Answer:
C. The vertex is [tex](-3,-1)[/tex], the domain is all real numbers, and the range is [tex]y\leq -1[/tex].
Step-by-step explanation:
We have been given a function [tex]f(x)=-2(x+3)^2-1[/tex]. We are asked to identify the vertex, domain and range of the given function.
We can see that our given parabola is in vertex form [tex]y=a(x-h)^2+k[/tex], where [tex](h,k)[/tex] represents vertex of parabola.
We can rewrite our given equation as:
[tex]f(x)=-2(x-(-3))^2-1[/tex]
Therefore, the vertex of our given parabola would be [tex](-3,-1)[/tex].
We know that parabola is a quadratic function and the domain of a quadratic function is all real numbers.
We know that range of a quadratic function in form [tex]f(x)=a(x-h)^2+k[/tex] is:
[tex]f(x)\leq k[/tex], when [tex]a<0[/tex] and,
[tex]f(x)\geq k[/tex], when [tex]a>0[/tex]
Upon looking at our given function, we can see that [tex]a=-2[/tex], which is less than 0, therefore, the range of our given function would be [tex]y\leq -1[/tex].
What is the answer to this question ?
How does the throughput of pci express version 3.0 compare to pcie version 2.0?
What is the reason for each step in the solution of the inequality?
−2(x+3)−4>4x+30
Select the reason for each step from the drop-down menus.
2nd picture is drop down box answers.
How many times does 1/2 fit into 30
How would you express as a unit rate: morag typed 60 words in one minute
Morag would be typing at a rate of 60 words per minute (wpm) in order to type 60 words in one minute.
What is Unit conversion?A statement of the connection between units that are used to alter the units of a measured quantity without affecting the value is called a conversion factor. A conversion ratio (or unit factor), if the numerator and denominator have the same value represented in various units, always equals one (1).
To express Morag's typing speed as a unit rate, we would divide the number of words by the number of minutes.
Therefore, the unit rate for Morag typing 60 words in one minute would be 60 words per minute (60 wpm).
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Write a problem saying a single load of laundry cost $2 and a double load costs $4 the machine only accept quarters what is the answer
The problem is: A laundromat charges $2 for a single load of laundry and $4 for a double load. The machine only accepts quarters. How many quarters are needed to wash two double loads of laundry?
To solve this problem, we first need to determine the total cost of washing two double loads. Since each double load costs $4, two double loads will cost 2 * $4 = $8.
Next, we need to determine how many quarters are in $8. Since there are 4 quarters in a dollar, there are 4 * 8
= 32 quarters in $8.
Therefore, 32 quarters are needed to wash two double loads of laundry
A boat was sailing for 4 hours and covered 224 miles. A jet is ten times as fast as the boat. Find the jet’s speed.
A 27 oz bottle of a new soda costs $2.25. What is the unit rate, rounded to the nearest tenth of a cent?
The unit rate is the cost per ounce of soda. By dividing the total cost by the total ounces, we get the price per ounce in dollars ($0.08333), and converting this to cents gives us $8.3 cents per ounce.
Explanation:The term unit rate refers to a rate in which the second term is 1. In this case, we want to find out how much 1 ounce of soda costs.
First, you want to divide the total cost of the bottle by the total ounces in the bottle. So you divide $2.25 by 27. The answer you get is the price of one ounce of soda in dollars. When calculating it, you get approximately $0.08333.
To get the rate in cents, convert the dollars to cents by multiplying by 100 (since 1 dollar is 100 cents). The answer ($8.33) is the cost of one ounce to the nearest tenth of a cent.
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What is 164% of 25? I have no idea and im in the middle of a test XDDD
Help with Precalc Vectors? Medal and fan v = (-7,8) Find the magnitude of vector v. Express your answer in radical form sqrt of 113 is what I got for that. second part: use the magnitude and the angle of vector v to write it in the form (lvlcostheta, lvl sin theta). I don't know how to determine the single angle for one vector. Earlier in a different part of the problem, I was asked to find the angle between two vectors, but I don't think you use that here since no second vector is provided. So how would I answer the second part? Thanks(:
To express the vector v = (-7,8) in the form (|v|cosθ, |v|sinθ), you first correctly found the magnitude |v| using the Pythagorean theorem. Then, to find the angle θ, use the arctangent function applied to the y/x components of your vector. Ensure the angle is in the correct quadrant by modifying it if necessary.
Explanation:To find the magnitude of the vector v = (-7,8), you've correctly done the Pythagorean theorem and found sqrt(113), which is the absolute value of your vector, written as |v|. Now, to express the vector in the form (|v|cosθ, |v|sinθ), we need to find the angle θ. This is found using the arctangent function (also known as the inverse tangent or atan). θ = atan(y/x), where x and y are the x and y components of your vector respectively. Be careful with the sign as atan can only return values from -π/2 to π/2.
In your case θ = atan(8/-7), but since this gives a negative answer and the vector is in the second quadrant where angles are positive, you should add π to your result to get the correct angle. Then you can substitute |v| and θ into the expressions for the x and y components.
