if f(x, y) = xy, find the gradient vector ∇f(3, 7) and use it to find the tangent line to the level curve f(x, y) = 21 at the point (3, 7).
Answers
if f(x, y) = xy, find the gradient vector ∇f(3, 7) and use it to find the tangent line to the level curve f(x, y) = 21 at the point (3, 7): This is the equation of the tangent line to the level curve f(x, y) = 21 at the point (3, 7).
To find the gradient vector ∇f(3, 7) for the function f(x, y) = xy, we first need to compute the partial derivatives of f with respect to x and y:
∂f/∂x = y
∂f/∂y = x
Therefore, the gradient vector ∇f is given by:
∇f = (∂f/∂x, ∂f/∂y) = (y, x)
Evaluating ∇f at the point (3, 7), we get:
∇f(3, 7) = (7, 3)
This is the gradient vector at the point (3, 7) on the level curve f(x, y) = 21.
To find the tangent line to the level curve at (3, 7), we can use the fact that the gradient vector ∇f is orthogonal to the level curve at each point. In other words, the tangent line at (3, 7) is perpendicular to the vector (7, 3).
Recall that the equation of a line in two dimensions can be written as:
y - y_0 = m(x - x_0)
where m is the slope of the line and (x_0, y_0) is a point on the line.
To find the slope of the tangent line at (3, 7), we can use the fact that it is perpendicular to the gradient vector ∇f(3, 7). The dot product of two orthogonal vectors is zero, so we have:
(7, 3) · (x - 3, y - 7) = 0
Expanding the dot product and solving for y, we get:
7(x - 3) + 3(y - 7) = 0
y - 7 = (-7/3)(x - 3)
This is the equation of the tangent line to the level curve f(x, y) = 21 at the point (3, 7).
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Related Questions
How do you find the height of a triangle when you know the base and area?
Answers
We can determine the height of a triangle when we know the base and area of the triangle using the area formula of triangle in base and height, that is
A = 1/2 × b × h
A triangle is a three sided polygon. The height of the triangle is the perpendicular distance of the not-included vertex to the considered base.
The formula to determine area of a triangle of base b and height h is
A = 1/2 × b × h
Therefore to determine the height of a triangle when base and area is given, we substitute the given values in appropriate units in the formula.
h = 2×A/ b
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A business study claimed that in any given month, for every 3 cars sold by Ford, 5 cars are sold by Subaru. How many cars will be sold by Ford if Subaru sells 10 cars?
And Can you identify a unit rate? Note: The unit rate can be for either manufacturer.
Also Does the ratio of cars sold by Ford and Subaru indicate a proportionāl relationship? Why or why not?
Answers
Answer:
b
Step-by-step explanation:
I just need a anwser
Answers
Answer:
y=3x
Step-by-step explanation:
Rewrite the function by completing the square. 4x^2-4x+1=0
Answers
Do in completing the square method.
\( {4x}^{2} - 4x + 1 = 0 \\ 4x^{2}-4x=-1 \\ \frac{4x^{2}-4x}{4}=\frac{-1}{4} \\ x^{2}+\frac{-4}{4}x=\frac{-1}{4} \\ x^{2}-x=\frac{-1}{4} \\ x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{1}{4}+\left(-\frac{1}{2}\right)^{2} \\ x^{2}-x+\frac{1}{4}=\frac{-1+1}{4} \\ x^{2}-x+\frac{1}{4}=0 \\ \left(x-\frac{1}{2}\right)^{2}=0 \\ \sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{0} \)
Now simplify.
\(x-\frac{1}{2}=0 \\ x-\frac{1}{2}=0 \\ \\ \rightarrow \: x=\frac{1}{2} \\ \rightarrow \: x=\frac{1}{2} \)
So, answer is..
\( \boxed{x=\frac{1}{2} }\)
Now rewrite the equation.
