Question

I) Find the directional derivative of π(π₯, π¦) = π₯ sin π¦ at (1,0) in the direction of the unit vector that make an angle of π/4 with positive π₯-axis.

190

likes949 views

Bud

4.6

56 Answers

To find the directional derivative of a function, we need to calculate the dot product of the gradient vector of the function and the unit vector in the given direction.

The gradient vector of the function f(x, y) is given by the partial derivatives of f with respect to each variable.

Let's find the partial derivatives of f(x, y):

\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (x \sin y) = \sin y

\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x \sin y) = x \cos y

Now, we can find the gradient vector:

\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right) = \left(\sin y, x \cos y\right)

The unit vector that makes an angle of π/4 with the positive π₯-axis is:

\mathbf{u}=\left(\cos\frac{\pi}{4},\sin\frac{\pi}{4}\right)=\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)

Now, we can calculate the dot product of the gradient vector and the unit vector:

\nabla f\cdot\mathbf{u}=\left(\sin y,x\cos y\right)\cdot\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)=\frac{\sqrt{2}}{2}\sin y+\frac{\sqrt{2}}{2}x\cos y

Substituting the coordinates of the point (1, 0) into the equation above:

\nabla f\cdot\mathbf{u}=\frac{\sqrt{2}}{2}\cdot\sin0+\frac{\sqrt{2}}{2}\cdot1\cdot\cos0=\frac{\sqrt{2}}{2}\cdot0+\frac{\sqrt{2}}{2}\cdot1\cdot1=\frac{\sqrt{2}}{2}

Answer: The directional derivative of f(x, y) = x sin y at (1, 0) in the direction of the unit vector that makes an angle of π/4 with the positive π₯-axis is \frac{\sqrt{2}}{2} .

The gradient vector of the function f(x, y) is given by the partial derivatives of f with respect to each variable.

Let's find the partial derivatives of f(x, y):

Now, we can find the gradient vector:

The unit vector that makes an angle of π/4 with the positive π₯-axis is:

Now, we can calculate the dot product of the gradient vector and the unit vector:

Substituting the coordinates of the point (1, 0) into the equation above:

Answer: The directional derivative of f(x, y) = x sin y at (1, 0) in the direction of the unit vector that makes an angle of π/4 with the positive π₯-axis is

Frequently asked questions (FAQs)

Question: Find the domain and range of the cube root function f(x) = βx.

+

Question: What is the equation of the parabola with its graph opening downwards, vertex at (2, -5), and passing through the point (1, -6)?

+

Question: In a circle with radius 10 cm, if an angle at the center measures 60Β°, what is the length of the arc it intercepts?

+

New questions in Mathematics