Gradient and directional derivative

MathProblemsBank banner

MathProblemsBank banner

Math Problems and solutions

Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to find a point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=x^{2}-y^{2}, \quad A(1 ; 1), \quad \vec{a}=\{2 ; 2\} . \]

2.3.16 Gradient and directional derivative

1.28 $

Problem: Function \( z=z(x, y) \), point \( A\left(x_{0} ; y_{0}\right) \) and vector \( \vec{a} \) are given. It is required to finc at point \( A \) : a) the gradient of the function and its value, b) the derivative of \( z \) in the direction of vector \( \vec{a} \). \[ z(x, y)=\ln \left(x^{2}+y^{2}\right), \quad A(3 ; 4), \quad \vec{a}=\{2 ;-1\} . \]

2.3.17 Gradient and directional derivative

1.28 $

Problem: Calculate the derivative of function \( f(x, y)=x^{3}+2 x^{2} y+2 \) at point \( A(2 ; 2) \quad \) in the direction of vector \( \overrightarrow{A B} \), where \( B=(4 ; 2) \).

2.3.18 Gradient and directional derivative

0.77 $

Problem: Calculate the gradient of function \( z=x y \cos x y \) at point \( M(0 ;-3) \).

2.3.19 Gradient and directional derivative

0 $

Problem: Find the gradient of function \[ f(x, y, z)=\frac{x^{2}+2 y}{2 y+z} \text { at point } A=(2 ; 4 ; 2) \text {. } \]

2.3.20 Gradient and directional derivative

0.77 $

Problem: Find the derivative of function \( \varphi \) at point \( A \) in the direction of vector \( \overrightarrow{A B} \) : \[ \varphi=\frac{y^{2}}{x}+\frac{y^{2}}{z}+\frac{z^{2}}{x^{2}}, \quad A(1 ; 0 ; 1), \quad B(2 ; 2 ; 3) . \]

2.3.21 Gradient and directional derivative

0.77 $

Problem: Calculate the derivative of function \( f(x, y)=x^{5}+x \arccos y^{2} \) at point \( A=(3 ; 0) \) in the direction of vector \( \vec{a}=(2 ; 2) \).

2.3.22 Gradient and directional derivative

0.77 $

Problem: Find the value and the direction of the gradient of field \( u=x y \) at point \( A(1 ; 1), B(1 ;-1) \).

2.3.23 Gradient and directional derivative

1.28 $

Problem: Find the gradient of scalar field \[ u(x, y, z)=y \sin z-x \cos y+x z-2 z^{2}+7 \] and determine whether this field is harmonic.

2.3.24 Gradient and directional derivative

1.03 $

Problem: For given scalar fields \( u, v \) find: a) the derivative of \( u \) in the direction of \( \vec{e} \) at point \( M \), b) the angle between gradients \( u \) and \( v \) at points \( N \) and \( M \), where \[ \begin{array}{l} v=6 \sqrt{6} x^{3}-6 \sqrt{6} y^{3}+2 z^{3}, \quad u=\frac{x z^{2}}{y} \\ M(1 ; 1 ; \sqrt{6}), \quad N\left(\frac{1}{\sqrt{6}} ; \frac{1}{\sqrt{6}} ; 1\right), \quad \vec{e}=(2 ; 3 ; 4) . \end{array} \]

2.3.25 Gradient and directional derivative

1.8 $

‹
1
2