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Calculate the mean of the number set 5 10 12 4 6 11 13 5
PLZ HELP I WILL MAKE YOU BRAINLIEST!! IT IS GOOD FOR YOUR PROFILE!
Rewrite as a square or a cube:
1 11/25
Answer:
1 11/25 = 36/25
= (6/5)^2
Step-by-step explanation:
what is the solution of -8/2y-8=5/y+4 - 7y+8/y^2-16? y = –4 y = –2 y = 4 y = 6
Answer:
d. 6
Step-by-step explanation:
just took the pretest:) have such a fantastic day loves, you're doing AMAZING!
Jeff invests an amount at 4% interest compounded annually. After 3 years, he has $1687.30. What was the original amount Jeff invested?
$1500
$1200
$1450
$750
What is the area of a triangle with verticies at (-2,1), (2,1) and (3,4)
how many solutions are in 6x+4x-6=24+9x
Jasmine finished the bike trail in 2.5 hours at an average rate of 2.5 miles per hour.Lucy biked the same trail at a rate of 6 1/5 mile per hour.How long did it take lucy to bike the trail?
2.5 hours at a rate of 2.5 miles per hour = 2.5 * 2.5 = 6.25 miles
the trail was 6.25 miles long
Lucy rode at 6 1/5 mile per hour
so 6.25 / 6 1/5 = 1.008 = 1.01 hours
Hi, How do you find the first and second derivatives of the function.
y=(x^2-7/63x) (x^4+1/x^3)
I think for the first derivative dy/dx it's 2/63x-3/63x^-3+4/9x^-5 but I'm not sure, and I have no clue for the second derivative d^2y/dx^2.
The first hand derivative of the function is [tex]6x^5 - \frac{35}{63}x^4 - \frac{1}{x^2} - \frac{4}{63x^3}[/tex] and the second derivative is [tex]30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]. To find the first derivative, apply the product rule to the given function. Then, differentiate the first derivative to obtain the second derivative. Simplify each step carefully.
To find the first derivative of the given function [tex]y = \left( x^2 - \frac{7}{63}x \right) \left( x^4 + \frac{1}{x^3} \right)[/tex], we'll use the product rule, which states that if [tex]y = u(x) \cdot v(x)[/tex], then [tex]y' = u' \cdot v + u \cdot v'[/tex].
First, define u(x) and v(x) as following:
[tex]u(x) = x^2 - \frac{7}{63}x = x^2 - \frac{1}{9}x[/tex][tex]v(x) = x^4 + \frac{1}{x^3}[/tex]Compute u'(x):
[tex]u'(x) = 2x - \frac{1}{9}[/tex]Compute v'(x):
[tex]v'(x) = 4x^3 + (-3)x^{-4} = 4x^3 - \frac{3}{x^4}[/tex]Apply the product rule: [tex]y' = u' \cdot v + u \cdot v'[/tex]
Thus,
[tex]y' = \left(2x - \frac{1}{9}\right)\left( x^4 + \frac{1}{x^3} \right) + \left( x^2 - \frac{1}{9}x \right) \left( 4x^3 - \frac{3}{x^4} \right)[/tex]Simplify this expression step-by-step to find the first derivative.
[tex]y' = \left(2x - \frac{1}{9}\right)\left( x^4 + \frac{1}{x^3} \right) + \left( x^2 - \frac{1}{9}x \right) \left( 4x^3 - \frac{3}{x^4} \right)[/tex][tex]y'[/tex] [tex]&= \left(2x \cdot x^4 + 2x \cdot \frac{1}{x^3} - \frac{1}{9} \cdot x^4 - \frac{1}{9} \cdot \frac{1}{x^3} \right)[/tex][tex]&\quad + \ \left( x^2 \cdot 4x^3 - x^2 \cdot \frac{3}{x^4} - \frac{1}{9}x \cdot 4x^3 + \frac{1}{9}x \cdot \frac{3}{x^4} \right)[/tex][tex]y'[/tex] [tex]&= 2x^5 + \frac{2}{x^2} - \frac{1}{9}x^4 - \frac{1}{9x^3} \\[/tex] [tex]&\quad + \ 4x^5 - \frac{3}{x^2} - \frac{4}{9}x^4 + \frac{1}{3x^3}[/tex][tex]y'[/tex] [tex]&= 6x^5 - \frac{1}{x^2} - \frac{5}{9}x^4 + \frac{2}{9x^3}[/tex][tex]y'[/tex] [tex]&= 6x^5 - \frac{5}{9}x^4 - \frac{1}{x^2} + \frac{2}{9x^3}[/tex]To find the second derivative, differentiate the first derivative, carefully differentiating each term:
[tex]y'' &= \frac{d}{dx}\left( 6x^5 \right) - \frac{d}{dx}\left( \frac{5}{9}x^4 \right) - \frac{d}{dx}\left( \frac{1}{x^2} \right) + \frac{d}{dx}\left( \frac{2}{9x^3} \right) \\[/tex][tex]y''[/tex] [tex]&= 30x^4 - \frac{5}{9} \cdot 4x^3 - \left( -2x^{-3} \right) + \left( -\frac{2}{9} \cdot 3x^{-4} \right) \\[/tex][tex]y''[/tex] [tex]&= 30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]So, for the function [tex]y = \left( x^2 - \frac{7}{63}x \right) \left( x^4 + \frac{1}{x^3} \right)[/tex], we have:
