Answers
Explanation:
Limit of the function:
Limit is the approximate value of the function at a defined value of x.
[tex]\begin{gathered} \lim _{x\to a}f(x)=L \\ As\text{ }x\rightarrow a\text{ then, f(x)}\rightarrow L \end{gathered}[/tex]a)
[tex]\lim _{x\to-\infty}(2x^3-2x)=-\infty[/tex]This limit is true. As x approaches negative infinity the function is also approached to negative infinity.
b)
[tex]\lim _{x\to\infty}(-2x^4+6x^3-2x)=-\infty[/tex]As x tends to infinity. The functions tend to have negative infinity.
This statement is true.
c)
[tex]\lim _{x\to\infty}(9x^5-6x^3-x)=-\infty[/tex]When x tends to infinity, the function will also go to infinity.
So, This statement is false.
Related Questions
A rectangle has a perimeter of 68 ft. The length and width are scaled by a factor 1.5The perimeter of the resulting rectangle is ___ ft.
Answers
Step 1: Given a rectangle has a perimeter of 68 ft.
Formula
[tex]\begin{gathered} Perimeter\text{= 2(L+B)} \\ 68\text{ = 2L + 2B} \\ \end{gathered}[/tex]Step 2: Find the perimeter when scaled by a factor 1.5
Then the perimeter of the resulting rectangle
[tex]\begin{gathered} P_1=\text{ 2L + 2B} \\ P_2\text{ = 1.5(2}L\text{ +2B)} \\ P_{2_{}_{}_{}_{}_{}_{}_{}_{}_{}}\text{ = 1.5 (68)} \\ P_2\text{ = 102} \end{gathered}[/tex]Hence the perimeter of the resulting rectangle is 102ft
Each participant must pay $14 to enter the race. Each runner will be given a T-shirt that cost race organizers $3.50. If the T-shirt was the only expense for the race organizers, which of the following expressions represents the proportion of the entry fee paid by each runner that would be donated to charity? is it $14.00÷($14.00-$3.50)
Answers
From the statement of the problem, we know that each participant:
• pays $14 to enter the race,
,• receives a T-shirt that cost $3.50 to the organizers.
The earnings minus the cost of the T-shirts will be donated to charity, so for each participant, the donation will be $14 - $3.50. So the proportion of the entry fee paid by each runner that would be donated to charity is:
[tex]\frac{14.00-3.50}{14.00}[/tex]Answer
A. ($14.00 - $3.50) / $14.00
write the division expression in words and as a fraction: h ÷ 16
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we have
[tex]h\div16[/tex]in words will be
h between 16
in fraction will be
[tex]\frac{h}{16}[/tex]Write the letter for the correct answer in the blank at the right of ea1. Which of the following sets of values completes the function tableInput (x)4x + 2Output (y)454(4) +24(5) +24(6) +26A. 16, 20, 24B. 18, 19, 20C. 18, 22, 26D. 0,1,22. Molly is buying packages of party favors for herbirthday nartyr Iaina tlR
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The missing column is the output for the function, so we just need to evaluate the expression in the second column.
For the second row, we have:
[tex]4(4)+2[/tex]We first evaluate the multiplication and then the addition:
[tex]4(4)+2=16+2=18[/tex]For the third row, we have the same steps:
[tex]4(5)+2=20+2=22[/tex]And for the last too:
[tex]4(6)+2=24+2=26[/tex]Thus, the outputs are 18, 22, 26, alternative C.
I need help with this practice problem It has an additional pic of a graph that I will include.
Answers
Given:
The function is given as,
[tex]f(x)=\sin (\frac{\pi x}{2})\text{ . . . . . . (1)}[/tex]The objective is to plot the graph of the function.
Explanation:
To find the maximum point, consider x = 1 in the equation (1),
[tex]\begin{gathered} f(1)=\sin (\frac{\pi(1)}{2}) \\ f(1)=\sin (\frac{\pi}{2}) \\ f(1)=1 \end{gathered}[/tex]Thus, the coordinate is (1,1).
To find the minimum point, consider x = -1 in equation (1).
[tex]\begin{gathered} f(-1)=\sin (\frac{\pi(-1)}{2}) \\ f(-1)=\sin (\frac{-\pi}{2}) \\ f(-1)=-\sin (\frac{\pi}{2}) \\ f(-1)=-1 \end{gathered}[/tex]Thus, the coordinate is (-1,-1).
To plot the graph:
The graph of the function will be,
Hence, the graph of the function is obtained.
