Answer:
The correct option is B.
Step-by-step explanation:
A function is called an exponential function if it has common ratio.
A function is called an linear function if it has common difference.
In option A.
[tex]\frac{f(2)}{f(1)}=\frac{6}{3}=2[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{9}{6}=\frac{3}{2}[/tex]
[tex]2\neq \frac{3}{2}[/tex]
Since the given table has different ratio, therefore it is not an exponential function. Option A is incorrect.
In option B.
[tex]\frac{f(2)}{f(1)}=\frac{6}{2}=3[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{18}{6}=3[/tex]
[tex]3=3[/tex]
Since the given table has common ratio, therefore it is an exponential function. Option B is correct.
In option C.
[tex]\frac{f(2)}{f(1)}=\frac{22}{10}=\frac{11}{5}[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{34}{22}=\frac{17}{11}[/tex]
[tex]\frac{11}{5}\neq \frac{17}{11}[/tex]
Since the given table has different ratio, therefore it is not an exponential function. Option C is incorrect.
In option D.
[tex]\frac{f(2)}{f(1)}=\frac{8}{7}[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{9}{8}[/tex]
[tex]\frac{8}{7}\neq \frac{9}{8}[/tex]
Since the given table has different ratio, therefore it is not an exponential function. Option D is incorrect.
Answer:
Table B represents an exponential function.
Step-by-step explanation:
An exponential function is a function which has common ratio. Using this fact we will evaluate the functions given in the form of a table.
Table A.
f(1) = 3
f(2) = 6
f(3) = 9
Now [tex]\frac{f(2)}{f(1)}=\frac{6}{3}=\frac{2}{1}[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{9}{6}=\frac{3}{2}[/tex]
Ratios are not equal so it's not an exponential function.
Table B.
f(1) = 2
f(2) = 6
f(3) = 18
[tex]\frac{f(2)}{f(1)}=\frac{6}{2}=\frac{3}{1}[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{18}{6}=\frac{3}{1}[/tex]
Here ratios are same therefore it's an exponential function.
Table C.
f(1) = 10
f(2) = 22
f(3) = 34
[tex]\frac{f(2)}{f(1)}=\frac{22}{10}=\frac{11}{5}[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{34}{22}=\frac{17}{11}[/tex]
Ratios are not equal therefore it's not an exponential function.
Table D.
f(1) = 7
f(2) = 8
f(3) = 9
[tex]\frac{f(2)}{f(1)}=\frac{8}{7}[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{9}{8}[/tex]
Ratios are not equal so it's not an exponential function.
Therefore Table B is the correct option.
simplify 5p2 5p3?
whats the answer
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Suppose p4(x) = 3 − 4x + 2x 2 − 3x 3 + 2x 4 is the degree 4 taylor polynomial centered at x = 0 for some function f. (i) what is the value of f(0)?
Hence, the Taylor polynomial centered at [tex]x=0[/tex] for some function [tex]f[/tex] is [tex]p^4(x)=3[/tex].
What is an equation?
The definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.
Here given equation is
[tex]p^4(x)=3-4x+2x^2-3x^3+2x^4 .......(1)[/tex]
Substitute [tex]x=0[/tex] in the equation [tex](1)[/tex], we get
[tex]p^4(x)=3-4(0)+2(0)^2-3(0)^3+2(0)^4 \\p^4(x)=3[/tex]
Hence, the Taylor polynomial centered at [tex]x=0[/tex] for some function [tex]f[/tex] is [tex]p^4(x)=3[/tex].
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Final answer:
The value of f(0) for the given Taylor polynomial p4(x) = [tex]3 - 4x + 2x^2 - 3x^3 + 2x^4[/tex] is 3, as the constant term in a Taylor polynomial centered at x = 0 represents f(0).
Explanation:
The question asks for the value of f(0) given a degree 4 Taylor polynomial for some function f, centered at x = 0. A Taylor polynomial of a function f centered at x = 0 is given by
f(x) = f(0) + f'(0)x + f''(0)x2/2! + f'''(0)x3/3! + ...
