Question

f(x,y)=3x2−4y2 . Find the directional derivative at the point (1,−1) in the direction of the vector u=(3,4)

247

likes
1236 views

Answer to a math question f(x,y)=3x2−4y2 . Find the directional derivative at the point (1,−1) in the direction of the vector u=(3,4)

Expert avatar
Miles
4.9
111 Answers
**

1. **Normalización de \mathbf{u}**:
El vector \mathbf{u} = (3,4) se normaliza de la siguiente forma:
\|\mathbf{u}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
Entonces, el vector unitario es:
\mathbf{\hat{u}} = \left( \frac{3}{5}, \frac{4}{5} \right)

2. **Cálculo del gradiente \nabla f**:
- Derivada parcial respecto a x:
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(3x^2 - 4y^2) = 6x
- Derivada parcial respecto a y:
\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(3x^2 - 4y^2) = -8y
Entonces,
\nabla f(x, y) = (6x, -8y)

3. **Evaluación del gradiente en el punto (1, -1)**:
\nabla f(1, -1) = (6 \cdot 1, -8 \cdot (-1)) = (6, 8)

4. **Cálculo de la derivada direccional**:
La derivada direccional en la dirección de \mathbf{\hat{u}} es el producto escalar:
D_{\mathbf{\hat{u}}}f(1, -1) = \nabla f(1, -1) \cdot \mathbf{\hat{u}} = (6, 8) \cdot \left( \frac{3}{5}, \frac{4}{5} \right)
= 6 \cdot \frac{3}{5} + 8 \cdot \frac{4}{5}
= \frac{18}{5} + \frac{32}{5}
= \frac{50}{5} = 10

La derivada direccional es 10.

Frequently asked questions (FAQs)
Math question: What is the dot product of vectors A = [3, -2] and B = [5, 4]?
+
Math question: What is the probability of rolling a fair 6-sided die and getting a 3, given that the die has already rolled a prime number?
+
How many congruence rules are there for triangles?
+
New questions in Mathematics
A particular employee arrives at work sometime between 8:00 a.m. and 8:40 a.m. Based on past experience the company has determined that the employee is equally likely to arrive at any time between 8:00 a.m. and 8:40 a.m. Find the probability that the employee will arrive between 8:05 a.m. and 8:30 a.m. Round your answer to four decimal places, if necessary.
Since one of the three integers whose product is (-60) is (+4), write the values that two integers can take.
Use the digits of 1,9,2,3 to come up with all the numbers 98 and 95
Reparameterize the curve r(t)= cos(t)i without (t)j (t)k by the arc length.
How many anagrams of the word STROMEC there that do not contain STROM, MOST, MOC or CEST as a subword? By subword is meant anything that is created by omitting some letters - for example, the word EMROSCT contains both MOC and MOST as subwords.
Suppose the Golf ball market is perfectly competitive and the functions are known: Q = 120 – 2Px – 2Py 0.2I Q = 2Px 40 Where I = Consumers' income ($200) and Py = Price of Good Y (40) Calculate the equilibrium elasticity: a) 1.6 b) -6 c) 6 d) 0.6
A study reports the following final notation: F (3, 32) = 9.50, p < .05. How many total participants were involved in this study? Group of answer choices 34 32 36
Calculate the minimum size of a simple random sample assuming a sampling error of 5% assuming that the population size is 100 elements
Engineers want to design seats in commercial aircraft so that they are wide enough to fit ​95% of all males.​ (Accommodating 100% of males would require very wide seats that would be much too​ expensive.) Men have hip breadths that are normally distributed with a mean of 14.4 in. and a standard deviation of 1.2 in. Find P95. That​ is, find the hip breadth for men that separates the smallest ​95% from the largest 5​%.
The probability of growing a seedling from a seed is 0.62. How many seeds do I need to plant so that the probability of growing at least one seedling is greater than or equal to 0.87?
Take the limit of (sin(x-4))/(tan(x^2 - 16) as x approaches 4.
Translate to an equation and solve. Let x be the unknown number: What number is 52% of 81.
Find sup { x∈R, x²+3<4x }. Justify the answer
Log0
P 13. Let P a point inside of a square ABCD. Show that the perpendicular lines drawn from A, B, C, respectively D, to BP, CP, DP, respectively AP are concurrent. Use geometric rotation.
A group of 17 people spent 9 days on vacation and spent R$776.34 on barbecue meat and the bill needs to be divided as follows: 6 people stayed for 9 days, 7 people stayed for 4 days, and 2 people stayed for 5 days and 2 people stayed 3 days, how much does each group have to pay for the days they stayed?
4m - 3t + 7 = 16
Emile organizes a community dance to raise funds. In addition to paying $300 to rent the room, she must rent chairs at $2 each. The quantity of chairs rented will be equal to the number of tickets sold. She sells tickets for $7 each. How much should she sell to raise money?
The following incoming payments show up at a tax inspection: 25 000€ on 19.01.2008, 140 000€ on 27.03.2008 and 19 000€ on a date that which is illegible, and 60 000€ on 15.06.2008. On which date did the payment of the 19 000€ appear, if on 30.06.2008 the money on the account (incl. interest at 4%) is 246 088.89€? Use simple interest and 30E/360 DCC. Solution: 45 days, 15.05.08
Sin(5pi/3)