There is point i.e., value of x where the function f(x) = x³ - 4x² - 7x +12 is not continuous.
Consider the polynomial function f(x) = x³ - 4x² - 7x +12
As we know if a function f be a function defined on an open interval containing point a where t a ∈ R be a constant, a function f is continuous at a if lim_(x→a) f(x) = f(a).
Here, the domain of polynomial function f(x) = x³ - 4x² - 7x +12 is set of all real numbers, as the function is well defined for all real values of x.
We know that every polynomial function is continuous in their domain.
This means, the polyomial function f(x) = x³ - 4x² - 7x +12 is continuous on set of all real numbers.
Therefore, there is no value of x where the polynomial function f(x) = x³ - 4x² - 7x +12 is not continuous.
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what is the minimum degree taylor polynomial about that you need to calculate to 3 decimal places?
The minimum degree of a Taylor polynomial required to calculate to 3 decimal places depends on the complexity of the function being approximated.
A Taylor polynomial is a way of approximating a function with a polynomial. The polynomial is built based on the function's value and its derivatives at a specific point.
To calculate a Taylor polynomial to 3 decimal places, the degree of the polynomial required depends on the complexity of the function and the desired accuracy.
If the function is smooth, with no sudden changes, a lower degree polynomial may be sufficient to approximate the function to 3 decimal places.
A general rule is to use a polynomial of degree at least two times the number of decimal places desired. So, for 3 decimal places, a polynomial of degree 6 or higher is required.
However, it is important to note that the minimum degree required can be determined by experimenting with different degrees of polynomials and checking the accuracy of the approximation.
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Over which interval is the graph of f(x) = Ex2 + 5x +
2
6 increasing?
O (-6.5, co)
• (-5, co)
• (-∞0, -5)
• (-00, -6.5)
"The function is increasing in the interval (-2.5,∞)".
What is a function?The binding of inputs to essentially one output is known as a function.It is useful to think of a function as a computer.The machine may only generate one output for each input and will only take certain inputs, which is referred to as the function's domain.In other terms, a function is a relationship between variables, and the type of relationship determines the function. Examples of such relationships include y = sinx and y = x +6.To determine whether the function is increasing or decreasing we are going to use the derivative test.
f(x) = x² + 5x + 6
f'(x) = 0
2x + 5 = 0
x = -5/2
Intervals associated with this point are (-∞,-2.5) and (-2.5,∞)
Now by checking in the interval (-∞,-2.5) at a point, let's say -3
Then,
f'(-3) = -1 so function is decreasing over this interval.
Now by checking in the interval (-2.5,∞) at a point, let's say 2
Then,
f'(2) = 1 so function is increasing over this interval.
Hence "The function is increasing in the interval (-2.5,∞)".
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The sign of the second derivative of a function can determine whether the function is increasing or decreasing. If the second derivative is positive, the function is concave up and therefore increasing.
Which interval is the graph of f(x) = Ex2 + 5x +26 increasing?For the function f(x) = Ex^2 + 5x + 26, the second derivative is 2E, which is positive for all values of E greater than 0. So for any value of E greater than 0, the function is increasing over its entire domain.Since the interval is not specified in the question, we cannot determine the exact interval over which the function is increasing. The answer options (-6.5, co), (-5, co), (-∞, -5), and (-00, -6.5) do not provide enough information to determine the interval.To learn more about graph refer:
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is 600,000 more or less than 11733.83
Answer:
600,000 is greater than 11,733.83. Or
600,000>11,733.83
Step-by-step explanation:
To compare the values of 600,000 and 11,733.83, we can simply look at the magnitude of the numbers.
600,000 is much larger than 11,733.83. This means that 600,000 is greater than 11,733.83
Tell whether the triangle with the given side lengths is a right triangle.
12 mm, 16 mm, 20 mm
The triangle that will be formed using the sides 12 mm , 16 mm and 20 mm will be a right trinagle.
What is a Right Triangle ?A right triangle, also known as a right-angled triangle, right-perpendicular triangle, orthogonal triangle, or previously rectangled triangle, is a triangle with one right angle, or two perpendicular sides. The foundation of trigonometry is the relationship between the sides and various angles of the right triangle.
The characteristics of a right-angled triangle.
