Q. 5 - Calculus

Question

Let $\left(x_0,y_0\right)$ be a point in the domain of the function $f(x, y)$ at which $f_x\left(x_0,y_0\right)$ and $f_y\left(x_0,y_0\right)$ both exist. Explain how these partial derivatives may be interpreted as the slopes of lines tangent to the surface defined by $f(x, y)$ at the point $\left(x_0,y_0\right)$.

Step-by-Step Solution

Answer

Ans: 

part (a). The partial derivative gives the rate of change of function when only the variable ' x ' changes by an infinitesimal value of ' h '. The value of ' y ' remains constant at $y_0$

part (b). The partial derivative gives the rate of change of function when only the variable ' y ' changes by an infinitesimal value of ' h '. The value of ' x ' remains constant at  $x_0$.

Step 1. Given information:

A function $f(x, y)$ in two variables has a point $\left(x_0,y_0\right)$ in its domain such that both $f_x\left(x_0,y_0\right)$ and $f_y\left(x_0,y_0\right)$ exist.

Step 2. Deriving the partial derivative of function:

The partial derivative of a function $f(x, y)$ in two variables at the point $\left(x_0,y_0\right)$ in its domain is defined by

$f_x\left(x_0,y_0\right)=\underset{h\to 0}{\lim }\frac{f\left(x_0+h,y_0\right)-f\left(x_0,y_0\right)}{h}$

• From the above definition, it is clear that the partial derivative gives the rate of change of function when only the variable ' x ' changes by an infinitesimal value of ' h '. The value of ' y ' remains constant at $y_0$.

Step 3. Finding the slope of the tangent line:

This rate of change can also be said to give the slope of the tangent line.

Combine the two statements to define the partial derivative $f_x\left(x_0,y_0\right)$ as, " the slope of the tangent line to the curve formed by the intersection of graph of $f(x, y)$ and plane $y=y_0$, at the point $\left(x_0,y_0\right)$."

Step 4. Deriving the partial derivative of a function:

The partial derivative of a function $f(x, y)$ in two variables at the point $\left(x_{0}, y_{0}\right)$ in its domain is defined by

$f_y\left(x_0,y_0\right)=\underset{h\to 0}{\lim }\frac{f\left(x_0,y_0+h\right)-f\left(x_0,y_0\right)}{h}$

From the above definition, it is clear that the partial derivative gives the rate of change of function when only the variable ' y ' changes by an infinitesimal value of ' h '. The value of ' x ' remains constant at $x_0$.

Step 5. Finding the slope of the tangent line:

This rate of change can also be said to give the slope of the tangent line.

Combine the two statements to define the partial derivative $f_y\left(x_0,y_0\right)$ as, " the slope of the tangent line to the curve formed by the intersection of graph of $f(x, y)$ and plane $x=x_0$, at the point $\left(x_0,y_0\right)$."