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>.
---
Learn more about multivariable calculus: https://brainly.com/question/17433118
---
Topic: Multivariable Calculus
Unit: Directional Derivatives
Help Algebra Question
What is the answer to this question ?
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).
Learn more about unit conversion here:
https://brainly.com/question/19420601
#SPJ2
I need help with this question... phenobarbital elixir contains 2 grams in 1 liter of elixir. How many milligrams of phenobarbital are contained in 1 teaspoon of elixir?
Answer:
9.86 milligrams
Step-by-step explanation:
It is given that 1 liter f elixir contains 2 grams of phenobarbital elixir.
Since 202.884 teaspoons = 1 liter
so 1 teaspoon of elixir = [tex]\frac{1}{202.884}[/tex] liter
∵ 1 liter of elixir contains phenobarbital = 2 gms.
∴ [tex]\frac{1}{202.884}[/tex] liter of elixir contains phenobarbital = [tex]\frac{2*1}{202.884}[/tex] = 0.009858 grams
or 0.009858 × 10³ milligrams
= 9.86 milligrams.
Therefore, 9.86 milligrams of phenobarbital will be required.
Final answer:
The milligrams of phenobarbital in one teaspoon of elixir, multiply the milligrams per milliliter (2 mg/mL) by the volume of one teaspoon (5 mL), resulting in 10 milligrams per teaspoon.
Explanation:
The milligrams of phenobarbital in one teaspoon of elixir by first understanding the concentration of the solution and then using the volume conversion. Since 1 liter of the phenobarbital elixir contains 2 grams (2000 mg), we need to find out how many milligrams are contained in a smaller volume, specifically in one teaspoon, which is 5 milliliters. milligrams, so 2 grams = 2000 milligrams.
Next, consider the volume conversion: There are 1000 milliliters in 1 liter. So if 2000 mg is in 1000 milliliters, there would be 2 mg in 1 milliliter (since 2000 ÷ 1000 = 2).
Finally, since 1 teaspoon equals 5 milliliters, we multiply the amount of milligrams in 1 milliliter by 5.
Mathematically: 2 mg/mL * 5 mL = 10 mg
Therefore, there are 10 milligrams of phenobarbital in 1 teaspoon of elixir.
What is the area of a triangle with verticies at (-2,1), (2,1) and (3,4)
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:
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
how many solutions are in 6x+4x-6=24+9x
Write twelve and sixteen thousand as a mixed decimal
n=6x35+4 what does n represent in a division problem
What is the area of 30 X 40 rectangle in square units
The area of the given rectangle is 1200 square units.
Given that, width = 30 units and length = 40 units.
What is the area of a rectangle?The formula of the area of a rectangle is used to find the area occupied by the rectangle within its boundary. The area of a rectangle is obtained by multiplying its length and width. Thus, the formula for the area, 'A' of a rectangle whose length and width are 'l' and 'w' respectively is the product "l × w".
Now, the area of a rectangle = 40 × 30
= 1200 square units
Therefore, the area of the given rectangle is 1200 square units.
To learn more about the area of a rectangle visit:
https://brainly.com/question/12019874.
#SPJ5
What are 5 numbers that have 3,5, and 7 as prime factors
he bought 8 pizzas to be shared equally among 40 people. what is the equivalent representation
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.
two pounds of peaches cost 4.50 5 pounds cost 11.25and 10 pounds cost 22.50 what is the constant of proportionality
4.50 /2 = 2.25
11.25/5 = 2.25
22.50/10 = 2.25
the constant is the price per pound which is 2.25
Answer:
c
Step-by-step explanation:
equivalent ratios number 1 to 50
At a speed of 45 yards per minute, a 120 pound swimmer burns 420 calories per hour and a 172 pound swimmer burns 600 calories per hour. write an equation that models the number of calories burned per hour as a function of the swimmer's weight.
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.
Learn more about Vector Representation here:https://brainly.com/question/37843560
#SPJ2
What is the meaning of the function of h = f(50, t)? describe the behavior of this function?
Simplyfy (x - 5 / x^3 + 27) + (2 / x^2 - 9)
√10, π, 3.5
order least to greatest
When the expression -5x^2+20x-17 is written in the form $a(x+d)^2+e$, where a, d, and e are constants, then what is the sum a+d+e?
PLEEEASE HELP
then difference between the cubes of two numbers is twice the value of the perfect square of a possitive Intiger
Kristina invests $5,000 at 15% simple interest for 2 years.How much interest did Kristina earn over the 2 years?
What number is 1/10 greater than 7962
Calculate the mean of the number set 5 10 12 4 6 11 13 5
If the test for a disease is accurate 65% of the time, how often will it come back positive if a patient doesn't have the disease?
Answer:
35% of the time
Step-by-step explanation:
A P E X
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