The right angle, or 90°, is always one angle.The hypotenuse is the side with the 90° angle opposite.The longest side is always the hypotenuse.The other two inner angles add up to 90 degrees.Base and perpendicular are the names of the other two sides that are close to the right angle.Right-angle triangles have an area that is equal to half of the product of their adjacent sides, hence Area of Right Angle Triangle = 1/2 (Base* Perpendicular).Three such triangles will result if we drop a perpendicular from the right angle to the hypotenuse.The radius of a circle whose circumference includes all three vertices is equal to one-half the length of the hypotenuse.The triangle is known as an isosceles right angled triangle, where the adjacent sides to the 90° are equal in length, if one of the angles is 90° and the other two angles are each equal to 45°.Let is consider the triangle is a right triangle so the base and height will be 12 mm and 16 mm respectivly.
Now using Pythagoras Theorem :
√(height² + base²) = hypotenuse
⇒√(12²+16²) = √(144+256) = √(400) = 20 mm = hypotenuse.
The triangle that will be formed using the sides 12 mm , 16 mm and 20 mm will be a right trinagle.
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What is the surface area of this rectangular prism?
The surface area of this rectangular prism 290 square inches.
What is a rectangular prism?A rectangular prism is a three-dimensional solid shape with six faces that including rectangular bases. A cuboid is also a rectangular prism. The cross-section of a cuboid and a rectangular prism is the same.
The surface area of a solid is defined as the sum of the areas of each of its faces (or surfaces), hence the name.
The surface area of rectangular prism formula is
Surface Area = 2(lw+lh+wh)
Here, length=5 inches, width= 5 inches and height=12 inches
Now, surface area = 2(5×5+5×12+5×12)
= 2(25+60+60)
= 2×145
= 290 square inches
Therefore, the surface area of this rectangular prism 290 square inches.
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The surface area of a rectangular prism is given by the expression 2(lh+bh+lb) where, l = length, h = height and b = width
What is a rectangular prism?A rectangular prism is a three-dimensional solid shape with six faces that including rectangular bases.
Given that, what is the surface area of this rectangular prism.
Let us consider a rectangular prism with l = length, h = height and b = width
Now, we will find its surface area,
Since, we know that, there are 6 rectangular faces in a rectangular prism
Therefore, for surface area of the rectangular prism will add the areas of all 6 faces,
Area of the faces = lb + bh + lh + lb + bh + lh (area of rectangle = length x width)
Area of the faces = 2(lb+bh+lh)
Hence, the surface area of a rectangular prism is given by the expression 2(lh+bh+lb) where, l = length, h = height and b = width
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what is the volume of the solid generated when the region in the first quadrant bounded by the graph of y=44−x2−−−−−√ and the x- and y-axes is revolved about the y-axis?
The volume of the solid generated is [tex]\pi (2/3) (44)^{3/2}.[/tex]
We have,
To find the volume of the solid generated by revolving the region in the first quadrant bounded by the graph of y = √(44 - x²) and the x- and y-axes about the y-axis, we can use the method of cylindrical shells.
The volume can be calculated using the following integral:
V = ∫[a,b] 2πx f(x) dx
where [a, b] is the interval of x-values that defines the region in the first quadrant.
In this case, the curve y = √(44 - x²) intersects the x-axis when y = 0, which gives us:
0 = √(44 - x²)
Squaring both sides and solving for x, we get:
x² = 44
x = ±√44
Since we're considering the region in the first quadrant, we take the positive square root:
x = √44 = 2√11
Now we can set up the integral:
V = ∫[0, 2√11] 2πx (√(44 - x²)) dx
Simplifying the integrand:
V = 2π ∫[0, 2√11] x (√(44 - x²)) dx
To evaluate this integral, we can make the substitution u = 44 - x²,
du = -2x dx:
V = 2π ∫[0, 44] (√u) (-du/2)
V = -π ∫[0, 44] (√u) du
Integrating:
V = -π [2/3 u^(3/2)] [0, 44]
V = -π [2/3 (44)^(3/2) - 0^(3/2)]
V = -π (2/3) (44)^(3/2)
Finally, taking the absolute value and simplifying:
V = π (2/3) (44)^(3/2)
So, the volume of the solid generated is π (2/3) (44)^(3/2).
Thus,
The volume of the solid generated is [tex]\pi (2/3) (44)^{3/2}.[/tex]
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Hillary works at a video store. She earns $8. 50 per hour plus 12% commission on all the weekly sales she makes. If she works 24 hours in a week, which expression best describes her sales (x) if she earns at least $324 during that week
Her sales are at least $1000 during that week.
