Answer:
[tex]y-3=-\frac{1}{2}(x-2)[/tex]
Step-by-step explanation:
The slope is
[tex]\frac{6-3}{-4-2}=-\frac{1}{2}[/tex]
So, using the point (2,3), the equation is
[tex]y-3=-\frac{1}{2}(x-2)[/tex]
The data set below has 7 values. Find the mean absolute deviation for the data set. If necessary, round your answer to the nearest hundredth. 14, 13, 16, 12, 17, 21, 26
Answer:
Hey stu132057, your the MAD for the data set given is
Step-by-step explanation:
Step 1: Find the mean of the data: (14+13+16+12+17+21+26)/7 = 17
Step 2: Find the difference between each data and mean:Difference between 14 and 17 is 3
Difference between 13 and 17 is 4
Difference between 16 and 17 is 1
Difference between 12 and 17 is 5
Difference between 17 and 17 is 0
Difference between 21 and 17 is 4
Difference between 26 and 17 is 9
Step 3: Add all the differences: 3+4+1+5+0+4+9 = 26
Step 4: Divide it by the number of data:26/7 = 3.7
So, the MAD = 3.7
-------------------------
Have a great day,
Nish
the product of 1540 and m is a square number. find the smallest possible value of m
The smallest possible value of m according to the task is; 1/1540.
What is the smallest possible value of m?Since it follows from the task content that the product of 1540 and m is a square number and the smallest possible small number is; 1.
The equation which holds true is; 1540 × m = 1
Consequently, m = 1/1540.
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if in Ashford the lights come on at 6:20pm and go off 11 1/2 hours later what time will it be?
A 75-gallon tank is filled with brine (water nearly saturated with salt; used as a preservative) holding 11 pounds of salt in solution. A salt solution containing 0.6 pounds of salt per gallon is added to the tank at the rate of 3gal/min. The contents of the tank are continuously and thoroughly mixed and drained out at thirteen quarts per minute. What is the amount of salt in the tank after an hour
Let [tex]A(t)[/tex] = amount of salt (in pounds) in the tank at time [tex]t[/tex] (in minutes). Then [tex]A(0) = 11[/tex].
Salt flows in at a rate
[tex]\left(0.6\dfrac{\rm lb}{\rm gal}\right) \left(3\dfrac{\rm gal}{\rm min}\right) = \dfrac95 \dfrac{\rm lb}{\rm min}[/tex]
and flows out at a rate
[tex]\left(\dfrac{A(t)\,\rm lb}{75\,\rm gal + \left(3\frac{\rm gal}{\rm min} - 3.25\frac{\rm gal}{\rm min}\right)t}\right) \left(3.25\dfrac{\rm gal}{\rm min}\right) = \dfrac{13A(t)}{300-t} \dfrac{\rm lb}{\rm min}[/tex]
where 4 quarts = 1 gallon so 13 quarts = 3.25 gallon.
Then the net rate of salt flow is given by the differential equation
[tex]\dfrac{dA}{dt} = \dfrac95 - \dfrac{13A}{300-t}[/tex]
which I'll solve with the integrating factor method.
