Step-by-step explanation:
To show that the function f(x,y) = 2x^2 − xy is differentiable at any point (a,b), we need to show that the partial derivatives of f with respect to x and y exist and are continuous at that point.
The partial derivative of f with respect to x is:
∂f/∂x = 4x - y
The partial derivative of f with respect to y is:
∂f/∂y = -x
We can see that both of these partial derivatives are continuous functions of x and y and are defined for all (x,y) in the domain of f. Therefore, f is differentiable at any point (a,b).
The derivative of f at (a,b) is the gradient vector of f evaluated at (a,b), which is given by:
Gradient vector of f(a,b) = ∇f(a,b) = ( ∂f/∂x, ∂f/∂y ) = (4a-b, -a)
So the derivative of f at (a, b) is (4a-b, -a).
solve this system of equations by using the elimination method.
2x+4y=8 2x+3y=4
Hey there!
Solution:Point Form: [tex](-4,4)[/tex]Equation Form: [tex]x=-4,y=4[/tex]
Hope this helps!
David cut a board 20 foot long into four equal pieces. Which expression does NOT equal the length of one of the pieces in feet? Responses
The expression that does not equal the length of one of the pieces in feet is C) 4/20
How to find the expression ?The expression that show the length of one of the pieces of feet would be the expressions that lead to the solution:
= Length of board cut by David / Number of equal pieces
= 20 / 4
= 5 ft each
Option A mirrors the equation above and so is equal to the length of one of the pieces. Option B also equals the equation above as the signs ÷ and / are both for division.
Option C does not equal the length of one of the pieces as it gives:
= 4 / 20
= 1 / 5 ft
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Options for this question are:
A)20/4
B)20 ÷ 4
C)4/20
In a wind tunnel experiment; the force on a projectile due to air resistance was measured at different velocities: Velocity (100 ftlsec) 0 2 4 6 8 10Force (100 Ib) 0 2.8 14 38. 74.4 128Find an interpolating polynomial for these data It is highly recommended that you use software or a calculator. F(v) = ____ Use the polynomial to estimate the force on the projectile when it is traveling at 750 ftlsec:
The estimated force on the projectile when it is traveling at 750 ft/sec is approximately 696.7 lbs.
The Possible Interpolating PolynomialOne possible interpolating polynomial for these data points is a polynomial of degree 3, also known as a cubic polynomial. Using software or a calculator to fit a cubic polynomial to the data points, the resulting equation is:
F(v) = -0.00000016v³ + 0.0004v² - 0.08v + 0.4
To estimate the force on the projectile when it is traveling at 750 ft/sec, we can substitute 750 for v in the equation above:
F(750) = -0.00000016(750³) + 0.0004(750²) - 0.08(750) + 0.4
F(750) ≈ 696.7
The estimated force on the projectile when it is traveling at 750 ft/sec is approximately 696.7 lbs.
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A bag contains 6 red marbles and 4 blue marbles. If two marbles are drawn at random and not replaced, what is the probability that two red marbles are drawn?
If two marbles are drawn at random and not replaced, the probability that two red marbles are drawn is; 1/3
What is the probability of selection?The parameters given for marbles in the bag are;
Number of red marbles = 6
Number of blue marbles = 4
Thus;
Total number of marbles = 6 + 4
Total number of marbles = 10 marbles
Now, probability of first selection being red marble = 6/10
Probability of second marble being red if first was not replaced = 5/9
Thus;
P(first two being red marbles) = (6/10) * (5/9)
= 1/3
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integrate both sides of the differential equation you found for dp to obtain an equation for p . your equation should then include a constant that depends on initial conditions. determine the value of this constant by assuming that the pressure at some reference height y0 is p0 .