Hi there,i am having some trouble solving the following two questions relating to extrema and intervals:
Answers
We have the following function:
[tex]y=ln|x^2+x-20|[/tex]The graph of this function is given by:
We know that a function is increasing is the first derivative is greater than zero. The derivative of the given function is given by
[tex]\frac{dy}{dx}=\frac{2x+1}{x^2+x-20}[/tex]Then, the condition is given by
[tex]\frac{dy}{dx}=\frac{2x+1}{x^{2}+x-20}>0[/tex]which implies that
[tex]\begin{gathered} 2x+1>0 \\ then \\ x<-\frac{1}{2} \end{gathered}[/tex]By means of this result and the graph from above, f(x) is increasing for x in:
[tex](-5,-\frac{1}{2})\cup(4,\infty)[/tex]Now, the function is decreasing when
[tex]\frac{dy}{dx}<0[/tex]which give us
[tex]\begin{gathered} \frac{dy}{dx}=\frac{2x+1}{x^{2}+x-20}<0 \\ 2x+1<0 \\ then \\ x>-\frac{1}{2} \end{gathered}[/tex]Then, by means of this result and the graph from above, the function is decreasing on the interval:
[tex](-\infty,-5)\cup(-\frac{1}{2},4)[/tex]In order to find the local extremal values, we need to find the second derivative of the given function, that is,
[tex]\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{2x+1}{x^2+x-20})[/tex]which gives
[tex]\frac{d^2y}{dx^2}=\frac{(x^2+x-20)(2)-(2x+1)(2x+1)}{(x^2+x-20)^2}[/tex]or equivalently
[tex]\frac{d^2y}{dx^2}=\frac{2(x^2+x-20)-(4x^2+4x+1)}{(x^2+x-20)^2}[/tex]which can be written as
[tex]\frac{d^2y}{dx^2}=\frac{2x^2+2x-40-4x^2-4x-1}{(x^2+x-20)^2}[/tex]then, we get
[tex]\frac{d^2y}{dx^2}=\frac{-2x^2-2x-41}{(x^2+x-20)^2}[/tex]From the above computations, the critical value point is obtained from the condition
[tex]\frac{dy}{dx}=\frac{2x+1}{x^{2}+x-20}=0[/tex]which gives
[tex]\begin{gathered} 2x+1=0 \\ then \\ x=-\frac{1}{2} \end{gathered}[/tex]This means that the critical point (maximum or minimum) is located at
[tex]x=-\frac{1}{2}[/tex]In order to check if this value corresponds to a maximum or mininum, we need to substitute it into the second derivative result, that is,
[tex]\frac{d^{2}y}{dx^{2}}=\frac{-2(-\frac{1}{2})^2-2(-\frac{1}{2})-41}{((-\frac{1}{2})^2+(-\frac{1}{2})-20)^2}[/tex]The denimator will be positive because we have it is raised to the power 2, so we need to check the numerator:
[tex]-2(-\frac{1}{2})^2-2(-\frac{1}{2})-41=-\frac{2}{4}+1-41=-40.5[/tex]which is negative. This means that the second derivative evalueated at the critical point is negative:
[tex]\frac{d^2y}{dx^2}<0[/tex]which tell us that the critical value of x= -1/2 corresponds to a maximum.
Since there is only one critical point, we get:
f(x) has a local minimum at x= DNE
f(x) has a local maximum at x= -1/2
Krista wants to paint her house. she buys 7 1/2. gallons of paint. she uses 3/5 of the paint on the front of her house and then buys 1. 1/2. more gallons of paints how many gallons of paint does she have left
Answers
We know that 7 1/2 is equivalent to 15/2.
If she uses 3/5, then it would remain 9/2 gallons.
[tex]\frac{15}{2}\cdot\frac{3}{5}=\frac{9}{2}=4.5[/tex]Then, she buys 1 1/2, which is equivalent to 1.5 or 3/2.
So, she has 6 gallons of paint.[tex]4.5+1.5=6[/tex]Before sketching the graph, determine where the function has its minimum or maximum value so you can place your first point there.
Answers
We will have the following:
*The zeros in the function are at the values:
[tex]\begin{gathered} -0.5|x-2|=-2\Rightarrow|x-2|=4 \\ x=-2 \\ x=6 \end{gathered}[/tex]So, the zeros are at x = -2 an x = 6.
*The x-intercepts are at the points:
[tex](-2,0)[/tex]And
[tex](6,0)[/tex]*The y-intercept is at the point:
[tex]y=-0.5|0-2|+2\Rightarrow y=1[/tex]So, the y-intercept is located at the point:
[tex](0,1)[/tex]Which equation represents a line which is parallel to the line Y = -4/5x -8?