[tex]\dfrac{dA}{dt} + \dfrac{13}{300-t} A = \dfrac95[/tex]
[tex]-\dfrac1{(300-t)^{13}} \dfrac{dA}{dt} - \dfrac{13}{(300-t)^{14}} A = -\dfrac9{5(300-t)^{13}}[/tex]
[tex]\dfrac d{dt} \left(-\dfrac1{(300-t)^{13}} A\right) = -\dfrac9{5(300-t)^{13}}[/tex]
Integrate both sides. By the fundamental theorem of calculus,
[tex]\displaystyle -\dfrac1{(300-t)^{13}} A = -\dfrac1{(300-t)^{13}} A\bigg|_{t=0} - \frac95 \int_0^t \frac{du}{(300-u)^{13}} [/tex]
[tex]\displaystyle -\dfrac1{(300-t)^{13}} A = -\dfrac{11}{300^{13}} - \frac95 \times \dfrac1{12} \left(\frac1{(300-t)^{12}} - \frac1{300^{12}}\right) [/tex]
[tex]\displaystyle -\dfrac1{(300-t)^{13}} A = \dfrac{34}{300^{13}} - \frac3{20}\frac1{(300-t)^{12}}[/tex]
[tex]\displaystyle A = \frac3{20} (300-t) - \dfrac{34}{300^{13}}(300-t)^{13}[/tex]
[tex]\displaystyle A = 45 \left(1 - \frac t{300}\right) - 34 \left(1 - \frac t{300}\right)^{13}[/tex]
After 1 hour = 60 minutes, the tank will contain
[tex]A(60) = 45 \left(1 - \dfrac {60}{300}\right) - 34 \left(1 - \dfrac {60}{300}\right)^{13} = 45\left(\dfrac45\right) - 34 \left(\dfrac45\right)^{13} \approx 34.131[/tex]
pounds of salt.
4. The figure shows a rectangular tank.
(a) The capacity of the tank is ___ L.
(b) After pouring 10L of water into the tank, the depth of water will be __ cm.
Answer:
20 L volume and 12.5 cm depth.
Step-by-step explanation:
See attached image.
Let S1 be the amount allowing 10% discount on the price of some goods and then adding 10% VAT. Let S2 be the amount adding 10% VAT and then allowing 10% discount. Which of the following is true? (i) S1 > S2 (ii) S1 < S2 (iii) S1 = S2 (iv) Cant be decided
Using decimal multipliers, the correct statement is given by:
(iii) S1 = S2.
What is a decimal multiplier?Increases of a% or decreases of a% re represented by decimal values, as follows:
The equivalent multiplier for an increase of a% is given by: [tex]\frac{100 + a}{100}[/tex]The equivalent multiplier for an decrease of a% is given by: [tex]\frac{100 - a}{100}[/tex]Hence, in the context of this problem:
An increase of 10% is represented by 1.1.A decrease of 10% is represented by 0.9.The options are given as follows:
S1 = 0.9 x 1.1 = 0.99.S2 = 1.1 x 0.9 = 0.99.They are equal, hence option (iii) is correct.
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What is the answer ?
∠ADB + ∠BDC = ∠ADC
39° + (3x - 4) = 8x + 5
3x - 4 = 8x + 5 - 39
3x - 4 = 8x - 34
34 - 4 = 8x - 3x
5x = 30
x = 6
∠ADC = 39° + (3(6) - 4) = 39 + 14 = 53°
Hope it helps!
Suppose cos(x) =1/(sqrt(5))
and sin(x) >0. what is the value of tan(2x)?
Answer:
[tex]tan(2\theta) = -\frac{4}{3}\\[/tex]
Step-by-step explanation:
So cos is defined as: [tex]cos(\theta) = \frac{adjacent}{hypotenuse}[/tex], meaning we can tell that the adjacent side is 1, and the hypotenuse is 5, from the fraction you gave.
Using this we can solve for the opposite side.
[tex]1^2 + b^2 = \sqrt{5}^2\\1+b^2 = 5\\b^2=4\\b=2[/tex]
Now it's important to note, that b can be a negative number, so we have to use the information that sin(x) > 0, to determine the length of this side.
The sin is defined as: [tex]sin(\theta) = \frac{opposite}{hypotenuse}[/tex], and since we we're solving for the opposite side, this means that the value +\- 2, is in the top, and since the hypotenuse is positive, this means that the opposite side is also positive.
This also tells us one more thing, since both cos(x) and sin(x) are positive, we are dealing with a angle in the first quadrant.