Let x stand in for her sales.
The equation to determine her sales is as follows, based on the data in the question:
Hillary works at a video store. She earns $8. 50 per hour plus a 12% commission on all the weekly sales she makes.
he works 24 hours a week if she earns at least $324 during that week
(8.50 × 24) + (12% × x) ≥ 324
204 + (0.12 × x) ≥ 324
204 + 0.12x ≥ 324
0.12x ≥ 324 - 204
0.12x ≥ 120
x ≥ 120/0.12
x ≥ 1000
Her sales are at least $1000
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What quantity of 62% acetic acid solution must be mixed with a 19% acetic acid solution to produce 334. 78 mL of 47% solution?
x = 217.9927 = amount of mL of the 62% acid solution
y = 116.7873 = amount in mL of the 19% acid solution
This can be solved using a system of two equations and two variables.
Let x = amount of mL of the 62% acid solution
Let y = amount in mL of the 19% acid solution
We know the total mL of the mixture of the two solutions is 600 mL,
so x + y = 334.78 ..... eq(1)
62% of solution x is acid which can be written as 0.62x and similarly acid from the second solution can be written as 0.19y. We know that 47% of the 334.78 mL solution will be acid or 0.47(334.78).
The acid in the mixture solution comes from the acid in solution x and solution y. So 0.62 x + 0.19 y = 0.47 (334.78) ... eq(2)
from equation (1)
x = 334.78-y
So, x = 217.9927
y = 116.7873.
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A bicycle store costs $4000 per month to operate. The store pays an average of $50 per bike. The average selling price of each bicycle is $150. How many bicycles must the store sell each month to break even
Answer:27
Step-by-step explanation:
Answer:
The store would need to sell 40 bikes to break even.
Step-by-step explanation:
In order to break even the store needs to make $4000 a month. They spend $50 per bike and sell them for $150, an easy equation to get their profit per bike is to subtract how .uch they pay for a bike from how much they make per bike which is $150-50=$100. So now we just need to divide $4000 by $100 to find out how many bikes they need to sell. $4000÷$100=40.
EASY 100 POINTS, WILL MARK FIRST ANSWER BRAINLIEST
The system of linear equations −3x + 2y = −6 and y equals negative one half times x plus 4 is graphed on a coordinate plane. Approximate the solution to the system.
(−1.5, −3.5)
(−1.5, 4.25)
(2.5, 1.5)
(3.5, 2.25)
Answer:
D. (3.5, 2.25)
Step-by-step explanation:
A solution to the system is where the two equations intersect.
looking at the graph, we see that the intersection point is between x= 3 and 4, and Y = 2 and 3.
Answer:
(3.5, 2.25)
Step-by-step explanation:
To find the coordinate of the solution to the system of linear equations −3x + 2y = −6 and y = -1/2x + 4, we need to solve the system. One method to do this is substitution. We can solve one of the equations for y, then substitute the expression for y into the other equation. This will give us an equation in terms of x, which we can then solve for x. Then, we can substitute the value of x back into one of the equations to find the corresponding value of y.
Using substitution, we can solve the system as follows:
y = -1/2x + 4
-3x + 2y = -6
Substitute y = -1/2x + 4 into -3x + 2y = -6:
-3x + 2(-1/2x + 4) = -6
-3x + (-x + 8) = -6
-4x + 8 = -6
-4x = -14
x = 3.5
Substitute x = 3.5 into y = -1/2x + 4:
y = -1/2(3.5) + 4
y = 2.25
Therefore, the solution to the system is approximately (3.5, 2.25).
Ruby read 12 pages of a book in 18 minutes. How many pages can she read per HOUR?
Answer: 40 pages/hour
Step-by-step explanation:
12pages / 18min * 60min/1hr
= 40pages / 1hr
OR
12 pages / 18min SIMPLIFY
2pages / 3 min
Multiply both by 20 to get how many pages per 60 minutes.
what will always be a solution to a homogenous set of equations
Homogeneous systems of linear equations can have either a unique solution or infinitely many solutions, depending on the rank of the matrix representing the system.
A homogeneous set of equations is a set of linear equations that has a trivial solution (that is, a solution consisting of only the zero vector). In other words, if the coefficients of the variables in a set of linear equations are all equal to zero, the set of equations is considered homogeneous.