Each coefficient in front of the powers of x represents the derivative of f at 0 divided by the factorial of the order of the derivative. For
p4(x) = [tex]3 - 4x + 2x^2 - 3x^3 + 2x^4[/tex]
the constant term, which is 3, represents f(0), because it corresponds to the value of the function at x = 0 before any derivatives are taken into account.
the original value of an investment is $1,800. if the value has increased by 7% each year, write an exponential to model the situation. then, find the value of the investment after 15 years
The value of the investment after 15 years is approximately [tex]$4,410.67.[/tex]
To model the situation where an investment of [tex]$1,800[/tex] increases by 7% each year, we can use the exponential growth formula:
[tex]\[ A = P(1 + r)^t \][/tex]
where:
-[tex]\( A \)[/tex]is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex]is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex]is the annual interest rate (in decimal form).
-[tex]\( t \)[/tex] is the time the money is invested for, in years.
Given:
[tex]- \( P = \$1,800 \)[/tex]
[tex]- \( r = 7\% = 0.07 \)[/tex](as a decimal)
-[tex]\( t = 15 \)[/tex] years
We substitute these values into the formula to get:
[tex]\[ A = 1800(1 + 0.07)^{15} \][/tex]
Now, we calculate the value of[tex]\( A \):[/tex]
[tex]\[ A = 1800(1.07)^{15} \][/tex]
Using a calculator, we find:
[tex]\[ A \approx 1800 \times (1.07)^{15} \approx 1800 \times 2.45037 \][/tex]
[tex]\[ A \approx \$4,410.67 \][/tex]
Therefore, the value of the investment after 15 years is approximately [tex]$4,410.67.[/tex]
Find the area of a circle with a diameter of 20 inches. use 3.14 for pi. (1 point) 62.8 in2 125.6 in2 188.4 in2 314 in2
What is 84%, percent of 300?
A long rope should be divided into pieces of 27 meters long each. if the rope measures 1215 meters, how many cuts should we make?
Answer:
yes
Step-by-step explanation:
Divide the following polynomials. Then place the answer in the proper location on the grid. Write your answer in order of descending powers of x. Write the quotient and remainder as a sum in this format: . Do not include parentheses in your answer. ( x^3 + y^3) ÷( x- y )
The quotient is [tex]\(x^2 + xy + y^2\)[/tex]s [tex]\(0\)[/tex]
Explanation:To divide the polynomial [tex]\(x^3 + y^3\) by \(x - y\),[/tex]division. The process yields a quotient of [tex]\(x^2 + xy + y^2\)[/tex]f [tex]\(0\)[/tex]s result indicates that the given polynomial is divisible by \(x - y\) without any remainder. The quotient represents the solution, showcasing the expression obtained when [tex]\(x^3 + y^3\)[/tex]y [tex]\(x - y\).[/tex] in the quotient is arranged in descending powers of [tex]\(x\)[/tex] ering to the instructions.
The remainder being [tex]\(0\)[/tex] irms the complete divisibility of the original polynomial by \(x - y\). This concise and ordered format aligns with the specified requirements for presenting the solution on the grid. In summary, the division of [tex]\(x^3 + y^3\) by \(x - y\)[/tex] [tex]\(x^2 + xy + y^2\)[/tex] f [tex]\(0\)[/tex]
Is a heart shape a quadrilateral
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Javier is evaluating the expression (-8)[10 + (-5) + (-8)]
Multiplying the numbers 10, -5, and -8 by a certain number and then adding the three products together will give Javier the fully simplified evaluated expression. What is that number?
The number that will give Javier the fully simplified evaluated expression is 24.
Explanation:To evaluate the expression (-8)[10 + (-5) + (-8)], we need to multiply the numbers inside the parentheses by a certain number and then add the three products together. Let's solve it step by step:
Multiplying -8 by 10: (-8) * 10 = -80Multiplying -8 by -5: (-8) * (-5) = 40Multiplying -8 by -8: (-8) * (-8) = 64Add the three products: -80 + 40 + 64 = 24Therefore, the number that will give Javier the fully simplified evaluated expression is 24.