So we can now define sin(x), using the opposite (2) and the hypotenuse (sqrt(5))
[tex]sin(\theta) = \frac{2}{\sqrt{5}}[/tex]
And we can rationalize the denominator for both the cosine and sine, by multiplying by the square root in the denominator so that
[tex]sin(\theta) = \frac{2\sqrt{5}}{5}\\\\cos(\theta) = \frac{\sqrt{5}}{5}[/tex]
Now we can define the value of tan(2 theta) using the double angle-identities such that:
[tex]tan(2\theta) = \frac{2\ tan(\theta)}{1-tan^2{\theta}}[/tex]
And we can also define tan(theta) using the definition that:
[tex]tan(\theta) = \frac{sin(\theta)}{cos(\theta)}[/tex]
So plugging in the values sin(theta) and cos(theta) we get the following:
[tex]tan(\theta) = \frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{5}}{5}}\\\\tan(\theta) = \frac{2\sqrt{5}}{5} * \frac{5}{\sqrt{5}}\\\\tan(\theta) = 2[/tex]
Btw in the last step, I just canceled out the 5 and sqrt(5) since they were both in the denominator and numerator
So now let's plug this value, 2 as tan(theta) into the equation
[tex]tan(2\theta) = \frac{2\ *2}{1-2^2}\\\\tan(2\theta) = \frac{4}{-3}\\tan(2\theta) = -\frac{4}{3}\\[/tex]
For the polynomial f(x)=x^3-kx^2+x+6, find the value of k if (x+1) is a factor of f(x).
The value of k if (x+1) is a factor of f(x) is -4
How to determine the value of k?The polynomial function is given as:
f(x)=x^3-kx^2+x+6
(x+1) is a factor of f(x)
So, we start by setting x + 1 to 0
x + 1 = 0
Solve for x
x = -1
Substitute x = -1 in f(x)=x^3-kx^2+x+6 and set the equation to 0
(-1)^3-k(-1)^2+(-1)+6 = 0
Evaluate the exponents
-1 - k - 1 + 6 = 0
Evaluate the like terms
k + 4 = 0
Solve for k
k = -4
Hence, the value of k if (x+1) is a factor of f(x) is -4
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PLEASE HELP!! i’ll give brainliest
A. ||
B.
C. neither they are skew lines
Answer:
I think it's A (parallel)
Answer:
A. its parallel
Step-by-step explanation:
all angles are equal... two sides are perpendicular and two are parallel
Show do i solve for the missing answers
Simplify the following polynomial expression.
(5x4 - 9x3 + 7x - 1) + (-8x4 + 4x2 - 3x + 2) - (-4x3 + 5x - 1)(2x - 7)
Answer:
5x^4 -37x^3 -6x^2 +41x -6
Step-by-step explanation:
We simplify this expression by removing parentheses and combining like terms. Parentheses are removed using the distributive property.
Form the productThe product of the final pair of polynomials in parentheses is ...
(-4x^3 +5x -1)(2x -7) = (-4x^3 +5x -1)(2x) +(-4x^3 +5x -1)(-7)
= -8x^4 +10x^2 -2x +28x^3 -35x +7
= -8x^4 +28x^3 +10x^2 -37x +7
Combine with remaining sums= (5x^4 -9x^3 +7x -1) + (-8x^4 +4x^2 -3x +2) - (-8x^4 +28x^3 +10x^2 -37x +7)
= (5 -8 -(-8))x^4 +(-9 -28)x^3 +(4 -10)x^2 +(7 -3 -(-37))x +(-1 +2 -7)
= 5x^4 -37x^3 -6x^2 +41x -6
help
4/10 as a decimal
Answer:
0.4
Step-by-step explanation:
4/10 is 4÷10 = 0.4
Consider a situation in which p(x) = and p(y) = . if p(x and y) is = , which best describes the events?
The correct option is (A) P(X) × P(Y) = P(X ∩ Y)
What is probability and example?
