx and e-kx.
The differential equation y' + ky = 0 is a first-order, linear differential equation.
The general solution y = c1ekx + c2e-kx is valid for all values of x.
The general solution y = c1ekx + c2e-kx is a sum of two linearly independent solutions to the differential equation.
The general solution y = c1ekx + c2e-kx is a particular solution if c1 and c2 are determined by initial or boundary conditions.
The general solution y = c1ekx + c2e-kx is a general solution if c1 and c2 are arbitrary constants.
The general solution y = c1ekx + c2e-kx is a unique solution if c1 and c2 are determined by initial or boundary conditions, otherwise, it is not unique.
The general solution y = c1ekx + c2e-kx is a solution to the homogeneous differential equation y' + ky = 0, it does not contain any particular solution for the non-homogeneous differential equation.
Hence, The c1 and c2 are arbitrary constants.
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The sum of a number k and t is 11
The mathematical sentence can be written as the equation below,
k + t = 11
How to write the sentence as an equation?Here we want to write the the mathematical sentence below:
"The sum of a number k and t is 11"
As an algebraic equation, so let's analyze each part of the mathematical sentence.
"sum" refers to an addition, in this case between two numbers represented by k and t, then we can write:
k + t
And that is equal to 11, then the equation will just be:
k + t = 11
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find the fixed point of the function f. [a fixed point of a function f is a real number c such that f(c)
The fixed point of the function f(x) = x^2 + 1 is c = 0.
f(x) = x^2 + 1
c = 0
To find the fixed point of the function f, we can set the equation f(x) = x^2 + 1 equal to x, since the fixed point is when f(x) is equal to x.
Therefore, we can solve for x by setting x^2 + 1 = x. Re-arranging the equation, we get x^2 - x + 1 = 0.
Using the quadratic formula, x = (-(-1) ± √(1 - 4(1)(1))/(2(1))
Simplifying this expression, x = (-1 ± √3)/2.
Since we are looking for a real number, we can eliminate the negative solution and set x = (1 + √3)/2.
Therefore, the fixed point of the function f is c = 0.
The fixed point of the function f(x) = x^2 + 1 is c = 0.
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Identify whether the statement represents an exponential function. Explain.A population of bacteria decreases by a factor of 18 every 24 hours.Yes, the statement is an exponential function because the population decreases daily by a proportional rate.Yes, the statement is an exponential function because the population decreases daily by a constant rate.No, the statement is not an exponential function because the population decreases daily by a proportional rate.No, the statement is not an exponential function because the population decreases daily by a constant rate.No, the statement is not an exponential function because it is a quadratic function.
Option A is correct i.e., Yes, the statement is an exponential function because the population decreases daily by a proportional rate.
Exponents in mathematics are usually defined as when a number or an expression is given a specific power. The number to which the power is assigned is multiplied power times itself to get the final value. It is depicted by raised symbol written beside the number or expression.
It is written as a∧b where a is the base and b is the power and is called b to the base a or a raised to bth power. An example of the same can be 2∧3= 2x2x2=8, [tex]a^{3}[/tex]=axaxa, etc.
There are some properties of exponents like
[tex]a^{b}[/tex] + [tex]a^{c}[/tex]= a∧bc E.g., [tex]2^{3}[/tex]+[tex]2^{4}[/tex]=[tex]2^{3.4}[/tex] =[tex]2^{12}[/tex][tex]a^{0}[/tex]=1 E.g., [tex]2^{0}[/tex]=1[tex]a^{(-3)}[/tex]=1/[tex]a^{3}[/tex] = [tex]3^{(-4)}[/tex]=1/[tex]3^{4}[/tex]To know more about exponents visit
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Suppose that h(x) is a continuous function and is defined for all x greater than or equal to 1. You are given that h(x) has critical points at x = 1, x = 3, and x = 5. If h'(x) is negative on the interval 1 < x < 3, positive on the interval 3 < x < 5, and positive on the interval x > 5, what can be said about the point (3, h(3))? O (3, h(3)) is an absolute maximum.O (3, h(3)) is an absolute minimum. O (3, h(3)) is neither an absolute maximum nor an absolute minimum. O Nothing can be determined about the point (3, h(3)).
An absolute maximum point is where the function reaches its highest value.
What is the difference between absolute maximum and absolute minimum?A function may have more than one location (x values) or point (ordered pairs) where these values occur, but there can be only one absolute minimum value and one absolute maximum value (in a given closed interval).
A point where the function achieves its highest value is known as an absolute maximum point. Similar to this, an absolute minimum point is the location at which the function's maximum value is obtained.
Minimum refers to the bare minimum that can be done. For instance, if anything requires seven dollars as the minimum payment, you cannot make a payment of six or less (you must pay at least seven). More than the bare minimum is acceptable, but not less. Maximum refers to the greatest quantity of anything.
Therefore, the correct answer is option C) (3, h(3)) is neither an absolute maximum nor an absolute minimum.
The complete question is:
Suppose that h(x) is a continuous function and is defined for all x greater than or equal to 1. You are given that h(x) has critical points at x = 1, x = 3, and x = 5. If h'(x) is negative on the interval 1 < x < 3, positive on the interval 3 < x < 5, and positive on the interval x > 5, what can be said about the point (3, h(3))?
A) (3, h(3)) is an absolute maximum.
B) (3, h(3)) is an absolute minimum.
C) (3, h(3)) is neither an absolute maximum nor an absolute minimum.
d) Nothing can be determined about the point (3, h(3)).
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