Answers
1) Since we have a line described by this equation y=-4/5x -8, and it is written in the slope-intercept form. It is convenient for us to rewrite those into the slope-intercept form.
a) 5y-4x=-10 Add 4x to both sides
5y = -10 +4x Divide both sides by 5
y= -10/5 +4/5x
y= 4/5x -2
b) 5x-4y = -20 Subtract 5x to both sides
-4y= -20 -5x Divide both sides by -4
y = 5 +5/4x
y= 5/4x +5
c) 5x +4y = -24
4y = -24 -5x
y = -6 -5/4x
d) 4x +5y= 35
5y= 35-4x
y= 7-4/5x
2) Since Parallel lines have the same slope, then the line parallel to Y = -4/5x -8 is y= 7-4/5x (4x +5y=35)
I need help with this problem I don’t understand it. The question is. Find the value of X___degreesY___degreesZ___degrees
Answers
SOLUTION
The fugure in the picture is a trapezoid.
Adjacent angles between the bases of a trapezoid are suplementary (add up to 180 degrees)
The bases of this trapezoid are the parallel sides PT and AR, so
[tex]x\degree+y\degree=180\degree[/tex]Now, since the trapezoid is an isosceles trapzoid, then the base angles will be equal, hence
[tex]x=53\degree[/tex]Hence x = 53 degrees
Then
[tex]\begin{gathered} x+y=180\degree \\ 53+y=180\degree \\ y=180-53 \\ y=127\degree \end{gathered}[/tex]hence y = 127 degrees
Also since it is an isosceles triangle the sides TR and AP would be equal
Hence z = 8
The variables x = 3/4,y= 2/9 z = 6 are related in such a way that zvaries jointly with x and y.Find z when x = --3 and y = 5.
Answers
Given:
x=3/4, y=2/9 and z=6
z varies jointly with x and y.
From the above data, we can obtain a relation connecting x and z.
The relation connecting z and x can be written as,
[tex]z=8x[/tex]Putting x=3/4 in the above equation,
[tex]z=8\times\frac{3}{4}=6[/tex]So, the relation z=8x is satisfied.
Similarly, the relation between z and y can be written as
Point A is located at (-3,5). Find its new coordinates after it is reflected along the x-axis then dilated using a scale factor of 4 with center of dilation at the origin.
Answers
We have (-3,5)
The rule for reflection around the x-axis is
[tex](x,y)\rightarrow(x,-y)[/tex]so the point after the reflection is around x-axis
[tex](-3,5)=(-3,-5)[/tex]for the dilatation we need to multiply the point find above by 4
[tex]A^{\prime}=(-3(4),-5(4))=(-12,-20)[/tex]How do I rearrange 2x + y = -3 Into a Y = mx + b
Answers
Solution
Given
[tex]2x+y=-3[/tex]Move the term 2x to the right-hand side and change its sign
[tex]y=-2x-3[/tex]Therefore, the required answer is
[tex]y=-2x-3[/tex]Jim has a new job and earns a salary of $46,000. Valorie has a new job and earns a salary of $59,000. Jim will receive a salary increaseof $2,700 per year and Valorie will receive a salary increase of $1,500 per year. Based on this information, which TWO statements arecorrect?es )A)It will take 10 years for Jim to earn the same salary as Valorie.B)It will take 12 years for Jim to earn the same salary as Valorie.D)When solved for x, 46,000x + 2,700 = 58,000x + 1,500 gives the number ofyears it will take Jim to earn the same salary as Valorie.When solved for x, 46,000 + 2,700x = 58,000 + 1,500x gives the number ofyears it will take Jim to earn the same salary as Valorie.When solved for x, 46,000x + 2,700x = 58,000x + 1,500x gives the numberof years it will take Jim to earn the same salary as Valorie.E)
Answers
Let's use the variable x to represent the number of years.
So, if the initial salary of Jim is $46,000 and it increases by $2,700 each year, after x years, his salary is:
[tex]46000+2700x[/tex]Doing the same for Valorire, her salary is:
[tex]58000+1500x[/tex]In order to find after how many years their salary will be the same, we can equate both salaries and calculate the value of x:
[tex]\begin{gathered} 46000+2700x=58000+1500x \\ 2700x-1500x=58000-46000 \\ 1200x=12000 \\ x=\frac{12000}{1200} \\ x=10 \end{gathered}[/tex]So let's check each option:
A.
True, it takes 10 years to they have the same salary.
B.
False, after 12 years Jim's salary is higher than Valorie's salary.
C.