Probability = the number of ways of achieving success. the total number of possible outcomes. For example, the probability of flipping a coin and it being heads is ½, because there is 1 way of getting a head and the total number of possible outcomes is 2 (a head or tail). We write P(heads) = ½ .We are given to consider a situation in which X and Y are two events such that
P(X) = 4/5, P(Y) = 1/4, P(X ∩ Y) = 1/5
We are to select the statement that best describes the events X and Y
We know that
any two events A and B are said to be independent if
P(A) × P(B) = P (A ∩ B)
We have, for events X and Y,
P(X) × P(Y) = 4/5 × 1/4 = 1/5 = P (X ∩ Y)
P(X) × P(Y) = P(X ∩ Y)
Thus, X and Y are independent because P(X) × P(Y) = P(X ∩ Y)
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The complete question is -
Consider a situation in which P(X) = 4/5 and P(Y) = 1/4. If P(X and Y) is = 1/5, which best describes the events?
They are independent because P(X) x P(Y) = P(X and Y).
They are independent because P(X) + P(Y) = P(X and Y).
They are dependent because P(X) x P(Y) = P(X and Y).
They are dependent because P(X) + P(Y) = P(X and Y).
Answer: a
Step-by-step explanation:
just took the test
In an animated film, a simple scene can be created by translating a figure against a still background. Write a rule for Independent Practice For See Exercises Example 8-9102113124 Extra Practice Skills Practice p. S5 Application Practice p. S28 the translation that maps the rocket from position I to position 2.
The rule of the translation that maps the rocket from position I to position 2 is 4 units right and 4 units up
How to determine the rule for the translation?The translation is added as an attachment
From the attached figure, we have the following corresponding coordinates:
Figure 1 = (0, 0)
Figure 2 = (4, 4)
The rule of translation is calculated as:
(x, y) = T<Figure 2 - Figure 1>
This gives
(x, y) = T<4 - 0, 4 - 0>
Evaluate
(x, y) = T<4, 4>
Hence, the rule of the translation that maps the rocket from position I to position 2 is 4 units right and 4 units up
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Complete question
In an animated film, a simple scene can be created by translating a figure against a still background. Write a rule for the translation that maps the rocket from position I to position 2.
Find the dimensions of a rectangle with an area of 2x^2-7x-10
Answer:
Step-by-step explanation:
I gotcha my dude.......
x-5 and 2x+3
The scale of a map is 1:250000. On the map a large forest has an area of 6cm². Calculate the actual area of the forest. Give your answers in square kilometers.
The distance on map exists 32 cm and actual area exists 37.5 km².
How to estimate the actual area of the forest in square kilometers?
Given: Scale of the map exists at 1:250000.
(a) Distance between two cities = 80 km
= 80000 m
= 8000000 cm
Distance on map = 8000000 [tex]*[/tex] 1/ 250000
= 32 cm
(b) Area of map = 6 cm²
Actual area = [tex]6(250000)^2[/tex] cm²
[tex]= 6 * 625 * 10^8[/tex] cm²
[tex]= 3750 * 10^8/ 10^{10}[/tex]
= 37.5 km²
Therefore, the distance on map exists 32 cm and actual area exists 37.5 km².
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Adult men have heights with a mean of 69. 0 inches and a standard deviation of 2. 8 inches. find the z-score of a man who is 71. 9 inches tall. (to 4 decimal places)?
The value of the z score is 1.03.
According to the statement
we have given that the value of mean and standard deviation and we have to find the value of the z score.
So, For this purpose we know that the
Z-score indicates how much a given value differs from the standard deviation. The Z-score, or standard score, is the number of standard deviations a given data point lies above or below mean.
And the given values are:
mean value = 69 inches
s.d value = 2.8 inches
And the value of x is 71.9 inches.
So, The Z score is
z = x - mean / standard deviation
substitute the values in it then
z = 71.9 - 69 / 2.8
then
z = 2.9 /2.8
z = 1.03
So, The value of the z score is 1.03.
