The directional derivative of [tex]f[/tex] at the given point in the direction indicated is [tex]\frac{5}{2}[/tex].
How to calculate the directional derivative of a multivariate function
The directional derivative is represented by the following formula:
[tex]\nabla_{\vec v} f = \nabla f (r_{o}, s_{o})\cdot \vec v[/tex] (1)
Where:
[tex]\nabla f (r_{o}, s_{o})[/tex] - Gradient evaluated at the point [tex](r_{o}, s_{o})[/tex].[tex]\vec v[/tex] - Directional vector.The gradient of [tex]f[/tex] is calculated below:
[tex]\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{\partial f}{\partial r}(r_{o},s_{o}) \\\frac{\partial f}{\partial s}(r_{o},s_{o}) \end{array}\right][/tex] (2)
Where [tex]\frac{\partial f}{\partial r}[/tex] and [tex]\frac{\partial f}{\partial s}[/tex] are the partial derivatives with respect to [tex]r[/tex] and [tex]s[/tex], respectively.
If we know that [tex](r_{o}, s_{o}) = (1, 3)[/tex], then the gradient is:
[tex]\nabla f(r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{s}{1+r^{2}\cdot s^{2}} \\\frac{r}{1+r^{2}\cdot s^{2}}\end{array}\right][/tex]
[tex]\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{1+1^{2}\cdot 3^{2}} \\\frac{1}{1+1^{2}\cdot 3^{2}} \end{array}\right][/tex]
[tex]\nabla f (r_{o}, s_{o}) = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right][/tex]
If we know that [tex]\vec v = 5\,\hat{i} + 10\,\hat{j}[/tex], then the directional derivative is:
[tex]\nabla_{\vec v} f = \left[\begin{array}{cc}\frac{3}{10} \\\frac{1}{10} \end{array}\right] \cdot \left[\begin{array}{cc}5\\10\end{array}\right][/tex]
[tex]\nabla _{\vec v} f (r_{o}, s_{o}) = \frac{5}{2}[/tex]
The directional derivative of [tex]f[/tex] at the given point in the direction indicated is [tex]\frac{5}{2}[/tex]. [tex]\blacksquare[/tex]
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What is the value of -5^6
Answer:
-15625
Step-by-step explanation:
Hope this helped !! :)
Have a good day.
Find the value of x: 4/5 (x+7) =20
(4/5 is a fraction not a division)
Answer: 18
Step-by-step explanation:
11) Ms. Ripa's family is planning to come visiting her from New York. She hopes to take
them to some of the National Parks in Utah while they are here, but she needs to save up to
buy their tickets. Adult tickets are $12 each, and child tickets are $7 each. If she has 5 adults
and 4 children coming to visit, how much money will she need to save up to buy all of their
tickets?
Orlando has a loan with an effective interest rate of 7.918%, compounded annually. which of the following must be true? i. in the effective rate formula, n is equal to one. ii. the nominal rate is 7.918%. iii. the federal funds rate is static. a. i and ii b. ii only c. iii only d. i, ii, and iii
The right answer will be both I and II
I. In the effective rate formula, n is equal to one.
II.The nominal rate is 7.918%.
What is compound interest?Compound interest is the interest on a loan or deposit that accrues on both the initial principal and the accumulated interest from previous periods.
The formula for the compound interest is given as:
[tex]A=p(1+\dfrac{r}{n})^{nt}[/tex]
Here
A= amonut
P= principle
r=rate of interest
t=time period
n=for how much time compunded
Thus we can see in the question it is compounded annually so n = 1 and the rate is given as 7.918%
Thus the right answer will be both I and II
I. In the effective rate formula, n is equal to one.
II.The nominal rate is 7.918%.
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Convert 14x + 6y = 36 to Y intercept form.
Answer:
0,6
Step-by-step explanation:
Answer: Y= -7/3x + 6
y=2x+1 and y=3x.