\(4 x ^ { 2 } - 4 x + 1 = 0 \\ \boxed{4\times \left(\frac{1}{4}\right)-4\times \left(\frac{1}{2}\right)+1=0 }\)
Verification :-
\(4 x ^ { 2 } - 4 x + 1 = 0 \\ 4\times \left(\frac{1}{4}\right)-4\times \left(\frac{1}{2}\right)+1=0 \\ 1-4\times \left(\frac{1}{2}\right)+1=0 \\ 1-\frac{4}{2}+1=0 \\ 1-2+1=0 \\ -1+1=0 \\ \underline{ 0 = 0} \: \\ \dashrightarrow \text{true}\)
What is the balance after 15 years in a savings account that earns 2% interest compounded bimonthly when the initial deposit is 1000?
Answers
The amount will be equal to $1348.07. The correct option is A.
What is compound interest?Compound interest is the interest levied on the interest. The formula for the calculation of compound interest is given as:-
\(\rm A = P(1+\dfrac{r}{n})^{nt}\)
Compound Interest Formula:
A = Account Balance
P = Principle/Initial Amount
r = Rate of Interest (decimal)
n = Number of times compounded (per year)
t = Number of Years
Given Information
Account Balance = ?
It is given that the balance after 15 years in a savings account earns 2% interest compounded bi-monthly when the initial deposit is 1000.
This is because we are gaining money, so the multiplier should be greater than 1. We already added 1, which is 100% so you simply add the 0.02 for the extra 2%.
Number of times compounded per year = 6
This is because it is being compounded bi-monthly, or once every 2 months. 12 months divided by 2 months is 6 months, so 6 times a year.
Number of years = 15
Solve by plugging the given values into the formula.
\(\rm A = P(1+\dfrac{r}{n})^{nt}\)
\(\rm A = 1000(1+\dfrac{0.02}{6})^{6 \times 15}\)
A = 1000 x (1.14)
A = $1348.07.
Therefore, the amount will be equal to $1348.07. The correct option is A.
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find a vector function, r(t), that represents the curve of intersection of the two cylinder x2 y2
Answers
To find a vector function that represents the curve of intersection of the two cylinders, we can consider the parametric equations for a circle in the xy-plane.
The equation of a circle with radius r centered at the origin is given by:
x = r * cos(t)
y = r * sin(t)
Since both cylinders have the same radius, we can use these equations to represent the curve of intersection. However, we also need to consider the z-coordinate. As the equation provided is incomplete, I assume it was cut off. Please provide the complete equation or additional information to proceed and find the curve of intersection.
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Evaluate 6 + x when x=3.
Answers
Answer: that equals 9
Step-by-step explanation:
Help me please!!!! I need help fast
Answers
Answer:
D
Step-by-step explanation:
120/360 x 2(pie)(10)
A boat is heading towards a lighthouse, where Yaritza is watching from a vertical distance of 139 feet above the water. Yaritza measures an angle of depression to the boat at point AA to be 21^{\circ}
∘
. At some later time, Yaritza takes another measurement and finds the angle of depression to the boat (now at point BB) to be 50^{\circ}
∘
. Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.
Answers
The distance from an angle of depression to the boat at 21° to angle of depression of 50° is 112 feet
How to determine the required distanceInformation from the question
A boat is heading towards a lighthouse, where Yaritza is watching from a vertical distance of 139 feet
Yaritza measures an angle of depression to the boat at point AA to be 21°
Yaritza takes another measurement and finds the angle of depression to the boat (now at point BB) to be 50°
the distance from point AA to point BB = ?
The direction of movements describes a right triangle of
opposite = ?
adjacent = 139 feet
The distance is calculated using tan, TOA let the angle be x
tan x = Opposite / Adjacent
for x = 21 point AA is solved
tan 21 = ? / 139
? = 139 * tan 21
? = 53.357
for x = 50 point BB is solved
? = 139 * tan 50
? = 165.654
the distance from point AA to point BB
= 165.654 - 53.357
= 112.297
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The function y = 14.75 +1.25(x - 2) can
be used to determine the cost in
dollars to play x games at a carnival. L
What is the rate of change of the cost
in dollars with respect to the number of
games played?