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graph the equation by translating y=|x-1|
i need help like super fast-
Answer:
Graph b
Step-by-step explanation:
y = |x| is a "v"at x = 0
y = |x-1| is a "v" shifted to the right by 1 unit
5) For the fraction 3/25, (a) write a percent and (b) write a decimal.
URGENT What is the discontinuity and zero of the function f(x)= (x^2 - 4x - 21)/(x + 7)
Answer:
when x=-7
Step-by-step explanation:
if u are solving a function like this, u must not have a division by zero so if we put x=-7 in the function we hav a division by zero
A 2 gallon container of disinfectant costs 22.72. What is the price per cup
Answer:
$0.71/cup
Step-by-step explanation:
1 gallon = 16 cups
2 gallons = 2 × 1 gallon = 2 × 16 cups = 32 cups
$22.72/(2 gal) = $22.72/(32 cups) = $0.71/cup
Find the equation of a line in slope-intercept form with a slope of 5 that contains the point (3,8).
Group of answer choices
y=5x−15
y=5x+8
y=5x−7
y=3x+8
Answer:
y = 5x - 7
Step-by-step explanation:
y = mx + c
y = 5x + c
to find c, input given coordinates
8 = 5 x 3 + c
8 = 15 + c
-c = 15 - 8
-c = 7
c = -7
put that all together and you get:
y = 5x - 7
The equation of the line with a slope of 5 that contains the point (3,8) is
y = 5x - 7.How to find the equation of the lineThe equation of a line in slope-intercept form is given by y = mx + b, where m represents the slope and b represents the y-intercept.
Given that the slope is 5 and the line contains the point (3,8), we can substitute these values into the equation and solve for the y-intercept (b).
Using the point-slope form of the equation:
y - y₁ = m(x - x₁)
We have:
y - 8 = 5(x - 3)
Expanding and simplifying:
y - 8 = 5x - 15
Now, let's isolate y:
y = 5x - 15 + 8
y = 5x - 7
Therefore, the equation of the line with a slope of 5 that contains the point (3,8) is y = 5x - 7.
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Evaluate the following integral (Calculus 2) Please provide step by step explanation!
Answer:
[tex]\displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x=-\dfrac{2}{x+1}+\text{C}[/tex]
Step-by-step explanation:
Fundamental Theorem of Calculus
[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]
If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.
Given integral:
[tex]\displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x[/tex]
Factor the denominator:
[tex]\begin{aligned}\implies x^2+2x+1 & = x^2+x+x+1\\& = x(x+1)+1(x+1)\\& = (x+1)(x+1)\\& = (x+1)^2\end{aligned}[/tex]
[tex]\implies \displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x=\int \dfrac{2}{(x+1)^2}\:\:\text{d}x[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{1}{a^n}=a^{-n}[/tex]
[tex]\implies \displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x=\int 2(x+1)^{-2}\:\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $ax^n$}\\\\$\displaystyle \int ax^n\:\text{d}x=\dfrac{ax^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]
Use Integration by Substitution:
[tex]\textsf{Let }u=(x+1) \implies \dfrac{\text{d}u}{\text{d}x}=1 \implies \text{d}x=\text{d}u}[/tex]
Therefore:
[tex]\begin{aligned}\displaystyle \int \dfrac{2}{x^2+2x+1}\:\:\text{d}x & = \int 2(x+1)^{-2}\:\:\text{d}x\\\\& = \int 2u^{-2}\:\:\text{d}u\\\\& = \dfrac{2}{-1}u^{-2+1}+\text{C}\\\\& = -2u^{-1}+\text{C}\\\\& = -\dfrac{2}{u}+\text{C}\\\\& = -\dfrac{2}{x+1}+\text{C}\end{aligned}[/tex]
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How many ways can a president, vice-president, secretary, and treasurer be chosen from a club with 9 members?
Answer:
3024
Step-by-step explanation:
Each chosen member (out of 9 members) will occupy a different position
out of these four (president , vice-president , secretary , treasurer).