Solve the equation: x^3 - 12x^2 + 48x - 64 = 0
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If the endpoints of the diameter of a circle are (−10, −8) and (−6, −2), what is the standard form equation of the circle? A) (x − 8)2 + (y − 5)2 = 13 B) (x + 8)2 + (y + 5)2 = 13 C) (x − 8)2 + (y − 5)2 = 13 D) (x + 8)2 + (y + 5)2 = 13
Answer: D) (x+8)2 +( y+5)2 = 13
Step-by-step explanation:
Got this right on USA test prep
If a data set has ssr = 400 and sse = 100, then the coefficient of determination is
To solve this problem you must apply the proccedure shown below:
1. You have that the data set has SSR=400 and SSE=100
2. Therefore you have the coefficient of determination is:
r²=SSR/SSTO
SSTO=SSR+SSE
3. Then, when you substitute the values, you obtain:
SSTO=400+100
SSTO=500
r²=400/500
4. So, you have that the result is:
r²=0.8
Therefore, as you can see, the answer for the exercise shown above is: the coefficient of determination is 0.8
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At 9AM, a freight train leaves Boston for NYC, traveling at an average rate of 24 miles per hour. At noon, a passenger train sets out on the same route, traveling at an average rate of 60 miles per hour. How far from Boston do the two trains pass each other?
___ miles
Final answer:
The freight and passenger trains will meet 120 miles from Boston at 2 PM, which is 2 hours after the passenger train departs.
Explanation:
To solve this problem, we need to figure out when the two trains will meet, given their different start times and speeds. Since the freight train leaves at 9 AM and the passenger train leaves at noon, there is a 3-hour difference between their departure times.
In those 3 hours, the freight train will cover a distance of 3 hours imes 24 miles/hour = 72 miles.
Both trains are now heading towards each other, with the passenger train moving faster. We need to find out how long it will take for the passenger train to catch up to the freight train, starting from the 72-mile head start.
Let's call the time it takes for the passenger train to catch up 't' hours. In 't' hours, the passenger train will cover 60t miles and the freight train will cover 24t miles.
Since the passenger train is trying to cover the lead that the freight train has plus the distance both are covering after noon, we can set up an equation: 60t = 72 + 24t. Solving for 't' gives us 't' = 72 / (60 - 24) = 2 hours. So, in 2 hours after noon, which is at 2 PM, the two trains will meet.
Now we find out how far from Boston they will meet by calculating the distance the passenger train travels in those 2 hours: Distance = 60 miles/hour imes 2 hours = 120 miles from Boston.
Which of the following terms correctly describes 4:1?
Answer:
The answer is Ratio
Step-by-step explanation:
Ratio, correctly describes the expression, 4:1. Option (A) is correct.
Let's discuss the given options.
(A) Ratio: In mathematics, ratio is a comparison of two numbers, it is expressed in the form of "a : b," where "a" and "b" are the quantities being compared. It shows the relative proportion between the values.
(B) Percent: When we want to represent a fraction or ratio as a number out of 100, a percent is used. It represents by the symbol %. For example, 40% = 40 out of 100 or 40/100.
(C) Factor: A number that evenly divides another number without producing a residual is referred to as a factor. For example, 1,2,3 are the factors of 6 because they divide 6 without any remainder.
(D) Fraction : A fraction represents a part of a whole. It is in the form of numerator divided by denominator. For example, 2/5 represents 2 parts out of 5 equal parts.
Now, take the given expression, which is 4:1. The definition of ratio fits well for this expression. It means that there are four units of one quantity for every one unit of the other quantity.
Hence, 4:1 is a ratio. Option (A) is correct.
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The complete question is as follows:
Which of the following terms correctly describes 4:1?
(A) Ratio
(B) Percent
(C) Factor
(D) Fraction.