The trivial solution, that is, the solution in which all variables are equal to zero, will always be a solution to a homogeneous set of equations. This is because, if all coefficients are equal to zero, then each equation in the set will reduce to 0 = 0, which is always true. Thus, the trivial solution is always a solution to a homogeneous set of equations.
In addition to the trivial solution, a homogeneous set of equations may also have non-trivial solutions, which are solutions that are not equal to only the zero vector. To find these non-trivial solutions, we typically use techniques such as Gaussian elimination, row reduction, or the use of matrices and determinants.
It is important to note that homogeneous systems of linear equations can have either a unique solution or infinitely many solutions, depending on the rank of the matrix representing the system.
Therefore, homogeneous systems of linear equations can have either a unique solution or infinitely many solutions, depending on the rank of the matrix representing the system.
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Exponential Functions
Using the details given, the exponential growth at rate of 10% over 14 years is 26582.5
What is an Exponential FunctionAn exponential function is a mathematical function in which the variable appears as an exponent. It can be written in the form of y = ab^x, where a is the base, b is a constant, and x is the variable. This function has the property that y grows or decays at an exponential rate as x increases or decreases.
y = a(1 + r)^t
In the given problem, we have the data given as;
a = 7000r = 10%t = 14 yearsSubstituting the values into the formula;
y = 7000(1 + 0.1)^14
y = 26582.5
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lian currently pays 640 per month to rent her apartment she has been told by her landlord that her rent will be increasing 15% next year. how much will she have to pay per month, next year in rent?
she has to pay $736 per month, next year in rent.
What is percentage?A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.
Given, lian currently pays 640 per month to rent her apartment she has been told by her landlord that her rent will be increasing by 15% next year.
Her rent is now = 640 per month
Since, After a year her rent will increase by 15 %
Thus,
Her rent next year per month = 640(1 + 15%)
Her rent next year per month = 640(1 + 0.15)
Her rent next year per month = 640* 1.15
Her rent next year per month = 736
Therefore, she has to pay $736 per month, next year in rent.
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i need help with thus..
Answer: -35/18
Step-by-step explanation: Hello!
The main thing that you need to know when dividing fractions is what the reciprocal is. Here is an example of finding the reciprocal:
1/2 divided by 2/3 = 1/2 x 3/2.
Basically, you flip the second fraction upside down (Put the number under the fraction line over the fraction line and put the number above the fraction line under the fraction line). Then, you multiply the two fractions. Then, you will find the answer. Here is how to do that with the problem you have:
5/3 divided by (-6/7) = ?
After finding the reciprocal, you would get:
5/3 x (-7/6)
To find the answer, multiply the numerators together (5 and -7) and the denominators (3 and 6) together.
For the new numerator, you get -35. For the new denominator, you get 18. Put the new numerator above the fraction line and the new denominator below the fraction line.
The answer is:
-35/18
please convert the following decimal numbers to binary numbers: (to receive full credit, you must show how you reach your answers. 1 point each, total 5 points): 1) 13 2) 121 3) 252 4) 1023 5) 2049
Answer:
1101 1111001111111001111111111100000000001Step-by-step explanation:
2/13=6 reminder 1
2/6= 3 reminder 0
2/3= 1 reminder 1
2/1= 0 reminder 1
13 becomes 1101 base two/ binary form
The conversion of number from decimal to binary:
[tex](13)_{10}=(1101)_{2}[/tex]
[tex](121)_{10}=(1111001)_{2}[/tex]
[tex](252)_{10}=(11111100)_{2}[/tex]
[tex](1023)_{10}=(1111111111)_{2}[/tex]
[tex](2049)_{10}=(100000000001)_{2}[/tex]
Recursively multiplying the given decimal number by two is one way to convert a decimal number to a binary number. The remainders are then recorded until we arrive at 0 as the final quotient. The remainders are then entered in reverse order to obtain the binary equivalent of the provided decimal number.
The number is 13.
13 ÷ 2 = 6, Remainder = 1
6 ÷ 2 = 3, Remainder = 0
3 ÷ 2 = 1, Remainder = 1
1 ÷ 2 = 0, Remainder = 1
So, [tex](13)_{10}=(1101)_{2}[/tex]
The number is 121.
121 ÷ 2 = 60, Remainder = 1
60 ÷ 2 = 30, Remainder = 0
30 ÷ 2 = 15, Remainder = 0
15 ÷ 2 = 7, Remainder = 1
7 ÷ 2 = 3, Remainder = 1
3 ÷ 2 = 1, Remainder = 1
1 ÷ 2 = 0, Remainder = 1
So, [tex](121)_{10}=(1111001)_{2}[/tex]
The number is 252.
