ANSWER
[tex]\frac{df}{dx}=-\frac{1}{(x-2)^2}[/tex]EXPLANATION
We want to find the derivative of the given function:
[tex]f(x)=\frac{1}{x-2}[/tex]First, we have to rewrite the function as follows:
[tex]f(x)=(x-2)^{-1}[/tex]Next, make the following substitution:
[tex]a=x-2[/tex]The function now becomes:
[tex]f(x)=a^{-1}[/tex]Apply the chain rule of differentiation:
[tex]\frac{df}{dx}=\frac{df}{da}\cdot\frac{da}{dx}[/tex]Therefore, we have that:
[tex]\frac{df}{da}=-1\cdot a^{-1-1}=-a^{-2}[/tex]and:
[tex]\frac{da}{dx}=1[/tex]Therefore, the differentiation of the function is:
[tex]\begin{gathered} \frac{df}{dx}=-a^{-2}\cdot1 \\ \Rightarrow\frac{df}{dx}=-(x-2)^{-2}\cdot1 \\ \frac{df}{dx}=-\frac{1}{(x-2)^2} \end{gathered}[/tex]Solve y^3= −125.
y = −5
y = ±5
y = −25
y = ±25
Answer:
A. y=-5Step-by-step explanation:
Use the order of operations.
PEMDAS stands for:
ParenthesesExponentsMultiplyDivideAddSubtractDo exponents first.
[tex]\sf{y^3=-125}[/tex]
[tex]\sf{x=\sqrt[3]{f\left(a\right)},\:\sqrt[3]{f\left(a\right)}\dfrac{-1-\sqrt{3}i}{2},\:\sqrt[3]{f\left(a\right)}\dfrac{-1+\sqrt{3}i}{2}}}[/tex]
[tex]\rightarrow \sf{y=\sqrt[3]{-125},\:y=\sqrt[3]{-125}\dfrac{-1+\sqrt{3}i}{2},\:y=\sqrt[3]{-125}\dfrac{-1-\sqrt{3}i}{2}}[/tex]
[tex]\sf{\sqrt[3]{-125}=\boxed{\sf{-5}}}[/tex]
Therefore, the correct answer is y=-5.
I hope this helps, let me know if you have any questions.
[tex]\huge\text{Hey there!}[/tex]
[tex]\mathsf{y^3 = -125}[/tex]
[tex]\large\text{Solve/take the cube root}[/tex]
[tex]\mathsf{(-125)^{^\dfrac{1}{3}}}\mathsf{ = y}[/tex]
[tex]\mathsf{y = (-125)^{^\dfrac{1}{3}}}[/tex]
[tex]\large\text{Simplify it}[/tex]
[tex]\mathsf{y = -5}[/tex]
[tex]\huge\text{Therefore, your answer is: \boxed{\mathsf{y = -5\ (\rm \bold{O}ption\ A.)}}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
URGENT!! ILL GIVE
BRAINLIEST!!!!! AND 100 POINTS!!!
Collinear points are two or more points that lie on the sameA. planeB. angleC. lineD. space
Collinear points are two or more points that lie on the same line.
For Example:
Point A, B and C
Consider the following algebraic expression:7s - 7Step 1 of 2: Identify the first term of the algebraic expression. Indicate whether the term is a variable term or a constant term. For avariable term, identify the variable and the coefficient of the term.
Given the algebraic expression below
[tex]7s-7[/tex]The first term of the algebraic expression is
[tex]7s[/tex]The first term "7s" is a variable term.
The variable of the first term is "s"
The coefficient of the variable term is 7
Andrew constructed a triangle so that the measurement of 1 and 2 were congruent. if angle 3 measured 70 degrees, what is the measure of angle 1?
Andrew constructed a triangle such that the measurements of angles m<1 and m<2 are congruent.
The above statement can be inferred from concept of congruency of triangles. The oppsoite sides of the two congruent angles in a traingles are also equal.