What is the solution to log^2 (2x^3 -8)-2log ^2 x= log^2x
The coordinates have opposite signs. Which choice includes all the quadrants that could contain the point?
Final answer:
A point with coordinates that have opposite signs could lie in either the second or fourth quadrant of a coordinate system. The second quadrant contains points where x is negative and y is positive, while the fourth quadrant contains points where x is positive and y is negative.
Explanation:
When discussing the coordinates with opposite signs in a coordinate system, we are referring to the locations where the x and y coordinates have different signs. In this case, we are considering which quadrants could contain a point with coordinates that have opposite signs. To visualize this, we must remember that the coordinate system is divided into four quadrants, with each axis serving as a boundary between them. The quadrants are usually numbered counterclockwise, starting with the top right quadrant as the first quadrant.
In the first (I) quadrant, both x and y are positive. In the second (II) quadrant, x is negative while y is positive. In the third (III) quadrant, both x and y are negative. And in the fourth (IV) quadrant, x is positive while y is negative. Therefore, if a point has coordinates with opposite signs, it could only be found in the second (II) and fourth (IV) quadrants, where one of the coordinates is positive and the other is negative.
Lisa has a scholarship that pays 75% of her tuition for all four years she attends college. What is the total amount the scholarship is worth if Lisa's classes cost 12,000 per year
Find the length and width of a rectangle that has the given perimeter and a maximum area. perimeter: 316 meters
A city had a parade for a winning basketball team. Since it was put together in five days, only 25% of the groups asked to be in the parade were able to participate. If the parade committee asked 128 groups to participate, how many were able to be in the parade?
A. 27
B. 103
C. 11
D. 32
Sherrod deposits $500 each year in a savings account earning 3% interest compounded annually. He makes no withdrawals. How much interest will the account earn after 3 years?
The formula for annual compound interest, including principal sum, is:
A = P (1 + r/n) (nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (500)
r = the annual interest rate (.03)
n = the number of times that interest is compounded per year (1)
t = the number of years the money is invested (3)
A=5000(1+.03/1)^1(3)
A=546.36
Interest (I) gained is A-P
I=546.36-500
I=46.36
A moving company charges $0.60 per pound for a move from New York to Florida. A family estimates that their belongings weigh about 4 tons. About how much would it cost the family to move from New York to Florida? $2,400 $6,000 $4,800 $10,000
List all the pairs of integers with a product of 30
Write the sum using summation notation, assuming the suggested pattern continues. 16 + 25 + 36 + 49 + ... + n2 + ...
Answer:
Hence, the sum using summation notation, assuming the suggested pattern continues [tex]16+25+36+49+......+n^2+.....[/tex] is:
[tex]\sum_{n=4}^{\infty}n^2[/tex]
Step-by-step explanation:
We have to write the sum using summation notation, assuming the suggested pattern continues:
[tex]16+25+36+49+......+n^2+.....[/tex]
Clearly we may also write this pattern as:
[tex]4^2+5^2+6^2+.....+n^2+......[/tex]
So, in terms of the summand it is written as:
[tex]\sum_{n=4}^{\infty}n^2[/tex]
( We have started our summation from 4 since the term in the summation starts with 16 which is 4^2 and goes to infinity )
ΔXYZ was reflected to form ΔLMN. Which statements are true regarding the diagram? Check all that apply. ΔXYZ ≅ ΔLMN ∠Y ≅ ∠M ∠X ≅ ∠L ∠Z ≅ ∠L YZ ≅ ML XZ ≅ LN
Answer:
A)ΔXYZ ≅ ΔLMN
B)∠Y ≅ ∠M
C)∠X ≅ ∠L
F)XZ ≅ LN
Step-by-step explanation: correct on edge 2020
A reflected triangle is congruent to the original triangle with corresponding angles and sides being equal. However, the orientation changes such that ∠X corresponds to ∠L, ∠Y to ∠M, and ∠Z to ∠N in the reflected triangle. Similarly, side YZ corresponds to MN, not ML, and XZ corresponds to LN.