At what x-values are the equations equal?
x = 1 and x = 3
x = 0 and x = 1
x = 1 and x = -1
x = 2 and x = 3
Answer:
x = 1 and x = 3
Step-by-step explanation:
Let's solve your system by substitution.
y=2x+1;y=3x
Step: Solvey=2x+1for y:
y=2x+1
Step: Substitute2x+1foryiny=3x:
y=3x
2x+1=3x
2x+1+−3x=3x+−3x(Add -3x to both sides)
−x+1=0
−x+1+−1=0+−1(Add -1 to both sides)
−x=−1
−x
−1
=
−1
−1
(Divide both sides by -1)
x=1
Step: Substitute1forxiny=2x+1:
y=2x+1
y=(2)(1)+1
y=3(Simplify both sides of the equation)
Answer:
x=1 and y=3
Solve the following inequality using the algebraic approach )pictire provided)
Answer:
x <4
Step-by-step explanation:
hope it helps
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HELPP 5 POINTS Here's a graph of a linear function. Write the equation that describes that function.
Express it in slope-intercept form.
Step-by-step explanation:
m = (y2-y1)/(x2-x1)
m = (4-0)/(0-(-8))
m = 4/8
y = mx + c
4 = (4/8)(0) + c
c = 4
Thus, the equation is y = (4/8)x + 4
5
A church has 3 bells. One rings after every 3 minutes, the other after
every
4
minutes and the third one after every 5 minutes. If they all ring together at
8:00 a.m., at what time will they ring together again?
You
Answer:
9:22
Step-by-step explanation:
Find the LCM Least common multiple of all three numbers (minutes)
3,4,5 LCM is 60
Divide each one into the LCM
60/3 = 20
60/4 = 15
60/5 = 12
Totals =82
add that to the time 8:00
82 is an Hour and 22 minute so it would be 9:22
I looked at other problems and this is how they solved it. I hope it helps
4/7 divided by 2 1/3 = ???
Answer:
[tex]\frac{12}{49}[/tex]
explanation:
Use the keep change flip method. This consists of literally flipping a fraction to make the denominators alike, allowing for faster solves and multiplication.
[tex]2\frac{1}{3}[/tex] is [tex]\frac{7}{3}[/tex], so flip 7/3 so it is 3/7, matching up with 4/7.
Next, multiply.
4 x 3
7 x 7
4 x 3 is 12. Write 12 as a numerator.
7 x 7 is 49. Write 49 as a denominator.
Therefore, the answer is [tex]\frac{12}{49}[/tex]
Simplifying? We cannot simplify this number due to a lack of factors.
WHOEVER ANSWERS FIRST WILL BE MARKED BRAINEST
Answer:
[tex]5(n + 4)^{\frac{3}{2} }[/tex]
I hope this helps!
A truck whose bed is 8.5 feet long, 5 feet wide, and 3.5 feet high is delivering sand for a sand sculpture competition. How many trips must the truck make to deliver 400 cubic feet of sand?
Answer:
The truck will need to make 3 trips.
Step-by-step explanation:
First, we need to find the volume that the truck can carry. Using the dimensions of the truck and the volume formula, we have:
[tex]V=lwh\\V=8.5\times5\times3.5\\V=148.75\text{ ft}^3[/tex]
Since we need to move a total of 400 cubic feet, we will divide that amount by the truckload capacity (volume).
[tex]400 \div 148.75 \approx 2.689[/tex]
In context, he will need to take three truck loads: two loads will be full, and the third will not be full.
A 160-inch board is cut into two pieces. One piece is four times the length of the other. Find the length of the shorter piece.
PLEASE HELP
Let the shorter piece be x. The longer piece is 4 times the length of the other, so it is 4x.
4x + x = 160.
5x = 160
x = 32
4x = 128
The length of the shorter piece is x, which is 32.
32 inches1750 litres of petrol is required to fill the tanks of 50 cars. If 1960 litres of petrol is available, how many more cars can be filled?
1750 litres.............50 cars
1960 litres.............n cars
so n x 1750 = 1960 x 50
n = 98000 : 1750
n = 56
56 cars tanks will be filled
Answer:
56
Step-by-step explanation:
1750 divided by 50 to find out how much litres each car gets. =35
so 35 litres per car so divide 1960 by 35. =56
56 cars can be filled
CHECK ANSWER.
the six extra cars so 35 multiplied by 6. =210
1750+210=1960!