From the above statement we can deduce the type of a triangle that Andrew drew as follows:
[tex]\text{Andrew drew a isoceles triangle}[/tex]An isoceles triangle has two equal sides and angles i.e congruent sides and interior angles. Hence,
[tex]m\angle1\text{ = m}\angle2\ldots\text{ Eq1}[/tex]The following information is given for the third interior angle m<3 of the isoceles triangle:
[tex]m\angle3\text{ = 70 degrees}[/tex]We need determine the angle measure of the angle 1. Recall that the sum of interior angles of a triangle is given as follows:
[tex]m\angle1\text{ + m}\angle2\text{ + m}\angle3\text{ = 180 degrees }\ldots\text{ Eq2}[/tex]Substitute Eq1 into Eq2 as follows:
[tex]\begin{gathered} m\angle1\text{ + m}\angle1\text{ + m}\angle3\text{ = 180} \\ \\ 2\cdot m\angle1\text{ + m}\angle3\text{ = 180} \end{gathered}[/tex]Substitute the angle measurement of angle ( 3 ) in the expression above and solve for angle ( 1 ) as follows:
[tex]\begin{gathered} 2\cdot m\angle1\text{ + 70 = 180} \\ 2\cdot m\angle1\text{ = 110} \\ m\angle1\text{ = }\frac{110}{2} \\ \\ m\angle1\text{ = 55 degrees }\ldots\text{ Answer} \end{gathered}[/tex]Given four numbers P, Q, R and S. The first three numbers form an arithmetic sequence while the last three form a geometric sequence. If the sum of the first and the fourth number is 16 and the sum of the second and the third number is 12, find these four numbers.
The most appropriate choice for arithmetic and geometric series will be given by-
P = 0, Q = 4 , R = 8, S = 16 or P = 15, Q = 9, R = 3, S = 1 are the required numbers
What is arithmetic and geometric series?
Arithmetic series are those series whose difference between two consecutive terms are same.
Geometric series are those series whose ratio between two consecutive terms are same.
Here,
P, Q and R forms an Arithmetic sequence
Let P = a - d , Q = a, R = a + d, Where a is the first term of the Arithmetic sequence and d is the common difference of the sequence.
Let S = b
Q, R and S forms a Geometric sequence
[tex]\frac{a + d}{a} = \frac{b}{a +d}[/tex]
[tex](a + d)^2 = ab\\a^2 + d^2 + 2ad = ab\\[/tex] ............... (1)
Now the sum of first and fourth number is 16
a - d + b = 16
b = 16 - a + d
Putting the value of b in (1),
[tex]a^2 + d^2 + 2ad = a(16 - a +d)\\a^2 + d^2 + 2ad = 16a -a^2+ad\\a^2+a^2 + d^2+2ad - ad - 16a = 0\\2a^2 + ad+d^2 -16a = 0[/tex]............ (2)
Sum of second and third number is 12
[tex]a + a + d = 12\\2a +d = 12\\d = 12-2a[/tex]
Putting the value of d in (2)
[tex]2a^2 + a(12 - 2a)+(12 - 2a)^2-16a = 0\\2a^2 + 12a - 2a^2+144-48a+4a^2 - 16a = 0\\4a^2-52a+144=0\\4(a^2-13a+36)=0\\a^2 -13a+36=0\\a^2-9a-4a+36=0\\a(a - 9)-4(a-9)=0\\(a-4)(a-9) = 0\\[/tex]
[tex]a - 4 = 0[/tex] or [tex]a - 9 = 0[/tex]
[tex]a = 4[/tex] or [tex]a = 9[/tex]
When a = 4,
[tex]d = 12 - 2\times 4\\d = 12 - 8\\d = 4[/tex]
[tex]b = 16 - 4+4\\b = 16[/tex]
P = 4 - 4 = 0
Q = 4
R = 4 + 4 = 8
S = 16
When a = 9,
[tex]d = 12 - 2\times 9\\d = 12 - 16\\d = -6[/tex]
[tex]b = 16 - 9-6\\b = 1[/tex]
P = 9 - (-6) = 15
Q = 9
R = 9 + (-6) = 3
S = 1
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Give a number in scientific notation that isbetween the two numbers on a number line.71 X 103 and 71,000,000
For this problem we have the following two numbers
[tex]71x10^3[/tex][tex]71000000[/tex]Let's convert the two numbers with scientific notation