False, the variable x should multiply the increase per year in the salary, not the initial salary.
D.
True, that's the equation and procedure we used.
E.
False, the variable x should multiply just the increase per year in the salary, not the initial salary.
So the correct options are A and D.
Given the triangle, select all statements that must be true
Answers
First we need to find the value of x
Remember that the sum of the interior angles of a triangle must be 180
(9x+3) +(5x-2)+(11x-21)=180
25 x - 20 = 180
x=8
the angle E =9(8)+3=75
the angle F= 11(8)-21=67
the angle D= 5(8)-2=38
the shortest side is EF
the longest side is DF
DF > DE >EF
the statements that are true are
d and e
134224454321₁ de dCHHERWhich system of linear inequalities is represented by thegraph?Oy²x +3 and 3x − y > 2O y ≥=x + 3 and 3x − y > 20 y ²Ox-x + 3 and 3x +y>2O y ≥x + 3 and 2x -y > 2
Answers
Given,
The graph of the inequalities is,
Required
The equation of the inequalities.
Here,
The y intercept of the red line is (0,3).
The y intercept of the grey line is(0,-2).
The slope of the red line is,
[tex]Slope=\frac{4-3}{3-0}=\frac{1}{3}[/tex]The slope of the grey line is,
[tex]\begin{gathered} Slope=\frac{4-(-2)}{2-0} \\ =3 \end{gathered}[/tex]So, the inequalities of the graph are
[tex]\begin{gathered} y\ge\frac{1}{3}x+3 \\ 3x-y>2 \end{gathered}[/tex]Hence, option A is correct.
A figure skating school offers introductory lessons at $25 per session. There is also a registration fee of $30. Write a linear equation in slope- intercept form that represents the situation.Part B: You want to take at least 6 lessons, what is the cost.
Answers
Given:
Registration fee = $30
Fee per session = $25
To write a linear equation in slope-intercept form, we have:
Using the general slope intercept form:
y = mx + b
where
y = Total fee
m = fee per session
x = number of lessons
b = registration fee
Therefore, we have:
y = 25x + 30
Part B.
To find the cost if you want to take at least 6 lessons:
y = 25(6) + 30
= 150 + 30
= 180
The cost of taking at least 6 lessons is $180
ANSWER:
a) y = 25x + 30
b) $180
6c + 14 = -5c + 4 + 9csolve for c
Answers
This is a simple question we just need to reorganize our equation and solve it to find the value of "c" as follows:
We can see while we are reorganizing our equation we move some numbers and variables from a term to another and when we do that we need to change its sign as we did above. Now let's keep going on our calculation:
As we can see, our final answer is c = -5
list all congruent pairs of congruent angles and write the ratios of the corresponding side lengths
Answers
Scale factor = 8/6 = 4/3 = 1.33333
Pairs of congruent angles;
m
m
m
Ratio of the coresponding side lengths
AC: BC = LN : MN
4.5 : 6 = 6 : 8
in CDE, J is the centroid. If JF=15 find EJ
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From the figure, J is the centroid. Hence, the lines DH, FE and CG are medians.
Therefore, we can apply the 2/3 rule, that is, the centroid is 2/3 of the way from the vertex to the opposite midpoint.
In other words, we can write
[tex]JE=\frac{2}{3}FE[/tex]since, we know that FE=FJ+JE, we have
[tex]JE=\frac{2}{3}(FJ+JE)[/tex]and, from this equation we can find JE since FJ=15:
[tex]JE=\frac{2}{3}(15+JE)[/tex]The, we obtain
[tex]\begin{gathered} JE=\frac{2}{3}(15)+\frac{2}{3}JE \\ JE-\frac{2}{3}JE=\frac{2}{3}(3\cdot5) \\ \end{gathered}[/tex]in which we moved (2/3)JE to the left hand side and we wrote 15 as 3*5. Now, it reads
[tex]\begin{gathered} \frac{3}{3}JE-\frac{2}{3}JE=2\cdot5 \\ \frac{1}{3}JE=10 \\ JE=3\cdot10 \\ JE=30 \end{gathered}[/tex]Therefore, JE=EJ=30.
Point Fis on line segment EG. Given EG = 5x + 7, EF = 5x, andFG = 2x – 7, determine the numerical length of FG.