Explanation:In the subject of mathematics, specifically geometry, when you reflect a shape, the original shape and its image are congruent. This means they have the same size and shapes, but their orientation might differ. Therefore, the statement ΔXYZ ≅ ΔLMN is indeed true.
In terms of the specific angles, a reflection does not change their measures; it simply changes their orientations. Therefore, ∠X is congruent to ∠L, ∠Y is congruent to ∠M, and ∠Z is congruent to ∠N. However, the statement ∠Z ≅ ∠L is incorrect as Z corresponds to N, not L, in the reflected triangle.
Similarly, side lengths also remain the same in a reflection. Thus, XY corresponds to ML, YZ corresponds to MN and XZ corresponds to LN, making the statements YZ ≅ MN and XZ ≅ LN true. But the statement YZ ≅ ML is incorrect as YZ corresponds to MN, not ML, in the reflected triangle.
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There are approximately 2.6 million deaths per year in country A. Express this quantity as deaths per minute.
Let f be a function of two variables that has continuous partial derivatives and consider the points a(7, 3), b(12, 3), c(7, 7), and d(15, 9). the directional derivative of f at a in the direction of the vector ab is 5 and the directional derivative at a in the direction of ac is 4. find the directional derivative of f at a in the direction of the vector ad. (round your answer to two decimal places.)
To answer this question, it is necessary to make use of the concepts of directional derivative and gradient of a function.
The solution is:
grad F( x, y ) ×ad(u) =64/10
The gradient of a function f (x,y) is a vector defined by:
grad f(x,y) = δf(x,y)/δx × i + δf(x,y)/δy × j
and directional derivative, in the direction of ab is defined by :
grad f( x,y) × Uᵃᵇ(u) (1) where Uᵃᵇ is a unitary vector in the direction of ab
according to that Uᵃᵇ (u) = Uᵃᵇ/|ab|
Then if the directional derivative of f (x,y) in the direction of the vector ab is 5.
A ( 7 , 3 ) B ( 12 , 3 ) then vector ab is:
ab = [ 12 - 7 , 3 - 3 ] ⇒ ab = [ 5 , 0 ] and |ab| = √ (5)² (0)² |ab| = 5
a unitary vector in the direction of ab is:
ab(u) = 5×i/ 5 and according to equation (1)
δf(x,y) / δx × i × 5 × i / |5| = 5
δf(x,y) / δx × i × 5 × i = 25
δf(x,y) / δx = 5 then f( x,y ) = 5 × x + ????
We go on to calculate the component on j of f(x,y)
Following the same procedure
ac = ( 7 , 7 ) - ( 7 , 3 ) ⇒ ac = [ 0 , 4 ] |ac| = √(4)² + (0)²
|ac| = 4
Unitary vector in the direction of ac(u) is:
ac/|ac| = 4 × j / 4
Then :
δf(x,y) / δy × j × + 4 × j / 4 = 4
δf(x,y) / δy × j × + 4 × j = 16
δf(x,y) / δy = - 4 and f(x,y) = -4×y
f(x,y) = 5 × i + 4 × j
Finally:
vector ad = [ ( 15 - 7 , 9 - 3 ) ] ⇒ ad = ( 8 , 6 )
Unitary vector in direction ad is
ad(u) = ( 8 ×i + 6 ×j ) / √ (8)² + (6)² ⇒ ad(u) = ( 8 ×i + 6 ×j ) /√ (8)² + (6)²
ad(u) = ( 8 ×i + 6 ×j ) /10
Now we have f (x,y ) = 5 × i + 4 × j and ad(u) = ( 8 ×i + 6 ×j ) /10
We can calculate the directional derivative of f(x,y) in the direction of ad with the use of equation (1)
grad F( x, y ) ×ad(u) = ( 5 × i + 4 × j ) × ( 8 ×i + 6 ×j ) /10
grad F( x, y ) ×ad(u) = 4 + ( 24/10)
grad F( x, y ) ×ad(u) =64/10
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