[tex]71x10^3=71000=7.1x10^4[/tex][tex]71000000=7.1x10^7[/tex]Now we just need to find a number between the two given we know that:
[tex]7.1x10^4<7.1x10^7[/tex]The final answer for this case would be any number between these two numbers and it could be:
[tex]7.1x10^6[/tex]also it could be:
[tex]9.5x10^5[/tex]Or any number between the two given
Answer:
The answer is B,D, And F
Step-by-step explanation:
7.1 × 103 = 7,100
7.1 × 105 = 710,000
Because 7,100 < 710,000 < 71,000,000 then 7.1 × 105 falls between 7.1 × 103 and 71,000,000
Answer the following.(a) An astronomer's infrared telescope is able to detect radiation with a wavelength of 1.96 x 10^-5 meters. Write this number in standardnotation(b) The diameter of Venus at its equator is approximately 12,100 kilometers. Write this number in scientific notation.(a) Imeters(b) kilometers
a)
We need to convert 1.96 x 10^-5 meters. into standard notation.
Now, on the scientific notation, the power of ten shows how many places the decimal point has been moved.
- If the exponent is positive then the decimal point has been moved to the left.
- If the exponent is negative, then the decimal point has been moved to the right.
In this case, the power is negative 5 . So, the decimal point has been moved 5 places to the right.
Hence:
1.96 x 10^-5 = 0.0000196
b)
a. Meters
First, we need to convert kilometers into meters using the rule of three:
If 1k = 1000 meters
Then 12,100k = x
Where x = (12,100k*1000m)/ 1k
x =12000000 meters
We need to convert from standard notation to scientific notation:
12000000.0 = 1.2x10^7 meters
The decimal point has been moved 7 places to the left, so the power of then is positive 7.
b) We need to convert 12,100 kilometers into scientific notation:
12,100 = 12,100.0
Converting into scientific notation
1.21x10^4 kilometers
The decimal point has been moved 4 places to the left. Hence, the power of the is positive 4.
if a figure has four corners then it is a quadrilateral and figure has four corners therefore it is a quadrilateral which statement illustrate this to be true the large attachment account example the law of syllogism the law contrapositive
The given conditions are true:
law of contrapositive
Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 5 sweaters for $99 or she is able to buy 6 shirts and 4 sweaters for $90. How much does a shirt cost? How much does a sweater cost?
Given :
The cost of 3 shirts and 5 sweaters is $99 .
The cost of 6 shirts and 4 sweaters is $90.
To determine :
The sweater cost and shirt cost each.
Explanation :
Let the shirt cost be x.
Let the sweater cost be y.
Then the equation formed is
[tex]3x+5y=99\ldots\ldots\ldots\ldots\ldots\ldots..1[/tex][tex]6x+4y=90\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots2[/tex]Solve the equation 1 and 2 to find x and y with the help of elimination method.
Multiply by 2 in equation 1.
[tex]6x+10y=198\ldots\ldots\ldots\ldots\ldots\ldots.\ldots.\ldots3[/tex]Solve equation 2 and 3 by subtraction,
[tex](6x+10y-198)-(6x+4y-90)=0[/tex][tex]6y-198+90=0[/tex][tex]6y=108[/tex][tex]y=18[/tex]The value of y is 18 and now substitute the value of y in equation 2 to find x.
[tex]6x+4y=90[/tex][tex]6x+4\times18=90[/tex][tex]6x=90-72[/tex][tex]6x=18\Rightarrow x=3[/tex]The value of x is 3.
Answer :
The cost of shirt is 3 dollar .