Draw a graph of y = 8x. From the graph, find the value of y when x = 1 ...... I need it fast..pls I will mark brainliest.
Answer:
Solving steps:
find the value of y when x = 1
=> y = 8(1)
multiply 8 by 1
=> y = 8
Hey there!
ORIGINAL EQUATION: y = 8x
This one should be quite easy to answer because we already have the value of x. (x = 1) all we have to do is find the y-value and where it is on the number line.
NEW EQUATION: y = 8(1)
SIMPLIFY IT!
y = 8
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
Susie ran 2.35 kilometers. What is the value of the 3 in 2.35?
A. 3 tens
B. 3 tenths
C. 3 hundredths
D. 3 thousands
Answer:
tenths
Step-by-step explanation:
the 2 is in the ones place and the 5 is in the hundreds meaning the 3 would be in the tenths
Need help finding the missing side of each triangle?
How do I put things in standard notation? Please explain :(
Answer:
[tex]in standard notation \: 9 { \times 10}^{ - 4} = 0.0009[/tex]
prove that 1-tan^2x/1+tan^2x=2cos^2x-1
Answer:
see explanation
Step-by-step explanation:
Using the trigonometric identities
tan x = [tex]\frac{sinx}{cosx}[/tex] , sin²x + cos²x = 1
Consider the left side
[tex]\frac{1-tan^2x}{1+tan^2x}[/tex]
= [tex]\frac{1-\frac{sin^2x}{cos^2x} }{1+\frac{sin^2x}{cos^2x} }[/tex] ← multiply numerator/ denominator of the whole fraction by cos²x
= [tex]\frac{cos^2x-sin^2x}{cos^2x+sin^2x}[/tex]
= [tex]\frac{cos^2x-sin^2x}{1}[/tex]
= cos²x - sin²x
= cos²x - (1 - cos²x)
= cos²x - 1 + cos²x
= 2cos²x - 1
= right side , thus proven
[tex]\frac{1 - \tan^2(x)}{1 + \tan^2(x)}\\= \frac{\frac{\cos^2(x)}{\cos^2(x)} - \frac{\sin^2(x)}{\cos^2(x)}}{\frac{\cos^2(x)}{\cos^2(x)} + \frac{\sin^2(x)}{\cos^2(x)}}\\= \frac{\frac{\cos^2(x) - \sin^2(x)}{\cos^2(x)}}{\frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)}}\\= \frac{\cos^2(x) - \sin^2(x)}{\cos^2(x)} \cdot \frac{\cos^2(x)}{\cos^2(x) + \sin^2(x)}\\= \frac{(\cos^2(x) - \sin^2(x)) \cdot \cos^2(x)}{\cos^2(x) \cdot (\cos^2(x) + \sin^2(x))}\\= \frac{\cos^2(x) - \sin^2(x)}{\cos^2(x) + \sin^2(x)}[/tex]
[tex]= \cos^2(x) - \sin^2(x)\\= \cos^2(x) - (1 - \cos^2(x))\\= \cos^2(x) - 1 + \cos^2(x)\\= 2\cos^2(x) - 1[/tex]
Ava and her three friends went out for lunch. Their total bill was $96. If each person paid the same amount, how much would each person pay?
$______
Answer:
$24
Step-by-step explanation:
Ava + her 3 friends = 4 people total.
The bill was $96
To split this evenly among 4 people we can divide the bill by 4.
96/4 = 24
Therefore each person would pay $24.
Hope this helps! :)
Suppose you decide to run at least 7.5 miles per week during the week to train for the cross-country season in the fall. During one week, you run 2 miles on Sunday and 2.5 miles on Wednesday. How many more miles, m, do you need to run before the week is over? *
Given :
You decide to run at least 7.5 miles per week during the week to train for the cross-country season in the fall.
During one week, you run 2 miles on Sunday and 2.5 miles on Wednesday.