252 ÷ 2 = 126, Remainder = 0
126 ÷ 2 = 63, Remainder = 0
63 ÷ 2 = 31, Remainder = 1
31 ÷ 2 = 15, Remainder = 1
15 ÷ 2 = 7, Remainder = 1
7 ÷ 2 = 3, Remainder = 1
3 ÷ 2 = 1, Remainder = 1
1 ÷ 2 = 0, Remainder = 1
So, [tex](252)_{10}=(11111100)_{2}[/tex]
The number is 1023.
1023 ÷ 2 = 511, Remainder = 1
511 ÷ 2 = 255, Remainder = 1
255 ÷ 2 = 127, Remainder = 1
127 ÷ 2 = 63, Remainder = 1
63 ÷ 2 = 31, Remainder = 1
31 ÷ 2 = 15, Remainder = 1
15 ÷ 2 = 7, Remainder = 1
7 ÷ 2 = 3, Remainder = 1
3 ÷ 2 = 1, Remainder = 1
1 ÷ 2 = 0, Remainder = 1
So, [tex](1023)_{10}=(1111111111)_{2}[/tex]
The number is 2049.
2049 ÷ 2 = 1024, Remainder = 1
1024 ÷ 2 = 512, Remainder = 0
512 ÷ 2 = 256, Remainder = 0
256 ÷ 2 = 128, Remainder = 0
128 ÷ 2 = 64, Remainder = 0
64 ÷ 2 = 32, Remainder = 0
32 ÷ 2 = 16, Remainder = 0
16 ÷ 2 = 8, Remainder = 0
8 ÷ 2 = 4, Remainder = 0
4 ÷ 2 = 2, Remainder = 0
2 ÷ 2 = 1, Remainder = 0
1 ÷ 2 = 0, Remainder = 1
So, [tex](2049)_{10}=(100000000001)_{2}[/tex]
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3. [1 points] a random variable x ∼ n(μ,σ2) is gaussian distributed with mean μ and variance σ2. given thatforanya,b∈r,wehavethaty =ax bisalsogaussian,finda,bsuchthaty ∼n(0,1).
The values of a and b such that y∼n(0,1) are a = ± (1/σ) and b = ± (μ/σ)
of a gaussian distribution.
Gaussian distribution (also known as normal distribution) is a bell-shaped curve, and it is assumed that during any measurement values will follow a normal distribution with an equal number of measurements above and below the mean value.
E(X) = μ, Var(X) = σ^2
Y = aX + b
We want E(Y) = 0 and Var(Y) = 1
E(Y) = a E(X) + b
0 = a μ + b
b = -a μ --- (1)
Var(Y) = (a^2) Var(X)
1 = (a^2) σ^2
a^2 = 1/σ^2
a = ± 1/σ --- (2)
From (1) and (2)
b = (± 1/σ) μ
So, a = ± (1/σ) and b = ± (μ/σ)
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The complete question is :
A random variable X ∼ N (µ, σ2 ) is Gaussian distributed with mean µ and variance σ 2 . Given that for any a, b ∈ R, we have that Y = aX + b is also Gaussian, find a, b such that Y ∼ N (0, 1)
also this one would help me out alot
The line that connects the coordinates (7,5) and (9,5) has the equation y = 5.
What is coordinates?A point or shape's location in a two-dimensional plane can be found using coordinates, which are a pair of numbers. The terms "x-coordinate" and "y-coordinate" are used to define a point's location on a 2D plane. A group of values that exhibit precise positioning On graphs, it is typically represented by a pair of numbers: the first number indicates the length along, while the second number indicates the length up or down. The point (12,5), for instance, is 12 units down and 5 units up. In order to determine a point's coordinates in a coordinate system, you must do the opposite.Due to the fact that the slope of the line across the two points (x1, y1) and (x2, y2) [tex]$\mathrm{m}=\frac{\mathrm{y}_2-\mathrm{y}_1}{\mathrm{x}_2-\mathrm{x}_1}$[/tex] Now, as shown below, we can determine the slope of the line through points (7,5) and (-9,5):
[tex]$\mathrm{m}=\frac{5-5}{-9-7}=\frac{0}{-16}=0[/tex]
As a result, the line's slope is o.