The value of this constant, by assuming that the pressure at some reference height y₀ is p₀, is p - p₀ = p.g(y₀ - y).
dp = -pg dy
At height y = y₀ and p = p₀
∫dp = ∫-pg dy
⇒p = -pgy + C Eqn(1)
Where c is the constant of integration
We know that y = y₀ and p = p₀
⇒p₀ = -pgy₀ + C
⇒C = p₀ + pgy₀
Put the value of C in Eqn(1)
⇒p = -pgy + p₀ + pgy₀
⇒p - p₀ = -pgy + pgy₀
⇒p - p₀ = pg(y₀ - y)
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the opportunity cost of producing one bushel of wheat in canada is equal to 1/2 tv set. the opportunity cost of producing one bushel of wheat in the us is 2 tv sets. if these two countries specialize according to comparative advantage and then trade with each other, .
If Canada specializes in producing wheat and the US specializes in producing TV sets, both countries will benefit from trade.
To determine this if Canada produces wheat, the opportunity cost is 1/2 TV set. If the US produces wheat, the opportunity cost is 2 TV sets. So, Canada has a comparative advantage in producing wheat. Similarly, the US has a comparative advantage in producing TV sets.
Let's say Canada produces 100 bushels of wheat, and the US produces 100 TV sets. If they trade with each other, Canada can exchange 50 bushels of wheat for 50 TV sets, and the US can exchange 50 TV sets for 50 bushels of wheat. Both countries would benefit from this trade.
Canada would end up with 50 TV sets and 50 bushels of wheat, while the US would end up with 50 bushels of wheat and 50 TV sets. Both countries would be better off than if they had not traded.
Opportunity cost in mathematics is a term that is used to describe the cost of one alternative in terms of the benefits forgone by the best alternative. It can also be represented in terms of mathematical equations.
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b) Five thousand three hundred and forty-five plus eight hundred and six in number
Answer:
I'm not sure if this is what you meant but 5,345 + 806 = 6,151
Step-by-step explanation:
Answer:6151
Step-by-step explanation:
5345 + 806 = 6151
4. Find the area of the irregular shape. Round to the nearest tenth. 8 cm 5 cm
90.2 cm²
241 cm²
65.1 cm²
72.3 cm²
Answer: Total area = 40 cm² + 25.12 cm² = 65.12 cm²
Step-by-step explanation:
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Answer:65.1
Step-by-step explanation:
The population of a small town increased from 25,056 people in 2002 36,720 in 2010 what is the percentage change in the population
well, we know in 2002 it was 25,056, then it ballooned to 36,720 in 2010, the difference is really 36720-25056 = 11664.
now, if we take 25,056(origin amount) to be the 100%, what's 11664 off of it in percentage?
[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 25056 & 100\\ 11664& x \end{array} \implies \cfrac{25056}{11664}~~=~~\cfrac{100}{x} \\\\\\ \cfrac{ 58 }{ 27 } ~~=~~ \cfrac{ 100 }{ x }\implies 58x=2700\implies x=\cfrac{2700}{58} \implies x\approx \stackrel{\%}{46.55}[/tex]
Please answer this question
The transformation with respect to dilation will only preserve the angle measure i.e. B.
What is dilation in mathematics?
Dilation is a process of changing the size of an object or shape by decreasing or increasing its dimensions by some scaling factors. For example, a circle with radius 10 unit is reduced to a circle of radius 5 unit. The application of this method is used in photography, arts and crafts, to create logos, etc. In Geometry, there are four basic types of transformations. They are:
RotationTranslationReflectionResizing or DilationSome features of shapes that remain unchanged during the dilation transformations are:
Each angle of the figure is the same.Midpoints of the sides of the figure remain the same as the midpoint of the dilated shape.Parallel and perpendicular lines in the figure remain the same as the parallel and perpendicular lines of the dilated figure.The images remain the same.Now,
As in dilation Each angle of the figure remains the same.
Hence,
The transformation with respect to dilation will only preserve the angle measure
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How do i solve for value of m
A farmer has enough food to feed 20 animals in his cattle for 6 days. How long
would the food last if there were 10 more animals in his cattle?
Answer:
4 daysb
Step-by-step explanation:
Here the number of animals and the number of days are in inverse proportion. Hence the food will last 4 days.
The food will last for 6 days for 20 animals.
If there were 10 more animals in the cattle, the total number of animals would be 20+10 = 30.
