szklanovqq

Answered

2022-11-13

Calculating Rate of Change

At the point (0,1,2) in which direction does the function $f(x,y,z)=x{y}^{2}z$ increase most rapidly? What is the rate of change of $f$ in this direction? At the point (1,1,0), what is the derivative of $f$ in the direction of the vector $2\hat{i}+3\hat{j}+6\hat{k}$?

I assumed that the rate of change is the same as the gradient of the function, namely $\u25bdf$. Calculating this gave me:

$\u25bdf=\frac{\mathrm{\partial}(x{y}^{2}z)}{\mathrm{\partial}x}\hat{i}+\frac{\mathrm{\partial}(x{y}^{2}z)}{\mathrm{\partial}y}\hat{j}+\frac{\mathrm{\partial}(x{y}^{2}z)}{\mathrm{\partial}z}\hat{k}$

$\text{}\text{}\text{}\text{}\text{}\text{}={y}^{2}z\text{}\hat{i}+2xz\text{}\hat{j}+x{y}^{2}\text{}\hat{k}$

Evaluating at point:

$\u25bdf(0,1,2)=2\text{}\hat{i}$

Hence, the function increases most rapidly in the x direction.

I am uncertain of how to approach solving the third part of the question, should I evaluate the rate of change at (1,1,0) and then find the difference between that and the vector $2\hat{i}+3\hat{j}+6\hat{k}$?

At the point (0,1,2) in which direction does the function $f(x,y,z)=x{y}^{2}z$ increase most rapidly? What is the rate of change of $f$ in this direction? At the point (1,1,0), what is the derivative of $f$ in the direction of the vector $2\hat{i}+3\hat{j}+6\hat{k}$?

I assumed that the rate of change is the same as the gradient of the function, namely $\u25bdf$. Calculating this gave me:

$\u25bdf=\frac{\mathrm{\partial}(x{y}^{2}z)}{\mathrm{\partial}x}\hat{i}+\frac{\mathrm{\partial}(x{y}^{2}z)}{\mathrm{\partial}y}\hat{j}+\frac{\mathrm{\partial}(x{y}^{2}z)}{\mathrm{\partial}z}\hat{k}$

$\text{}\text{}\text{}\text{}\text{}\text{}={y}^{2}z\text{}\hat{i}+2xz\text{}\hat{j}+x{y}^{2}\text{}\hat{k}$

Evaluating at point:

$\u25bdf(0,1,2)=2\text{}\hat{i}$

Hence, the function increases most rapidly in the x direction.

I am uncertain of how to approach solving the third part of the question, should I evaluate the rate of change at (1,1,0) and then find the difference between that and the vector $2\hat{i}+3\hat{j}+6\hat{k}$?

Answer & Explanation

Miah Carlson

Expert

2022-11-14Added 17 answers

Hint: The directional derivative of f, in the direction of vector $\overrightarrow{u}$, is just:

$\mathrm{\nabla}f\cdot \overrightarrow{u},\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\mathrm{\nabla}f\cdot \frac{\overrightarrow{u}}{\Vert \overrightarrow{u}\Vert}$

(there are different conventions, according to context).

$\mathrm{\nabla}f\cdot \overrightarrow{u},\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\mathrm{\nabla}f\cdot \frac{\overrightarrow{u}}{\Vert \overrightarrow{u}\Vert}$

(there are different conventions, according to context).

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