Find the y-intercept now by using the slope and one of the two points.
y = mx + b
5 = (0)(7) + b
b = 5
Equation in slope intercept form is written as follows:
y = mx + b
y = (0)x + 5
y = 5
Consequently, y = 5 is the line's equation.
The complete question is:
Find the equation of the line that passes through the points (7,5) and (−9,5)
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The slope of the line given by the coordinates of two points (x1, y1) = (-7, 5) and (x2, y2) = (-7, 9) is undefined. The slope of the line for coordinates (2, 3) and (2, -3) is undefined.
What is slope of the line?A line's slope in mathematics is defined as the ratio of the change in the y coordinate to the change in the x coordinate.
Both the net change in the y-coordinate and the net change in the x-coordinate are denoted by y and x, respectively.
Given that,
The coordinates of two points are:
(x1, y1) = (-7, 5) and
(x2, y2) = (-7, 9).
The slope of the line is given by the following equation:
m = (y2 - y1) / (x2 - x1)
Substituting the value of the given coordinates we have:
m = (9 - 5) / (-7 - (-7))
m = (4)/ -7 + 7 = undefined
The slope of the line given by the coordinates of two points (x1, y1) = (-7, 5) and (x2, y2) = (-7, 9) is undefined.
Similarly, for the coordinates (2, 3) and (2, -3):
m = (y2 - y1) / (x2 - x1)
m = (-3 -3) / (2 - 2)
m = undefined.
Hence, the slope of the line for coordinates (2, 3) and (2, -3) is undefined.
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Question 3/3
What is the area of this composite figure?
Circle = r²
Triangle = 1/2 bh
On solving the provided question, we can say thatarea of the figure = area of triangle + area of semi-circle = 12 + 14.13 = 26.13 yd sq.
What is area?The size of an area on a surface can be expressed as an area. The area of an open surface or the boundary of a three-dimensional object is referred to as the surface area, whereas the area of a planar region or planar region refers to the area of a form or planar layer. The total amount of space filled by a planar (2-D) surface or shape of an object is known as its area. Draw a square on a piece of paper using a pencil. a character with two dimensions. The area of a shape on paper is the space it takes up. Imagine that the square is composed of more compact unit squares.
area of semi-circle = 1/2 * [tex]\pi *r^2[/tex] = 0.5*3.14*3*3 = 14.13yd sq.
area of triangle = 1/2*b*h = 0.5*6*4 = 12sq. yd
area of the figure = area of triangle + area of semi-circle = 12 + 14.13 = 26.13 yd sq.
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consider the function f(x,y) = x 2y, defined on d, i.e., the region bounded by the parabolas y= 2x2 and y= 1 x2.
The function f(x,y) is defined for all (x,y) pairs that lie within the region d, which is bounded by the parabolas y= 2x2 and y= 1 x2. It returns the product of x and 2 to the power of y.
The function f(x,y) is a two-variable function defined on the region d, which is bounded by the parabolas y= 2x2 and y= 1 x2. In order to calculate the output of this function, we must first identify the values of x and y that lie within the region d. We can do this by graphing both parabolas and finding the intersection point, which is when x=1 and y=2. This tells us that any (x,y) pairs with x<1 and y<2 will be within the region d. Now that we have identified the region, we can calculate the output of the function by multiplying x and 2 to the power of y. For example, if x=0.5 and y=1, then f(x,y)= 0.52 = 0.25.
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let f(x) be a function with domain double-struck r. (a) show that e(x) = f(x) f(−x) is an even function. (simplify your answers completely.) e(x) = f(x) f(−x)
The given equation e(x)=f(x)f(-x) is an even function since e(x)=e(-x) and f(x)^2=f(-x)^2.
let f(x) be a function with domain double-struck r. (a) show that e(x) = f(x) f(−x) is an even function. (simplify your answers completely.) e(x) = f(x) f(−x) = f(x) f(x) = f(x)^2 e(x) = f(x)^2
e(x) = e(-x) Therefore, e(x) is an even function.
(-x) is an even function, we need to show that e(x)=e(-x).
We start by replacing -x with x in the equation:
e(x)=f(x)f(-x)
e(x)=f(x)f(x)
e(x)=f(x)^2
Now, we replace x with -x in the equation:
e(-x)=f(-x)f(-x)
e(-x)=f(-x)^2
Since the left-hand side of both equations are the same (e(x)=e(-x)), and the right-hand sides of both equations are the same (f(x)^2=f(-x)^2), we can conclude that e(x) is an even function.