$2.50 per game
$1.25 per game
$14.75 per
game
$2.00 per
game
Answers
Answer:
$1.25 per game
Step-by-step explanation:
Annabeth works at a flower shop. Today, she spent 4(1)/(4) hours arranging flowers and 3(3)/(8) hours repotting flowers. How long did Annabeth work at the flower shop today?
Answers
Answer:
7 \(\frac{5}{8}\)
Step-by-step explanation:
4 \(\frac{1}{4}\) + 3\(\frac{3}{8}\) we need a common denominator and that would be 8
4 \(\frac{1}{4}\) = 4\(\frac{2}{8}\) (\(\frac{1}{4}\) x \(\frac{2}{2}\) makes the equivalent fraction \(\frac{2}{8}\))
4\(\frac{2}{8}\) + 3\(\frac{3}{8}\) = 7\(\frac{5}{8}\)
aldo has scored 77, 88, 77, 87, and 63 on his previous five tests. what score does he need on his next test so that his average (mean) is 79?
Answers
To maintain an average of 79, Aldo must obtain a score of 82 in his upcoming exam.
To get the solution, let's calculate the total of the first five scores.
Total of the first five scores = 77 + 88 + 77 + 87 + 63 = 392
Now we know that there are a total of six scores, so to get the mean, we will divide the total by six.
Mean = Total/Number of ScoresTherefore, 79 = 392 + x/6
We need to find the value of x that will satisfy the above equation.
Now we will solve for x.79 = 392 + x/6
(Multiply both sides by 6) 6 * 79 = 6 * 392 + x6 * 79 = 2352 + xx = 6 * 79 - 392x = 474 - 392x = 82
Therefore, Aldo needs to score 82 on his next test to achieve an average of 79.
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Elizabeth gave 10% of the cost as a tip to the delivery person who delivered her car after service at a cost of $297. What will be the total cost with tip to the nearest cent?
Answers
Answer:
What did it mean when the Dutch “kowtowed” to the Chinese emperor or his representatives?
Step-by-step explanation:
Answer:
326.7
Step-by-step explanation:
you can divide 297 by 10 this gives you 10% then add to the original amount
What is 33 1/3% of $2,940
Answers
Answer:
$980
Step-by-step explanation:
33 1/3% x
------------ = ------------- Cross multiply
100% $2,940
33 1/3 x 2,940 = 100x Multiply
98000 = 100x Divide 100 to both sides
980 = x
Using the remainder theorem, which quotient has a remainder of 20?
OA (3r4 – 573 + 5x + 2) = (x - 2)
O B. (3r3 + 702 + 5x + 2) = (x - 2)
Oc. (13 + 5x2 4r + 6) = (x + 5)
OD. (74 – 812 + 16) = (x + 4)
Answers
Answer:
The answer is A. (3x^4-5x^3+5x+2) =(x-2)
Step-by-step explanation:
we see that the polynomial that gives the remainder as 20 is
(3\(x^{4}\) - 5x³ + 5x + 2) ÷ (x - 2)
Option A is our correct answer.
What is the remainder theorem?The remainder theorem states that when a polynomial p(x) is divided by
(x - a), then the remainder = f(a).
We have,
(3\(x^{4}\) - 5x³ + 5x + 2) ÷ (x - 2) ____(1)
We can directly find the remainder as:
x - 2 = 0
x = 2
Putting x = 2 in (1) we get,
= 3 x 16 - 5 x 8 + 5 x 2 + 2
= 48 - 40 + 10 + 2
= 20
The remainder is 20.
(3x³ + 7x² + 5x + 2) ÷ (x - 2) ) _______(2)
x - 2 = 0
x = 2
Putting x = 2 in (2) we get,
= 3 x 8 + 7 x 4 + 5 x 2 + 2
= 24 + 28 + 10 + 2
= 64
The remainder is 64.