So, here we have to calculate the Number of Permutations of 9 members Taken 4 at a Time :
= 9P4
= 9 × 8 × 7 × 6
= 3024
Hans is planting a garden with snapdragons and daisies. the table shows some possible combinations of the two plants. if hans plants 29 daisies, how many snapdragons will he plant?
The linear equation y = -x+45 models the scenario.
Solving the linear equation, it exists found that Hans will paint 16 snapdragons.
What is linear equation?Since the rate of change exists always the exact, this question exists modeled by a linear equation.
Linear equation: y = mx + b
Where, m exists the slope and b exists the y-intercept.
To find the slope, we have to get two points (x, y), and the slope exists given by the change in y divided by the change in x.
Points: (11, 34) and (12, 33).
Change in y: 33 - 34 = -1
Change in x: 12 - 11 = 1.
Slope: m = -1/1 = -1
The equation of the line exists y = -x + b
Replacing one of the points, the y-intercept can be found.
Point (11, 34) means that when x = 1, y = 34.
y = -x + b
34 = -1+b
b = 45
Therefore, the equation y = -x+45 models the scenario.
29 daisies mean that y = 29, we have to estimate the value of x for which y = 29.
y = -x+45
simplifying the equation,
29 = -x+45
x = 45 - 29 = 16
The value of x = 16
Therefore, Hans will plant 16 snapdragons.
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Answer:
The equation ✔ x = 45 – y models the scenario.
Hans will plant✔ 16 snapdragons
Step-by-step explanation:
Can someone please help me with this?
Answer: [tex]\Large\boxed{f(-9)=-189}[/tex]
Step-by-step explanation:
f(x) = -3x² - 6x
Requirements of the question
Find the value of f(-9)
Substitute values into the given function
f(x) = -3x² - 6x
f(9) = -3 (-9)² - 6 (-9)
Simplify the exponent
f(9) = -3 (81) - 6 (-9)
Simplify by multiplication
f(-9) = (-243) - (-54)
Simplify by subtraction
[tex]\Large\boxed{f(-9)=-189}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Evaluate the following series:
This is a telescoping sum. The K-th partial sum is
[tex]S_K = \displaystyle \sum_{k=1}^K \left(\frac1{\sqrt{k+1}} - \frac1{\sqrt{k+3}}\right) \\\\ ~~~= \left(\frac1{\sqrt2} - \frac1{\sqrt4}\right) + \left(\frac1{\sqrt3} - \frac1{\sqrt5}\right) + \left(\frac1{\sqrt4} - \frac1{\sqrt6}\right) + \left(\frac1{\sqrt5} - \frac1{\sqrt7}\right) + \cdots \\\\ ~~~~~~~~+ \left(\frac1{\sqrt{K-1}} - \frac1{\sqrt{K+1}}\right) \\\\ ~~~~~~~~+ \left(\frac1{\sqrt K} - \frac1{\sqrt{K+2}}\right) + \left(\frac1{\sqrt{K+1}} - \frac1{\sqrt{K+3}}\right)[/tex]
[tex]\displaystyle = \frac1{\sqrt2} + \frac1{\sqrt3} - \frac1{\sqrt{K+2}} - \frac1{\sqrt{K+3}}[/tex]
As [tex]K\to\infty[/tex], the two trailing terms will converge to 0, and the overall infinite sum will converge to
[tex]\displaystyle \sum_{k=1}^\infty \left(\frac1{\sqrt{k+1}} - \frac1{\sqrt{k+3}}\right) = \lim_{k\to\infty} S_k = \boxed{\frac1{\sqrt2} + \frac1{\sqrt3}}[/tex]
By the limit comparison test, the expression √[1 / (1 + 1 / k)] - √[1 / (1 + 3 / k)] has a limit, then the expression [1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k] has a limit and the series ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)] is convergent.
Is the series convergent?