The food will last for the same 6 days but now it would be divided among 30 animals.
Therefore, the food would last for 6 days / 30 animals = 0.2 days per animal.
So, the food will last for 0.2 days if there were 10 more animals in the farmer's cattle.
A bank loaned out $12,000, part of it at the rate of 6% per year and the rest at the rate of 16% per year. If the interest received totaled$1000, how much was loaned at 6%?
The amount loaned at 6% was $9200. The solution has been obtained by making and solving the equations.
What is an equation?
A claim that two expressions are equal and connected by the equals sign "=" is the mathematical definition of an equation. The two expressions need not be entirely composed of variables; one expression may have variables while the other may not.
We are given that a bank loaned out $12,000 and a part of it at the rate of 6% per year and the rest at the rate of 16% per year.
Let the amount loaned at 6% be 'x'
So, amount loaned at 16% will be '12,000 - x'
The total interest received was $1000.
Therefore,
⇒6%x + 16%(12000-x) = 1000
⇒0.06x + 1920 - 0.16x = 1000
⇒920 = 0.1x
⇒9200 = x
Hence, the amount loaned at 6% was $9200.
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what is the best use of
Trigonometry is best used in navigation to estimate where to point the compass to get a straight line. It will be simple to pinpoint a location as well as find distance and see the horizon using a compass and trigonometric functions in navigation.
What is Trigonometry?The area of mathematics known as trigonometry examines the correlation between the ratios of a right-angled triangle's sides to its angles. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are used to study this relationship. The concept of trigonometry was developed by the Greek mathematician Hipparchus, and the word trigonometry is a derivative of Latin from the 16th century.
One of the most significant area of mathematic is rigometry. The words "Trigonon" and "Metron," which mean respectively "triangle" and "measure," are combined to form the word "trigonometry." It is the investigation of how a right-angled triangle's sides and angles relate to one another. Therefore, using formulas and identities based on this relationship aids in determining the measure of a right-angled triangle's unknown dimensions.
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Full question:
What is the best use of trigonometry?
graph the ellipse give by the equation 49^2 + 16^2 = 784. Rewrite the equation i standard form.
Answer:
The equation 49^2 + 16^2 = 784 can be rewritten as (x/24.5)^2 + (y/8)^2 = 1.
The standard form of an ellipse is (x/a)^2 + (y/b)^2 = 1, where a and b are the lengths of the major and minor axes, respectively. In this case, a = 24.5 and b = 8.
To graph the ellipse, we can use the equation in the standard form and plot points that satisfy the equation. The center of the ellipse is (0,0) and it has a major axis of 24.5 and minor axis of 8.
The graph of this ellipse will look like an oval, with the center at the origin, major axis along the x-axis and minor axis along the y-axis.
select the proper inverse operation to check the answer to 14+17=31
Answer:
31-17=14
Step-by-step explanation:
When you reverse it it equals the same answer
Answer:
31-14=17
Step-by-step explanation:
When We Reverse It,Will Be it Self.
Which is the area, A
A
, of this triangle, using the formula A=12ab⋅sin(C)
A
=
1
2
a
b
·
sin
C
?
The area of triangle, using the formula A is 146.9 in^2
How to find a area of triangle using trigonometry?Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It can be used to calculate the area of a triangle by using the length of its sides.
The most commonly used formula to calculate the area of a triangle is known as Heron's formula. This formula uses the lengths of the triangle's three sides, known as a, b, and c. Then use the formula A = 1/2 x[tex]\sqrt{(a + b+ c)(-a + b + c)(a - b + c)(a + b -c)}[/tex] to calculate the area of the triangle.
To calculate the area of a triangle using Heron's formula, first measure the length of each side of the triangle. Then use the lengths of the sides to calculate s using the formula above. Once you have calculated s, plug the values for a, b, and c into Heron's formula to calculate the area of the triangle.