The given equation e(x)=f(x)f(-x) is an even function since e(x)=e(-x) and f(x)^2=f(-x)^2.
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A population has u = 80 and a standard deviation of o = 5. 2. What are the mean and standard deviation of the sampling distribution
& if a sample of 100 were taken? (2 points)
CH-80,02 -0. 52
CH-80,0 -0. 052
H-80,0 - 5. 2
H-80,0 - 52
H-80. 0 - 520
The mean of the sampling distribution is 80 and the standard deviation is 0.52.
What is Normal Distribution?A normal distribution in statistics is defined as the type of continuous probability distribution where the data are arranged in a symmetrical bell shaped graph.
Given that,
Population mean, μ = 80
Population standard deviation, σ = 5.2
Mean of a sampling distribution in a normally distributed sample is same as the population mean and the standard deviation of the sampling distribution = σ / √n, where n is the sample size.
Sample size, n = 100
Mean of the sampling distribution = 80
Standard deviation of the sampling distribution = 5.2 / √100 = 5.2/10 = 0.52.
Hence mean and the standard deviation of the sampling distribution is 80 and 0.52 respectively.
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Let F(t) be the outside temperature (*F)t hours after 3 A.M. Explain the meaning of F(4) < F(5)
F(4) < F(5) means that the temperature at 7 A.M. is less than the temperature at 8 A.M.
What is an inequality?An inequality is used to compare two or more expressions.
For example -
ax + b > c
ax + b < c
Given is that F(t) is the outside temperature (*F)t hours after 3 A.M.
It is given that -
F(4) < F(5)
This means that the temperature at 7 A.M. is less than the temperature at 8 A.M.
Therefore, F(4) < F(5) means that the temperature at 7 A.M. is less than the temperature at 8 A.M.
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The perimeter of a rectangle is 40 cm. The length is 14 cm.
Let x = width of the rectangle.
Ravi says he can find the width using the equation 2(x + 14) = 40.
Fran says she can find the width using the equation 2x + 28 = 40.
Answer the questions to solve the equations and to compare the steps and solutions.
1. Which of these is the most helpful first step for solving Ravi's equation, 2(x + 14) = 40? (1 point)
Circle the best answer.
Add 14 to both sides
Subtract 14 from both sides
Divide both sides by 2
Multiply both sides by 2
2. What would your next step be? (1 point)
3. Solve Ravi's equation, 2(x + 14) = 40, to find the width of the rectangle. Show your work. (1 point)
4. Which of these is the most helpful first step for solving Fran's equation, 2x + 28 = 40? (1 point)
Circle the best answer.
Multiply both sides by 2
Subtract 28 from both sides
Divide both sides by 2
Add 28 to both sides
5. What would your next step be? (2 points)
6. Solve Fran's equation, 2x + 28 = 40, to find the width of the rectangle. Show your work. (2 points)
7. The two equations have different solution steps. Do they have the same solution? Use the distributive property to show why this answer makes sense. (2 points)
The solution is given below.
What is equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign. In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.
here, we have,
The perimeter of a rectangle is 40 cm. The length is 14 cm.
Let x = width of the rectangle.
Ravi says he can find the width using the equation 2(x + 14) = 40.
Fran says she can find the width using the equation 2x + 28 = 40.
now, we get,
1. Divide both sides by 2
2(x+14) = 40
x+14 = 20
2. Isolate the x term by subtracting 14 from both sides
3. x = 6. The width of the triangle is 6 cm.
4. Isolate the x term by subtracting 28 from both sides
2x + 28 = 40
2x = 12
5. Divide both sides by 2
6. x = 6
7. The two equations have the same solution, because by the distributive rule, 2(x+14) = 2x+28.
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What is the equation of the line that passes through the point (-7,-8)and has a slope of 0?
Answer:
because the line has a slope of 0, you know that it is horizontal
therefore, the equation is y=-8
Step-by-step explanation:
Based on past experience, a bank believes that 8% of the people who receive loans will not make payments on time. The bank has recently approved 600 loans. Describe the sampling distribution model of the proportion of clients in this group who may not make timely payments. Find the mean/standard error of the sampling distribution of the proportion.
The mean/standard error of the sampling distribution of the proportion is 0.0111.
What is meant by standard deviation?Its root-mean The square root of the mean of the squares of all the values in a series calculated from the arithmetic mean is the square deviation, also referred to as the standard deviation and denoted by the symbol.