Herein we have a series that involves radical components. First, we simplify the expression given:
∑ [1 / √(k + 1) - 1 / √(k + 3)] = ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)]
The convergence of the series can be proved by the limit comparison test, where each component of the subtraction of the series is compared with a series that is convergent. We notice that both 1 / √(k + 1) and 1 / √(k + 3) resembles the expresion 1 /√k. Then, we have the following subtraction of ratios:
[1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k]
√k / √(k + 1) - √k / √(k + 3)
√[k / (k + 1)] - √[k / (k + 3)]
Then, by using the limit property for rational functions we find the following result for n → + ∞:
√[1 / (1 + 0)] - √[1 / (1 + 0)]
√1 - √1
1 - 1
0
By the limit comparison test, the expression √[1 / (1 + 1 / k)] - √[1 / (1 + 3 / k)] has a limit, then the expression [1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k] has a limit and the series ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)] is convergent.
Remark
The statement is incomplete and complete form cannot be found, therefore, we decided to determine if the series is convergent or not.
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The sum ∫2−2() ∫52()−∫−1−2() can be written as a single integral in the form ∫() determine and
We have
[tex]\displaystyle \int_{-2}^2 f(x) \, dx - \int_{-2}^{-1} f(x) \, dx = \int_{-1}^2 f(x) \, dx[/tex]
so that
[tex]\displaystyle \int_{-2}^2 f(x) \, dx + \int_2^5 f(x) \, dx - \int_{-2}^{-1} f(x) \, dx \\\\ ~~~~~~~~~~~~ = \int_{-1}^2 f(x) \, dx + \int_2^5 f(x) \, dx \\\\ ~~~~~~~~~~~~ = \boxed{\int_{-1}^5 f(x) \, dx}[/tex]
Identify the standard form of the equation by completing the square.
4x2 − 9y2 − 8x + 36y − 68 = 0
Answer:
[tex]\dfrac{(x-1)^2}{9}-\dfrac{(y-2)^2}{4}=1[/tex]
Step-by-step explanation:
Given equation:
[tex]4x^2-9y^2-8x+36y-68=0[/tex]
This is an equation for a horizontal hyperbola.
To complete the square for a hyperbola
Arrange the equation so all the terms with variables are on the left side and the constant is on the right side.
[tex]\implies 4x^2-8x-9y^2+36y=68[/tex]
Factor out the coefficient of the x² term and the y² term.
[tex]\implies 4(x^2-2x)-9(y^2-4y)=68[/tex]
Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:
[tex]\implies 4\left(x^2-2x+\left(\dfrac{-2}{2}\right)^2\right)-9\left(y^2-4y+\left(\dfrac{-4}{2}\right)^2\right)=68+4\left(\dfrac{-2}{2}\right)^2-9\left(\dfrac{-4}{2}\right)^2[/tex]
[tex]\implies 4\left(x^2-2x+1\right)-9\left(y^2-4y+4\right)=36[/tex]
Factor the two perfect trinomials on the left side:
[tex]\implies 4(x-1)^2-9(y-2)^2=36[/tex]
Divide both sides by the number of the right side so the right side equals 1:
[tex]\implies \dfrac{4(x-1)^2}{36}-\dfrac{9(y-2)^2}{36}=\dfrac{36}{36}[/tex]
Simplify:
[tex]\implies \dfrac{(x-1)^2}{9}-\dfrac{(y-2)^2}{4}=1[/tex]
Therefore, this is the standard equation for a horizontal hyperbola with:
center = (1, 2)vertices = (-2, 2) and (4, 2)co-vertices = (1, 0) and (1, 4)[tex]\textsf{Asymptotes}: \quad y = -\dfrac{2}{3}x+\dfrac{8}{3} \textsf{ and }y=\dfrac{2}{3}x+\dfrac{4}{3}[/tex][tex]\textsf{Foci}: \quad (1-\sqrt{13}, 2) \textsf{ and }(1+\sqrt{13}, 2)[/tex]