To Find :
How many more miles, m, do you need to run before the week is over.
Solution :
Total number of miles we travelled till Wednesday :
t = miles run on Sunday + miles run on Wednesday
t = 2 + 2.5 miles = 4.5 miles.
Now, number of miles remaining :
R = Total target - t
R = 7.5 - 4.5 miles
R = 3 miles.
Therefore, we need to run 3 more miles before the week is over.
Hence, this is the required solution.
Please Answer Number 7
Answer: 32ft²
Step-by-step explanation:
The area of a trapezoid is given by the formula: A = [(b1+b2)/2] x h, where b1 and b2 are the two bases, and h is the height.
In our situation, b1 = 3, and b2 = 5. The height is given as 2.
Plugging these numbers into our formula:
A = (b1+b2)/2 x h
A = [(3+5)/2] x 2
A = (8/2) x 2
A = 4 x 2
A = 8ft²
8ft² is the area of one trapezoid. If Mr. Smith wants to fit 4 of these shapes into the room, the combined area will be 4 x 8 = 32ft².
Have a nice day :)
In a particular right triangle, the two legs have lengths of 40 inches and 42 inches. What is the area of the triangle?
Answer:
i don't know I don't have ideas about it
34. Is this polygon it is convex or concave.
Is 8397 divisible by 9 ? Explain your answer
Answer:
Yes
Step-by-step explanation:
9 can go into 8397 evenly without any remainders.
Answer:
8397/9= 933
8397 is divisible by 9 if you divide 8397 by 9 and the quotient is an integer. The easiest way to determine if 8397 is divisible by 9 is to calculate the sum of the digits in 8397. If the sum of the digits is divisible by 9, then 8397 is divisible by 9.
8+3+9+7= 27
27 is divisible by 9
Step-by-step explanation:
Determine the unit rate of a marathon runner who travels 9/2 miles in 3/4 hour.
Answer:
The marathon runner runs 1 mile every 10 minutes.
Step-by-step explanation:
9/2 miles is equal to 4.5 miles. 3/4 hours is equal to 45 minutes.
How many triangles can be constructed with side lengths of 1.8 cm, 3.4 cm, and 5.6 cm?
0
1
more than one
Answer: One triangle
Step-by-step explanation:
-The triangle's dimensions are defined and fixed.
-Given the fixed dimensions, only one unique triangle can be constructed.
-Hence, only one triangle of the dimensions 7.2cm, 6.9cm and 12.8 cm can be formed.
Answer:
1
Step-by-step explanation:
The correct shape would be a scalene triangle, which can't be bent and twisted and still have the same side lengths.
A computer store is having a sale. A laptop originally cost $700 and is now being sold at a fifth off of the original price. What is the sale price of the laptop?
Answer:
$560
Step-by-step explanation:
1/5th the original price just means 20%
20% of 700 is 140
700*.20= 140
Since it's 20% off that means subtraction
700-140=560
Hope this helps!
The measure of angle Q is 70 degrees. Find the measure of angle P.
Answer:40
Step-by-step explanation: because 70 times 2 = 140 and you add 40 will equal 180 because every triangle equals 180
The measure of angle m<P is 40 degrees
The given triangle is an isosceles triangle where:
m<R = m<Q
Taking the sum of the interior angles, we will have:
m<P + ,m<Q + m<R = 180
70 + 70 + m<P = 180
m<P = 180 - 140
m<P= 40 degrees
Hence the measure of angle m<P is 40 degrees
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A bird flew from a point on the ground directly to the edge of the roof of a building. The
height of the building is 70 feet, and the angle of elevation the bird's flight path made with the
ground is 30°. What is the total distance the bird flew in feet?
The total distance the bird flew in feet is 140 feet
Angle of elevation and depressionGiven the following parameters
The angle of elevation = 30 degrees
Height = 70 feet
Requiredtotal distance the bird flew in feet (D)
Using the SOH CAH TOA identity
sin 30 = 70/D
D = 70/sin30
D = 140eet
Hence the total distance the bird flew in feet is 140 feet
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