The cost of sweater is 18 dollar.
y - y1 = m (x - x1 ) write an equation in point slope form given point ( 4, -3 ) and m = 1
The point-slope form of a line is:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Replacing with m = 1 and the point (4, -3):
y - (-3) = 1(x - 4)
y + 3 = x - 4
Laura, a sandwich maker, produces 80 sandwiches on average per day. How many sandwiches will she produce in pdays?Number of sandwiches =
Number of sandwiches = 80p
Explanation:Given:
Laura produces 80 sandwiches per day
To find:
The number of sandwiches that will be produced in p days
1 day = 80 sandwiches
p days = 80 × p
p days = 80p
This means that she will produce 80p number of sandwiches in p days
Identify an equation in point slope form for the line perpendicular to y=1/4 x-7that passes through -2,-6
The equation in the point slope form for the line perpendicular to y = (1/4)x-7 that passes through the point (-2,-6) is y+6 = -4(x+2)
The given equation of the perpendicular line
y = (1/4)x -7
The equation is in the slope intercept form of the line
y = mx+b
Where m is the slope of the line
By comparing the given equation with the slope intercept form
The slope of the line m = 1/4
The slope of its perpendicular line = -1/m
= -4
The point slope form is
[tex](y-y_1)=m(x-x_1)[/tex]
The point is given that (-2,-6)
Substitute these values in the equation
(y-(-6) = -4(x-(-2)
y+6 = -4(x+2)
Hence, the equation in the point slope form for the line perpendicular to y = (1/4)x-7 that passes through the point (-2,-6) is y+6 = -4(x+2)
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can someone please help me find the value of X?
Answer:
x = 100degrees
Explanation:
Using the theorem;
The angle at the vertex is the half of the difference of its intercepted arcs
Angle at the vertex = 15 degrees
angle at the intercepted arcs = 70degrees and x degrees
According to the theorem;
15 = 1/2(x-70)
Cross multiply
15 * 2 = x - 70
30 = x - 70
Add 70 to both sides
30 + 70 = x - 70 + 70
100 = x
Swap
x = 100degrees
Hence the value of x is 100degrees
In ∆PQR, p=13 inches, q=18 inches and r= 12 inches. Find the area of ∆PQR to the nearest square inch.
Given data:
The first side of the triangle is p=13 inches.
The second side of the triangle is q=18 inches.
The third side of the triangle is r= 12 inches.
The semi-perimeter is,
[tex]\begin{gathered} s=\frac{p+q+r}{2} \\ =\frac{13\text{ in+18 in+12 in}}{2} \\ =21.5\text{ in} \end{gathered}[/tex]The expression for the area of the triangle is,
[tex]\begin{gathered} A=\sqrt[]{s(s-p)(s-q)(s-r)_{}} \\ =\sqrt[]{21.5\text{ in(21.5 in-13 in)(21.5 in-18 in)(21.5 in-12 in)}} \\ =\sqrt[]{(21.5\text{ in)(8.5 in)(3.5 in)(9.5 in)}} \\ =77.95in^2 \end{gathered}[/tex]Thus, the area of the given triangle is 77.95 sq-inches.
For the parabola given by 4y – 9 = x2 – 6x, find the vertex and focus.
Solution
Gievn the equation below
[tex]4y-9=x^2-6x[/tex]To find the vertex and focus of the given equation, we apply the parabola standard equation which is
[tex]4p(y-k)=(x-h)^2[/tex]Where p is the focal length and the vertex is (h,k)
Rewriting the equation in standard form gives
[tex]\begin{gathered} 4y-9=x^2-6x \\ 4y=x^2-6x+9 \\ 4y=x^2-3x-3x+9 \\ 4y=x(x-3)-3(x-3) \\ 4y=(x-3)^2 \\ 4(1)(y-0)=(x-3)^2 \end{gathered}[/tex]Relating the parabola standard equation with the given equation, the vertex of the parabola is
[tex]\begin{gathered} x-3=0 \\ x=3 \\ y-0=0 \\ y=0 \\ (h,k)\Rightarrow(3,0) \\ p=1 \end{gathered}[/tex]Hence, the vertex is (3,0)
The focus of the parabola formula is
[tex](h,k+p)[/tex]Where
[tex]\begin{gathered} h=3 \\ k=0 \\ p=1 \end{gathered}[/tex]Substitute the values of h, k and p into the focus formula
[tex](h,k+p)\Rightarrow(3,0+1)\Rightarrow(3,1)[/tex]Hence, the focus is (3, 1)
Select the correct answer.