(x³ + 5x² - 4x + 6) ÷ (x + 5) ______(3)
x + 5 = 0
x = -5
Putting x = -5 in (3) we get,
= - 125 + 5 x 25 - 4 x (-5) + 6
= -125 + 125 + 20 + 6
= 20 + 6
= 26
The remainder is 26.
(\(x^{4}\) - 8x² + 16) ÷ (x + 4) _______(4)
x + 4 = 0
x = -4
Putting x = -4 in (4) we get,
= 256 - 8 x 16 + 16
= 256 - 128 + 16
= 144
The remainder is 144.
Thus we see that the polynomial that gives the remainder as 20 is
(3\(x^{4}\) - 5x³ + 5x + 2) ÷ (x - 2)
Option A is our correct answer.
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A two digit number is such that the one's digit is four more than the tens digit and the sum of the digits is 14. find the number??
Answers
Answer:
Wyzant
SOLVE IT
Chi Y. asked • 07/27/18
The sum of the digits of a 2-digit number is 14. If it’s digit are reversed, the original number is increased by 18. What is the number?
There is no format. I just the answer ASAP.
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2 Answers By Expert Tutors
By:
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Andy C. answered • 07/27/18
TUTOR 4.9 (27)
Math/Physics Tutor
SEE TUTORS LIKE THIS
Here is the algebraic solution:
T is the tens digit
O is the ones digit
T+O = 14
So O = 14 - T <---- this is the first equation
The original number is 10*T + O
If the digits are reversed the original number is increased by 18.
The number if the digits are reversed is:
10*O + T = 10*T + O + 18
10*(14-T) + T = 10*T + (14-T) + 18 <--- substitutes O with 14-T per first equation above in bold
140 - 10T + T = 10T + 14 - T + 18 <--- distributive left side
140 - 9T = 9T + 32
140 = 9T + 9T + 32
140 - 32 = 9T + 9T
108 = 18T
108/18 = T
T= 6
THe tens digit is 6.
The ones digit must be 8.
Reversing 86 - 68 = 18
The original number is 68.
Consider an annual rate is 9% with quarterly compounding: R
4
=9%. Calculate the Equivalent annual rates with 4.1 monthly compounding, R
12
4.2 continuous compounding, r
E
Answers
The equivalent annual rate with monthly compounding is approximately 9.322%, and the equivalent annual rate with continuous compounding is approximately 9.252%. The exact equivalent annual rate (rE) cannot be determined based on the given information.
To calculate the equivalent annual rate with different compounding periods, we can use the formulas for compound interest.
a. Monthly Compounding (R12): The formula for monthly compounding is given by:
R12 =\((1 + r/n)^{n - 1}\)
Where r is the annual rate and n is the number of compounding periods per year. In this case, the annual rate (R4) is 9% and the compounding periods (n) is 4 (quarterly compounding).
R12 = \((1 + 0.09/12)^{12 }- 1\)
≈ 9.322%
Therefore, the equivalent annual rate with monthly compounding is approximately 9.322%.
b. Continuous Compounding (R4.2): The formula for continuous compounding is given by:
R4.2 = \(e^{(r*t)} - 1\)
Where r is the annual rate and t is the time period. In this case, the annual rate (R4) is 9% and the time period (t) is 1 year.
R4.2 = \(e^{(0.09*1)} - 1\)
≈ 9.252%
Therefore, the equivalent annual rate with continuous compounding is approximately 9.252%.
c. Exact Equivalent Annual Rate (rE): The exact equivalent annual rate cannot be determined based on the given information. To calculate the exact equivalent annual rate, we would need to know the compounding periods per year for the monthly compounding and the specific interest rate for the continuous compounding. Without these additional details, the exact equivalent annual rate (rE) cannot be determined.
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Please help and show work only do the left side
Answers
Writing linear equations given two points can be a useful skill when graphing linear equations.