Answers
To solve this, we will follow the steps below:
EG = EF + FG -------------------------------------(1)
From the question,
EG = 5x + 7 EF = 5x FG = 2x – 7
Substituting the above into equation(1)
5x+7 = 5x + 2x-7
5x+7 = 7x - 7
Collect like-term
7+7 = 7x-5x
14 = 2x
Divide both-side of the equation by 2
14/2 = 2x/2
7 = x
x=7
Substituting the value of x in poin FG
Point FG = 2x – 7 = 2(7) - 7 = 14 - 7 =7
Therefore, the numerical length of FG is 7
Step 1 of 2: Reduce the rational expression to lowest terms x/x^2 - 4xStep 2 of 2: Find the restricted values of X, if any, for the given rational expression.
Answers
We have the following expression:
[tex]\frac{x}{x^2-4x}[/tex]Step 1. Reduce the rational expression to the lowest tem
By factoring the variable x, we get
[tex]\frac{x}{x(x-4)}[/tex]We can cancel x out as long as x is different from zero. Then one restricted value is x=0. So, If x is different from zero, our expression can be reduced to
[tex]\frac{x}{x^2-4x}=\frac{1}{x-4}[/tex]but x must be different from 4.
Step 2. Find the restricted values of x.
Since x can not be zero or four, the restricted values are x=0 and x=4
if LN=3x-12, LM=20 and MN=x-2 find MN
Answers
LN + NM = LM
(3x - 12) + NM = 20
NM= 20 - 3x +12
Now replace NM = MN
x - 2= 20 - 3x + 12
Now find x
x + 3x = 20 + 12 + 2
4x = 34
x= 34/4= 17/2= 8.5
Now replace x in MN
MN= x-2 = 8.5 - 2= 6.5
Then
MN= 6.5
Write a two variable equation to represent how many miles Eman can walk over any time interval. Use x as your variable.
Answers
a dart hits the dartboard shown find the probability that it lands in the shaded regions? I don't know how to do this help
Answers
The probabiity of a dart landing on the shaded regions is 0.2125
Here, we want to find the probability that the dartboard hits the shaded regions
Mathematicaly, all we have to do is to calculate the portion of the board that is shaded
All we have to do is to subtract the area of the 4 circles from the area of the given square
Area of the square is the square of its side
We have this as;
[tex]\text{Area of square = 4}\times4=16m^2[/tex]The circles are similar
Thus, their areas are same
We only need to calculate the area of one of the circles and multiply by four
As we can see, 4 m represents the diameters of two circles combined
The diameter of one of the circles will thus be 2 m
The radius we will need in the calculations is half of this
We have the radius as 2/2 = 1 m
The area of the circles is thus;
[tex]4\times\pi\times r^2\text{ = 4}\times3.142\text{ }\times1^2=12.6m^2[/tex]The area of the shaded part is thus;
[tex]16-12.6=3.4m^2[/tex]So, the probability of the dart hitting the shaded portion will be;
[tex]\frac{3.4}{16}\text{ = 0.2125}[/tex]The equation for a proportional relationship is y=531. The graph of therelationship passes through the point (1,58) which represents the
Answers
The given equation is
[tex]y=5.8x[/tex]Where 5.8 is the constant of proportionality.
Additionally, the point (1, 5.8) represents the constant of proportionality because it's telling the constant ratio of change between variables.
Therefore, the answer is a constant ratio of change.
Simplify (combine like terms): -8x^4 + 7x^3 + 5x + 4x^4 - 6x^3 +2
Answers
We will have the following:
[tex]-8x^4+7x^3+5x+4x^4-6x^3+2=(-8x^4+4x^4)+(7x^3-6x^3)+5x+2[/tex][tex]=-4x^4+x^3+5x+2[/tex](See image below) The point A(-15) is rotated 270° clockwise about the origin. The coordinates of A' are?
Answers
Given:
A (-1,5) rotated 270 clockwise.
Find: A' coordinates
Sol:.
(x,y) Rotated by 270 become (y,-x).
A = (-1,5)
[tex]\begin{gathered} A=(-1,5) \\ A^{\prime}=(5,-(-1)) \\ A\text{'=(5,1)} \end{gathered}[/tex]You need to have $80,000 saved in 10 years. If your account earns 5% compounded monthly, how much would you need to deposit each month to reach your goal? Round answer to the nearest cent.
Answers
To determine the compound interest on monthly:
The following graph shows the proportional relationship between the distance a train travels and the amount of time it spends traveling.Which statements about the graph are true? Choose all answers that apply:A.) The y-coordinate of point A represents the total distance a train travels in 500 minutes.B.) The train travels 375 km in 3 hours.C.) None of the above
Answers
By looking at the graph:
A) FALSE
y- coordinate of point A Represents the total distance a train travel in 4 hours .
4 hours = 240 minutes
B) TRUE
When x = 3 (3 hours ) y= 375
Statement B is true