What is the value of this logarithmic e ession?
log2 16 - log₂ 4
Answer:l og2(16)=x log 2 ( 16 ) = x in exponential form using the definition of a logarithm. If x x and b b are positive real numbers and b b does not equal ...
Step-by-step explanation:
Use the graph of f to describe the transformation that results in the graph of g. Then sketch the graphs of g and f.
EXPLANATION
Since we have the function:
[tex]f(x)=(\frac{1}{3})^x[/tex]Graphing this an the transformation into a graph calculator we have:
We can see that the transformated function is translated 2 units to the right, and that It is also translated 4 units down.
Thus, the transformations are the following:
g(x) is the graph of f(x) translated 2 units to the right and 4 units down.
Use the data below to complete the following calculationm=76,37,27
Answer:
Σmf = 23347
Σm²f = 1621631
(Σmf)² = 545082409
Explanation:
The symbol Σ means that we need to sum all the products of m and f.
So, Σmf is equal to:
Σmf =76(94) + 37(92) + 27(63) + 98(62) + 62(81)
Σmf = 7144 + 3404 + 1701 + 6076 + 5022
Σmf = 23347
Then, to find Σm²f, we need to find the square of m, so:
Σm²f = (76)²(94) + (37)²(92) + (27)²(63) + (98)²(62) + (62)²(81)
Σm²f = 5776(94) + 1360(92) + 729(63) + 9604(62) + 3844(81)
Σm²f = 542944 + 125948 + 45927 + 595448 + 311364
Σm²f = 1621631
Finally, (Σmf)² is equal to:
(Σmf)² = (23347)²
(Σmf)² = 545082409
Therefore, the answers are:
Σmf = 23347
Σm²f = 1621631
(Σmf)² = 545082409
Choose SSS, SAS, or neither to comparethese two triangles.A) SSSB) SASC) neither
Answer:
C. Neither
Explanation:
The SSS Congruence Rule states that if the three sides of a triangle are equal to the three sides of another triangle, then the two triangles are congruent.
The SAS Congruence Rule states that if the two sides and the included angle of one triangle are equal to the two sides and the included angle of another triangle, then the two triangles are congruent.
Notice that in the given triangles, there are two congruent sides and a non-included angle, since this does not satisfy any of the rules stated above, SSS Congruence rule or SAS Congruence rule, we'll choose "neither" as the correct answer.
18
If p percent of an adult's daily allowance of
potassium is provided by x servings of Crunchy
Grain cereal per day, which of the following
expresses p in terms of x ?
Express p in terms of x : p = 5x
What is Percent?
A percentage is a figure or ratio stated as a fraction of 100 in mathematics. Although the abbreviations "pct.", "pct.", and occasionally "pc" are also used, the percent symbol, "%," is frequently used to indicate it. A % is a number without dimensions and without a measurement system.
If 5% of an adult's daily potassium requirement is provided by each serving of Crunchy Grain cereal, then x servings will offer x times 5%.
Five times as many servings, or p, of potassium are required for an adult's daily requirement.
As a result,
p = 5x can be used to describe the proportion of potassium in an adult's daily allotment.
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The numerator of the sum 1+1/3+2 is (a) 1 (b) 2 (c) 5 (d) 6.