First derivative [tex](y')[/tex] [tex]&= 6x^5 - \frac{5}{9}x^4 - \frac{1}{x^2} + \frac{2}{9x^3}[/tex]Second derivative [tex](y'')[/tex] [tex]&= 30x^4 - \frac{20}{9}x^3 + \frac{2}{x^3} - \frac{2}{3x^4} \\[/tex]Find the gradient of the function at the given point. function point f(x, y, z) = x2 + y2 + z2 (3, 9, 8)
Answer:
[tex]\displaystyle \nabla f(3, 9, 8) = 6 \hat{\i} + 18 \hat{\j} + 16 \hat{\text{k}}[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Multivariable Calculus
Differentiation
Partial DerivativesDerivative NotationGradient: [tex]\displaystyle \nabla f(x, y, z) = \frac{\partial f}{\partial x} \hat{\i} + \frac{\partial f}{\partial y} \hat{\j} + \frac{\partial f}{\partial z} \hat{\text{k}}[/tex]
Gradient Property [Addition/Subtraction]: [tex]\displaystyle \nabla \big[ f(x) + g(x) \big] = \nabla f(x) + \nabla g(x)[/tex]
Gradient Property [Multiplied Constant]: [tex]\displaystyle \nabla \big[ \alpha f(x) \big] = \alpha \nabla f(x)[/tex]
Step-by-step explanation:
Step 1: Define
Identify.
[tex]\displaystyle f(x, y, z) = x^2 + y^2 + z^2[/tex]
[tex]\displaystyle P(3, 9, 8)[/tex]
Step 2: Find Gradient
[Function] Differentiate [Gradient]: [tex]\displaystyle \nabla f = \frac{\partial}{\partial x} \Big( x^2 + y^2 + z^2 \Big) \hat{\i} + \frac{\partial}{\partial y} \Big( x^2 + y^2 + z^2 \Big) \hat{\j} + \frac{\partial}{\partial z} \Big( x^2 + y^2 + z^2 \Big) \hat{\text{k}}[/tex][Gradient] Rewrite [Gradient Property - Addition/Subtraction]: [tex]\displaystyle \nabla f = \bigg[ \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial x}(y^2) + \frac{\partial}{\partial x}(z^2) \bigg] \hat{\i} + \bigg[ \frac{\partial}{\partial y}(x^2) + \frac{\partial}{\partial y}(y^2) + \frac{\partial}{\partial y}(z^2) \bigg] \hat{\j} + \bigg[ \frac{\partial}{\partial z}(x^2) + \frac{\partial}{\partial z}(y^2) + \frac{\partial}{\partial z}(z^2) \bigg] \hat{\text{k}}[/tex][Gradient] Differentiate [Derivative Rule - Basic Power Rule]: [tex]\displaystyle \nabla f = 2x \hat{\i} + 2y \hat{\j} + 2z \hat{\text{k}}[/tex][Gradient] Substitute in point: [tex]\displaystyle \nabla f(3, 9, 8) = 2(3) \hat{\i} + 2(9) \hat{\j} + 2(8) \hat{\text{k}}[/tex][Gradient] Evaluate: [tex]\displaystyle \nabla f(3, 9, 8) = 6 \hat{\i} + 18 \hat{\j} + 16 \hat{\text{k}}[/tex]∴ the gradient of the function at the given point is <6, 18, 16>.
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Topic: Multivariable Calculus
Unit: Directional Derivatives
Help Algebra Question
Simplyfy (x - 5 / x^3 + 27) + (2 / x^2 - 9)
Simplify 6 - 23 + (-9 + 5) · 2
A. -10
B. -12
C. 6
D. -8
I've been told the answer is A. -10, but I need to know how to get that answer.
Thanks.
at least u tried to help but the answer is -10 bro
How do i know that the variable x has a uniform distribution function?
Two numbers total 53 and have a difference of 25. Find the two numbers.
equivalent ratios number 1 to 50
How do you solve an inequality
Answer:
you solve an inequality by doing the inverse equation on each side of the equation.
Step-by-step explanation:
Kristina invests $5,000 at 15% simple interest for 2 years.How much interest did Kristina earn over the 2 years?
In an x-y plot of an experiment what is usually plotted on the x axis?
a. the independent variable, which is the parameter that was manipulated.
b. th
n=6x35+4 what does n represent in a division problem
John throws a rock straight down with speed 12 m/s from the top of a tower. the rock hits the ground after 2.37 s. what is the height of the tower? (air resistance is negligible)