A = 1/2ab⋅sin(C)
= 1/2(21 in. )(14 in.)⋅sin(88)
= 1/2(294 in.2 )⋅sin(88)
= 1/2(294 in.2 )⋅0.9945
A = 146.9 [tex]in^{2}[/tex]
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Please help
2 (x- 5) =9 - 3x + 6 + 8 + 3x + 7
Responses
x=10
x-20
x=40
x=80
Answer:
Step-by-step explanation:
2(x-5)=9-3x+6+8+3x+7
2x-10=9-3x+14+3x+7 {Adding the constant no.s +6 and +8}
2x-10=30-3x+3x {Adding all the constants +14 , +7, +9}
2x-10=30-0 {+3x and -3x cancels out}
2x-10=30
2x=30-10 {coefficient with variable i.e 2x on one side}
2x=20 {20 divides by 2 if we bring coefficient 2 to right side}
x=10 Ans
7 The equation, A equal to 2,400 plus quantity 1 plus 0.031 over 4 end quantity all raised to the power of 4 times t, represents the amount of money earned on a compound interest savings account. What does the value 0.031 represent?
The value 0.031 represents the interest rate, which means the annual compounded interest rate is 3.1%.
The value 0.031 represents the interest rate, which means the annual compounded interest rate is 0.31%.
The value 0.031 represents the investment period, which means the investment is invested for 0.031 years.
The value 0.031 represents the investment period, which means the investment is invested for 3.1 years
The quantity which represents the 0.031 in the given formula of compound interest, the value represents the annual compound interest rate 3.1%, So option A is correct
What is compound interest?A loan or deposit amount is subject to compound interest as interest. In everyday life, it is the most prevalent idea. Both the principal and the interest accrued over time affect the compound interest for a given amount. Between compound and simple interest, this is the primary distinction.
[tex]A = P[1 + r/n]^{nt}[/tex]
Where,
A = compound interest
P = Principle Amount
r = Interest rate
n = number of period for which interest is calculated
t = time
Given that,
A equal to 2,400 plus quantity 1 plus 0.031 over 4 end quantity all raised to the power of 4 times t
[tex]A = 2400[1 + 0.031/4]^{4}[/tex]
So, here we can see that r is equal to 0.031
It represents the annual compound interest rate of the compound interest
Hence, it is a annual compound interest which has value of 3.1%
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List all real values of xx such that f(x)=0. If there are no such real x, type none in the answer blank. If there is more that one real xx, give a comma separated list (e.g. 1,2).
f(x)=19x2+10x
x =
The all the real values of x , such that for the function f(x) = 19x² + 10x satisfies f(x) = 0 are 0 , -10/19 .
The function is ⇒ f(x) = 19x² + 10x ;
taking x common from both terms and writing it in factor form ,
we get ;
⇒ x(19x + 10) = 0 ;
we know that the product of any two number is zero only when one of them is 0 ,
that means ;
⇒ x = 0 or 19x + 10 = 0 ;
⇒ x = 0 or 19x = -10 ;
⇒ x = 0 or x = -10/19 .
Therefore , all the real values of x such that f(x) = 0 are 0 , -10/19 .
The given question is incomplete , the complete question is
List all real values of x such that the function f(x)=0. If there are no such real x, type none in the answer blank. If there is more that one real x, give a comma separated list (e.g. 1,2).
f(x) = 19x² + 10x .
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what does the equation y=x^2 represent as a curve in R^2
The equation y = x² represents a parabola as a curve
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The equation y = x² represents a parabola as a curve
The equation is given as:
y = x²
The above equation has a degree of 2
Polynomials that have a degree of 2 are referred to as quadratic equations
The curve of a quadratic equations are called parabola
Hence, the equation y = x² represents a parabola as a curve
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what is the inequality for 3<-5n+2n
Answer:
n< -1
Step-by-step explanation:
1. Rewrite so n is on the left side of the inequality.
−5n+2n>3
2.Add 5n and 2n
-3n >3
3. Divide each term then simplify.
15. Reasoning The Annual Mug Race is the
longest river sailboat race in the world. The
event is run along the St. Johns River, which
is 310 miles long. About how many times as
long as the race is the river?
Using expression {River=x(race)} River's length is 7.38 times the race's length, approximately.
What are expressions?
Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between. The mathematical operators can be of addition, subtraction, multiplication, or division. For example, x + y is an expression, where x and y are terms having an addition operator in between. In math, there are two types of expressions, numerical expressions - that contain only numbers; and algebraic expressions- that contain both numbers and variables.
e.g. A number is 6 more than half the other number, and the other number is x. This statement is written as x/2 + 6 in a mathematical expression. Mathematical expressions are used to solve complicated puzzles.
We know that
The river is 310 miles long.
The race is 42 miles long.
So, if we want to know know how many times as long as the race is the river, we just need to use this expression
River=x(race)
Where x represents the factor that expresses the ratio between both, solving for x we have
x=river/race=310/42≈7.38
Therefore,
River's length is 7.38 times the race's length, approximately.
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Right question:-
The Annual Mug Race 42 mile is the
longest river sailboat race in the world. The
event is run along the St. Johns River, which
is 310 miles long. About how many times as
long as the race is the river?
If log5 125 = x, then x =
Answer:
X=3
Step-by-step explanation:
If you enter log5(125)= x into your calculator that should be your answer
The volume of a rectangular box is 2x(2x + 5)(2x- 3). (Drawing is not to
scale.)
Height 2x - 3
Length 2x + 5
Width 2x
Which statement about the volume of the box is true?
O A. The volume is the product of the length, 2x 5, and the width, 2x.
O B. The volume is the sum of the length, 2x + of 5, the width, 2x, and the
height, 2x - 3
• C. The volume does not depend on the height, 2x - 3.
D. The volume is the product of the area of the base, 2x(2x + 5), and
The volume is the product of the area of the base, 2x(2x+5), and the height, 2x-3.
What is Volume?
A measurement of three-dimensional space is volume. It is frequently expressed quantitatively using SI-derived units, as well as several imperial or US-standard units. Volume and the notion of length are connected.
What is Area?
The size of a patch on a surface is determined by its area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.
Given that;
Height= 2x - 3
Length= 2x + 5
Width= 2x
We have the following equation for the rectangular box in this instance:
2x(2x + 5)(2x- 3)
We may come up with an equivalent formula for the volume by using the distributive property of multiplication.
We have then ;
[tex](4x^2+10x)(2x-3)[/tex]
Once more writing, we have:
[tex](8x^3+20x^2)+(-12x^2-30x)[/tex]
Including related terms, we have:
8x^3+8x^2-30x
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write the abbreviation for each of the following: match the words in the left column to the appropriate blanks in the sentences on the right. make certain each sentence is complete before submitting your answer.
1. The abbreviation for femtogram is - fg.
2. The abbreviation for decimeters is - dm.
3. The abbreviation for microliter is - μL or ul.
4. The abbreviation for femtosecond is - fs.
1. Femtogram (fg): A unit of mass measurement in the International System of Units (SI). It is equal to [tex]10^{-15}[/tex] grams and is commonly used to measure the mass of small particles or atoms.
2. Decimeter (dm): A unit of length measurement in the International System of Units (SI). It is equal to 0.1 meters and is commonly used to measure small distances.
3. Microliter (μL or ul): A unit of volume measurement in the International System of Units (SI). It is equal to [tex]10^{-6}[/tex] liters and is commonly used to measure very small volumes of liquids.
4. Femtosecond (fs): A unit of time measurement in the International System of Units (SI). It is equal to [tex]10^{-15}[/tex] seconds and is commonly used to measure very short periods of time, such as in atomic and molecular physics, chemistry, and photonics.
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The complete question is -
write the abbreviation for each of the following: match the words in the left column to the appropriate blanks in the sentences on the right. make certain each sentence is complete before submitting your answer.
1. The abbreviation for femtogram is -
2. The abbreviation for decimeter is -
3. The abbreviation for microliter is -
4. The abbreviation for femtosecond is -
Briefly explain the three lines of evidence for the Earth's relative age. Write your answer in the essay box below.