The expression is given as,
[tex]\frac{1}{2}+\frac{1}{3}[/tex]Note that the denominator of both the fractions are prime numbers. So their lowest common multiple, LCM(2,3) will be the product of the numbers,
[tex]undefined[/tex]Henry had a batting average of 0.341 last season (out of 1000 at-bats, he had 341 hits). Given that thisbatting average will stay the same this year, answer the following questions. What is the probability that his first hit will not occur until his 5th at-bat? Answers. 0.64. 0.083. 0.129. 0.166
The probability of success (a hit) is given by:
p = 0.341
The complement (a failure) of this probability is:
q = 1 - 0.341 = 0.659
Then, we can construct a probability distribution for the first hit until his nth at-bat:
[tex]P(x=n)=p\cdot q^{n-1}[/tex]For his 5th at-bat, we have n = 5, then:
[tex]\begin{gathered} P(x=5)=0.341\cdot(0.659)^{5-1}=0.341\cdot0.659^4 \\ \\ \therefore P(x=5)=0.064 \end{gathered}[/tex]Lin is traveling from Japan to several other countries. The conversion table shows exchange rates between different currencies.
2,000 Yen 16 Euros
40 Euros = 3,125 Indian Rupees
What is the rate of yen per Indian rupee?
0.625
0.64
1.5625
1.6
The rate of yen per Indian rupee is 1.6
How do you convert currency to another currency?
Divide your current currency by the exchange rate if you are aware of it.
Assume, for instance, that you want to change 100 USD into EUR and the USD/EUR exchange rate is 0.631. Simply multiply 100 by 0.631 to do this, and the result is 63.10 EUR, which is the amount you will receive.
Given, 2000 Yen = 16 Euros
and, 40 Euros = 3,125 Indian Rupees
Find rate of yen per India rupee
1 Euro = 2000/16
1 Euro = 125 yen
And, 1 Euro = 3125/40
1 Euro = 78.125 India rupee
Now, find 1 yen = ? Indian rupee
we know, 1 Euro = 125 yen and 1 Euro = 78.125 India rupee
put the values of euro in India rupee in 1 Euro = 125 yen, we get
78.125 India rupee = 125 yen
1 India rupee = 125 / 78.125
1 Indian rupee = 1.6 Yen
Therefore, the rate of yen per Indian rupee is 1.6
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what's the answer?[tex] - 4 \sqrt{15 \times - \sqrt{3} } [/tex]
In decimal form this is equal to -17.22.
During Thanksgiving Break, 68% of a school's students ate green bean casserole. Out of 650 students, how many ate green bean casserole?
650 --- total
650*.68=442
442 students ate green bean casserole
.68 represents the percentage
so for example, if they asked me for 50% of 1000
we need to multiply 1000*0.5
if they asked for 60% we will multiply 1000*0.6
Determine the measures of angles x, y, and z: x = 75°95°105°° y = 75°95°105°° z = 75°95°105°°
Consider the figure,
So, we have, Two angles are called complementary when their measures add to 90 degrees. Two angles are called supplementary when their measures add up to 180 degrees.
So, here, [tex]<\text{AHB}+z=180[/tex]
Therefore, z can be calculated as,
[tex]z=180-<\text{AHB}=180-105=75[/tex]Now, the angles DHC and
which number of pounds of bananas and total cost of the bananas could be used as the missing values in the table
Given :
The table show a proportional relationship between the number of pounds and the total cost
so,
Solve for 5x - 3y = -45the equations beside it are the answer choices.
You have the following equation:
5x - 3y = -45
In order to solve the previous equation for y, you proceed as follow:
5x - 3y = -45 subtract 5x both sides
- 3y = -45 - 5x multiply by -1 both sides
(-1)(-3y) = (-1)(-45 - 5x)
3y = 45 + 5x divide by 3 both sides
y = 45/3 + 5/3 x order the right side
y = 5/3 x + 15
Hence, the solution for y is y = 5/3 x + 15
What are the coordinates of M', the image of M (2,4), after a counterclockwiserotation of 90° about the origin?
If M(h,k) is rotated 90° counterclockwise about the origin, the new position would be M'(k, -h)
M(2, 4)-> M'(4, -2)