The standard deviation reveals the degree of data dispersion. It expresses how far away from the mean each observed value is.
Almost 95% of the values in any distribution will be within two standard deviations of the mean. The degree of data dispersion from the mean is described by the term "standard deviation" (or " σ").
The data tend to cluster around the mean when the standard deviation is low; when it is high, the data are more dispersed.
The formula to calculate the standard error of sampling distribution of sample proportion is:
[tex]$ SE(p) =\sqrt{\frac{p(1-p)}{n} }[/tex]
It is given that the bank believes that 8% of people who receive loans will not make payments on time. That is,
Population proportion, p =0.08
Sample size, n= 800
Using the formula defined :
[tex]$ SE(p) =\sqrt{\frac{0.08(1-0.08)}{600} }[/tex]
[tex]$ SE(p) =\sqrt{0.000123}[/tex]
SE(p) = 0.0111
Thus, the mean/standard error of sampling distribution of the proportion is 0.0111.
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Find the slope of the line containing the points ( 3, 4 ) and ( 3, - 6 ).
The slope of the line containing the points ( 3, 4 ) and ( 3, - 6 ) is -8.
What is the slope of a line which passes through points ( p,q) and (x,y)?The equation of the straight line has its slope and given point.
If we have a non-vertical line that passes through any point(x1, y1) has gradient m. then general point (x, y) must satisfy the equation
y-y₁ = m(x-x₁)
Its slope would be:
[tex]m = \dfrac{y-q}{x-p}[/tex]
Slope of parallel lines are same. Slopes of perpendicular lines are negative reciprocal of each other.
Given;
The two points ( 3, 4 ) and ( 3, - 6 ).
Now, to find the slope
m=-6-4/4-3
m=-8/1
m=-8
Therefore, the slope will be -8.
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Consider the vector A == Axi + Ayj + Azk. Find the components AhA2' A3 of the vector A in a skewed coordinate system whose axes have directions specified by the following unit vector triad:
the components of A in this system will be: Ah = Axi1 + Ayj1 + Azk1, A2 = Axi2 + Ayj2 + Azk2, and A3 = Axi3 + Ayj3 + Azk3. To find the components of A in a skewed coordinate system, we first need to express the vector A in terms of the unit vectors of the system.
i = i1i2i3
j = j1j2j3
k = k1k2k3
Ah = Axi1 + Ayj1 + Azk1
A2 = Axi2 + Ayj2 + Azk2
A3 = Axi3 + Ayj3 + Azk3
To find the components of A in a skewed coordinate system, we first need to express the vector A in terms of the unit vectors of the system. The unit vectors of the skewed coordinate system are given by i = i1i2i3, j = j1j2j3, and k = k1k2k3.
Therefore, the components of A in this system will be: Ah = Axi1 + Ayj1 + Azk1, A2 = Axi2 + Ayj2 + Azk2, and A3 = Axi3 + Ayj3 + Azk3.
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One year old baby often weighs 250% of its birth weight. What should a one year old baby weigh if it was 8 pounds at birth?
Answer:
20 pounds
Step-by-step explanation:
250% = 2.5
8 times 2.5 = 20 pounds
So, a one-year-old baby weighs 20 pounds.
adding and subtracting mixed numbers with unlike denominators worksheets
Adding and subtracting mixed numbers with unlike denominators requires converting them into equivalent fractions with a common denominator. for example 3 3/5 + 2 1/7 = 5 8/35.
3 3/5 + 2 1/7 = 5 8/35, 5 1/5 - 3 2/3 = 1 7/15, 4 1/4 + 2 2/7 = 6 11/28, 6 3/4 - 3 5/6 = 2 13/24, 1 1/2 + 3 4/7 = 4 11/14, 2 5/6 - 1 1/4 = 1 7/12, 5 1/2 + 3 3/4 = 9 1/4, 3 1/5 - 2 3/7 = 0 16/35, 4 2/3 + 2 4/5 = 6 17/15 , 2 3/4 - 1 5/6 = 1 1/12
To add or subtract mixed numbers with different denominators, you must convert them into identical parts with common factors. Similarities can be found by looking at the most unusual differences between the two denominators. Mixed numbers are converted to bad parts, added or subtracted, and returned to perfect mixed numbers. How to find the common factor, that is,
Switch completely to unwise parts, extend or subtract, then switch completely to mixed numbers. Consider exact calculations of mixed numbers with different denominators.
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