WILL GIVE BRAINLIEST
Answer:
There are several lines of evidence that scientists use to determine the relative age of the Earth and its various features. These include:
Stratigraphy: This refers to the study of rock layers and the sequence in which they were formed. Scientists use stratigraphy to determine the relative ages of different rock layers and the fossils they contain. The principle of superposition states that in any undisturbed sequence of rocks, the oldest rock layers will be at the bottom and the youngest will be at the top.
Fossil record: Fossils are the remains of plants and animals that lived in the past. By studying the fossils in rock layers, scientists can determine the relative ages of different rock formations and the organisms that lived in them. Fossils of extinct species found in lower rock layers are older than those found in higher rock layers.
Radiometric dating: This is a method that uses the decay of radioactive isotopes in minerals to determine the absolute age of rocks and minerals. Scientists can use radiometric dating to determine the age of a rock or mineral, and by comparing the ages of different rocks and minerals, they can determine the relative ages of different rock formations and the Earth's surface features.
Please help
Select the correct choices that make the statement true.
Look at image
For the given image, the value of iota (i) is -i and 1.
What does the arithmetic term "iota" mean?
Thus, the development of complex numbers has given rise to imaginary roots. The sign I which stands for Iota, is used to represent -1. (Imaginary number). In essence, I refers to the imaginary component, often known as the iota. I's value is -1. A square root with a negative value represents an imaginary value. Imaginary numbers can be calculated using any of the standard arithmetic operators. An imaginary number has a negative value when squared.
Complex numbers are expressed using the imaginary unit number, where I is defined as imaginary or unit imaginary. Here, we'll go over the chart and principles for imaginary numbers that are employed in mathematical computations.
The value of iota (i²) = -1
The given expressions can be represented in the following way -
i³ = i² × i = -1 × i = -i
i⁸ = i² × i² × i² × i² = -1 × -1 × -1 × -1 = 1
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NO LINKS!!!
Please help me with #53.
Answer:
53
a. Graph of equation attached
b.
Domain:
[tex]-\infty \: < x < \infty \:[/tex]
Interval notation: [tex]\left(-\infty \:,\:\infty \:\right)[/tex]
Range:
[tex]f(x) > 1\\\\[/tex]
Interval notation: [tex]\left(1,\:\infty \:\right)[/tex]
c. See explanation below and on attached graph
d. See explanation below
Step-by-step explanation:
53
The function to be used for answering questions is
[tex]f(x) = 8\left(\dfrac{1}{4}\right)^{x+4}+1[/tex]
a. Graph
Graph is plotted using online graphing tool. 2 points identified are (-4, 9) and (-3, 3)
b. Domain and Range
First let's understand what domain and range are
Definition of domain
The domain of a function, y = f(x), is the set of input(x) values for which the function is real and defined
The function has no undefined points nor domain constraints. You can see that the graph goes to infinity for negative and positive x values Therefore, the domain is :
[tex]-\infty \: < x < \infty \:[/tex]
which can be expressed in interval notation as:
[tex]\left(-\infty \:,\:\infty \:\right)[/tex]
Definition of range
The range of a function y = f(x) is the set of values for y over the domain for which the function is defined
As [tex]x \rightarrow \infty[/tex], [tex]y \rightarrow 1[/tex]
y never quite reaches 0 but its value approaches 1 and therefore the range is
[tex]f(x) > 1\\\\[/tex]
In interval notation:
[tex]\left(1,\:\infty \:\right)[/tex]
c. End Behavior
The end behavior of a function f(x) refers to how the function behaves when the variable x increases or decreases without bound. In other words, the end behavior describes the ultimate trend in the graph of f(x) as [tex]x \rightarrow \pm \infty[/tex]
In this particular function we see that as [tex]x\to \:+\infty \:,\:f\left(x\right)\to \:1\\[/tex]
and as [tex]x\to \:-\infty \:,\:f\left(x\right)\to \:\infty\\[/tex]
d. Point (-3, 3)
By pure chance I chose this point as one of the points asked for in part a because I was looking for integer values I don't think it matters that there is repetition
(-3, 3) is plotted on the graph as point A
3. Evaluate f(-3)
We have
[tex]f(x) = 8\left(\dfrac{1}{4}\right)^{x+4}+1[/tex]
To find f(-3) substitute -3 for x in the above function:
[tex]f(-3) = 8\left(\dfrac{1}{4}\right)^{-3+4}+1\\\\= 8\left(\dfrac{1}{4}\right)^1} + 1\\\\\\= 8\left(\dfrac{1}{4}\right) + 1\\\\= 2 + 1 \\\\= 3[/tex]
So f(-3) = 3
Answer:
a) See attachment 1.