Write a linear equation that passes through the given two points?To write a linear equation given two points, you need to first find the slope of the line. You can do this by finding the change in the y-value and dividing it by the change in the x-value. From there, you can use the slope to solve for the y-intercept and write the equation.For example, if the two points are (3, 4) and (0,5), you would find the slope by calculating the change in the y-value (5-4 = 1) and dividing it by the change in the x-value (0-3 = -3).The slope would then be 1/-3, which can be simplified to -1/3. To solve for the y-intercept, you can plug in one of the points and solve for b. In this case, you would plug in (3, 4) and solve for b, giving you b = 5. Now that you have the slope and y-intercept, you can write the equation as y = -1/3x + 5.y = -1/2xy = 5/3x + 5/3y = 3/5x + 1y = -1/2x - 2y = 6/5x + 5y = -4/4x - 8y = -3/5x - 7/5y = -2x - 4y = 5/6x + 7y = -1/2x + 4To learn more about linear equation refer to:
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A recipe needs cup of butter for every 2 cups of flour. if you plan to use 3 cups of flour, how much butter will you need?
Answers
You will need amount of 1.5 cups of butter if you plan to use 3 cups of flour in your recipe.
To calculate how much butter is needed for 3 cups of flour in a recipe, start by identifying that the recipe requires a ratio of 1 cup of butter for every 2 cups of flour. To get the amount of butter needed for 3 cups of flour, multiply 1 cup of butter by 1.5. This will give you 1.5 cups of butter, which is the amount needed for 3 cups of flour. Therefore, you will need 1.5 cups of butter if you plan to use 3 cups of flour in your recipe. To ensure accuracy, it is recommended to measure the ingredients carefully and double check your calculations before proceeding.
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Gina Wilson All Things Algebra Unit 4: Congruent Triangles Homework 7: Proofs Review: all methods
page 1
Answers
To identify which triangles are congruent measure the angles and sides of each of the triangles and compare them.
What is the meaning of congruent?In mathematics, the word congruent is used if two figures have the same size and shape, although their orientation can be different.
How to know if two triangles are congruent?In the case of triangles two factors should be considered:
Length of each side.Angles.This means that to determine if the triangles are congruent, you must measure each side and angle and make sure these are the same in each triangle.
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A car travels 165 miles in 3 hours. If the car travels at the same speed, how far will the car go in 8 hours?
Answers
Answer:
440
Step-by-step explanation:
Solve by setting up the equation
165miles/3hours = x/8hours
Make sure miles or hours are on the top or bottom and not cross cross. Since you're looking for miles after driving 8 hours.
you multiply diagonally
so 165 times 8 and 3 times X
so you have 1320=3x
then divide by three on each side to get x by itself
then x=440
210 oranges, 252 apples, 294 pears are equally packed in cartons so that no fruit is left. What is the biggest possible number of cartons needed?
Answers
Answer:
42
Step-by-step explanation:
Given that,
210 oranges, 252 apples, 294 pears are equally packed in cartons so that no fruit is left.
The factor of 210 = 2×3×5×7
The factor of 252 = \(2^{2}\times 3^{2}\times 7\)
The factor of 294 = \(2\times 3\times 7^{2}\)
We need to find the biggest possible number of cartons needed. It can be calculated by taking HCF of the above numbers.
The HCF of 210, 252 and 294 = 2×3×7
= 42
Hence, 42 is the biggest possible number of cartons needed.
can anyone help me please? thanks ^^
Answers
Answer:
Yes; they are congruent
SSS
Step-by-step explanation:
Here, we want to check if the given triangles are congruent
From what we have, we can see that all three sides are marked for both triangles
So the sides of the triangles are income
So therefore, the triangles are congruent by SSS
On a coordinate plane, triangle X prime Y prime Z prime has points (negative 4, 5), (negative 6, negative 1), and (negative 2, negative 1).