b) Domain: (-∞, ∞)
Range: (1, ∞)
c) As x → -∞, f(x) → ∞
As x → ∞, f(x) → 1⁺
d) See attachment 2.
e) See below.
Step-by-step explanation:
Given exponential function:
[tex]f(x)=8\left(\dfrac{1}{4}\right)^{x+4}+1[/tex]
Part aTo find the y-intercept, input x = 0 and solve for y:
[tex]\implies f(0)=8\left(\dfrac{1}{4}\right)^{0+4}+1[/tex]
[tex]\implies f(0)=8\left(\dfrac{1}{4}\right)^{4}+1[/tex]
[tex]\implies f(0)=8\left(\dfrac{1}{256}\right)+1[/tex]
[tex]\implies f(0)=\dfrac{1}{32}+1[/tex]
[tex]\implies f(0)=\dfrac{33}{32}[/tex]
[tex]\implies f(0)=1.03125[/tex]
Therefore, the y-intercept is (0, 1.03125).
Substitute x = -2 into the function to find a second point on the curve:
[tex]\implies f(-2)=8\left(\dfrac{1}{4}\right)^{-2+4}+1[/tex]
[tex]\implies f(-2)=8\left(\dfrac{1}{4}\right)^{2}+1[/tex]
[tex]\implies f(-2)=8\left(\dfrac{1}{16}\right)+1[/tex]
[tex]\implies f(-2)=\dfrac{1}{2}+1[/tex]
[tex]\implies f(-2)=\dfrac{3}{2}[/tex]
[tex]\implies f(-2)=1.5[/tex]
Therefore, a second point on the curve is (-2, 1.5).
Part bThe domain of a function is the set of all possible input values (x-values).
The domain of the given function is unrestricted:
Interval notation: (-∞, ∞)The range of a function is the set of all possible output values (y-values).
As the exponential part of the function is always positive, the range of the given function is restricted:
Interval notation: (1, ∞)Part cAs x approaches negative infinity, the function approaches positive infinity:
As x → -∞, f(x) → ∞As x approaches positive infinity, the function approaches f(x) = 1.
As x → ∞, f(x) → 1⁺Part dSee attachment 2.
Part eSubstitute x = -3 into the function:
[tex]\implies f(-3)=8\left(\dfrac{1}{4}\right)^{-3+4}+1[/tex]
[tex]\implies f(-3)=8\left(\dfrac{1}{4}\right)^{1}+1[/tex]
[tex]\implies f(-3)=8\left(\dfrac{1}{4}\right)+1[/tex]
[tex]\implies f(-3)=2+1[/tex]
[tex]\implies f(-3)=3[/tex]
The ball thrown upwards hit the roof and returns back to the ground.The upward moment is given with : s= -t2+3t+4
and downward :s= - 2t2+t+7
How high is the roof from the ground ?
The height of the roof will be 10.75 units.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is to find the height of the roof.
The upward motion is modelled by the equation -
S = - t² + 3t + 4
The height of the roof is equal to maximum height attained. For the maximum height attained -
dS/dt = 0
d/dt (- t² + 3t + 4) = 0
- 2t + 3 = 0
2t = 3
t = 3/2 .... Eq(1)
The height of the roof will be -
S(max) = - t² + 3t + 4
S(3/2) = - (3/2)² +(3 x 3/2) + 4
S(3/2) = 9/4 + 9/2 + 4
S(3/2) = 2.25 + 4.5 + 4
S(3/2) = 10.75 units.
Therefore, the height of the roof will be 10.75 units.
To solve more questions on algebraic expressions, visit the link below -
brainly.com/question/1041084
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