Triangle XYZ was dilated using the rule DO,0.25 (x, y) → (0.25x, 0.25y).
The image is shown in the diagram. What are the coordinates of Z of the pre-image?
(–8, –4)
(–8, 2)
(–2, –1)
(–0.25, –0.5)
Answers
Answer:
(–8, –4)
Step-by-step explanation:
Transformation is the movement of a point from its initial location to a new location. Types of transformation are rotation, reflection, translation and dilation.
Dilation is the reduction or enlargement of the size of an object by a factor.
Given that triangle XYZ was dilated using the rule (x, y) ⇒ (0.25x, 0.25y) to produce triangle X'Y'Z' with points at X'(-4, 5), Y'(-6, -1) and Z'(-2, -1)
Since (x, y) ⇒ (0.25x, 0.25y)
(x, y) ⇒ 0.25(x, y)
4(x, y) ⇒ (x, y)
We can find the points of XYZ by multiplying by 4
hence:
X = 4(-4, 5) = (-16, 20)
Y = 4(-6, -1) = (-24, -4)
Z = 4(-2, -1) = (-8, -4)
Answer: a (-8,-4)
Step-by-step explanation:
Layla read 1 book in 4 months. What was her rate of reading, in books per month? Give your answer as a whole number or a FRACTION in simplest form.
Answers
Answer:
1/4 books per month.
Step-by-step explanation:
1 book in 4 months means you have to do 1 divided by 4 which equals .25. After converting to a fraction you get 1/4.
How many solutions does the equation 12(2x+6)=6−x have?
Answers
Answer:x=0
Step-by-step explanation:
I need an answer for this
Answers
Answer:
Step-by-step explanation:
18+12+9
Answer:
756 cm^2
Step-by-step explanation:
9 x 12 = 108 triangles
9 x 18 = 162 back side
9^2 + 12^2 = c^2
9^2 + 12^2 = 225
9^2 + 12^2 = 15^2
length of the hypotenuse is 15 so
15 x 18 = 270 slanted top
18 x 12 = 216 bottom
216 + 270 + 162 + 108 = 756 cm^2
Aaron buys erasers for his pencils. Each eraser costs $0.20. The total cost is $1.20. How many erasers does Aaron buy? Solve this problem any way you choose.
Answers
Two of the most common graphical charting techniques are ____.A) vertical charts and horizontal charts
B) bar charts and pie charts
C) line charts and series charts
D) printed charts and screen charts
Answers
The two most common graphical charting techniques are bar charts and line charts. (Option B)
The two most common graphical charting techniques are bar charts and line charts.
Bar charts are used to represent categorical data by displaying rectangular bars of different heights, where the length of each bar represents a specific category and the height represents the corresponding value or frequency.
Line charts, on the other hand, are used to represent the trend or relationship between data points over a continuous period or interval. Line charts connect data points with straight lines to show the progression or change in values over time or other continuous variables.
Option B) bar charts and line charts accurately represent the two most common graphical charting techniques used in data visualization. Vertical and horizontal charts (option A) are not specific chart types but rather describe the orientation of the axes. Series charts (option C) and printed charts and screen charts (option D) are not commonly used terms in the context of charting techniques.
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You deposit $20,000 in an account that pays 7.77% annual interest. Find the balance after 2 years when the interest is compounded monthly.
Answers
Answer:
The account balance after 24 months will be $23,092.70
Step-by-step explanation:
Given data
P= $20,000
r= 7.77%= 0.077
t= 2 years
A= ?
n= 24
The compound interest formula is
\(A= P(1+t)^t\)
Inserting our values to solve for A we have
\(A= 20000(1+\frac{0.077}{24} )^2^*^2^4\\\\A= 20000(1+0.003 )^2^*^2^4\\\\A= 20000(1.003 )^2^*^2^4\\\A= 20000(1.003 )^4^8\\\A= 20000*1.15463517818\\\A= 23092.7035636\\\A= 23,092.70\)