Question

Suppose you are standing on the surface \(z=f(x, y)\) at the point \((a, b, f(a, b)) .\) Interpret the meaning of \(f_{x}(a, b)\) and \(f_{y}(a, b)\) in terms of slopes or rates of change.

Short answer

Answer: The partial derivatives \(f_x(a,b)\) and \(f_y(a,b)\) represent the slopes of the tangent to the surface \(z=f(x,y)\) in the \(x\) and \(y\) directions, respectively, at the point \((a, b, f(a, b))\). They indicate how fast the function is changing with respect to \(x\) and \(y\) at this point. A positive value of \(f_x(a,b)\) means the function is increasing in the \(x\)-direction, while a negative value means it is decreasing. Similarly, a positive value of \(f_y(a,b)\) means the function is increasing in the \(y\)-direction, while a negative value means it is decreasing.

Step-by-step solution

Definition of partial derivatives

Recall the definitions of partial derivatives of a function \(z=f(x, y)\) with respect to \(x\) and \(y\). The partial derivative with respect to \(x\) is defined as the rate of change of \(f\) with respect to \(x\) while keeping \(y\) constant, and similarly for \(y\). Mathematically, these are defined as: $$ f_x(a, b) = \lim_{h\to 0} \frac{f(a+h, b) - f(a, b)}{h}, $$ and $$ f_y(a, b) = \lim_{k\to 0} \frac{f(a, b+k) - f(a, b)}{k}. $$

Interpretation of \(f_x(a, b)\)

The partial derivative \(f_x(a, b)\) tells us how fast the function \(z=f(x, y)\) is changing with respect to \(x\) at the point \((a, b, f(a, b))\) while keeping \(y\) constant. In other words, if we move along the \(x\)-axis direction starting from the point \((a, b, f(a, b))\), the slope of the tangent to the surface \(z = f(x, y)\) in the \(x\)-direction will be given by \(f_x(a, b)\). A positive value of \(f_x(a, b)\) indicates that the function is increasing in the \(x\)-direction, while a negative value indicates that it is decreasing in the \(x\)-direction.

Interpretation of \(f_y(a, b)\)

Similarly, the partial derivative \(f_y(a, b)\) tells us how fast the function \(z=f(x, y)\) is changing with respect to \(y\) at the point \((a, b, f(a, b))\) while keeping \(x\) constant. If we move along the \(y\)-axis direction starting from the point \((a, b, f(a, b))\), the slope of the tangent to the surface \(z = f(x, y)\) in the \(y\)-direction will be given by \(f_y(a, b)\). A positive value of \(f_y(a, b)\) indicates that the function is increasing in the \(y\)-direction, while a negative value indicates that it is decreasing in the \(y\)-direction. In summary, \(f_x(a, b)\) and \(f_y(a, b)\) give the slopes of the tangent to the surface \(z=f(x, y)\) in the \(x\) and \(y\) directions, respectively, at the point \((a, b, f(a, b))\). These values indicate how fast the function is changing with respect to \(x\) and \(y\) at this point.

Key concepts

Rate of Change

The rate of change is a key concept when dealing with functions of multiple variables. In our case, the function is dependent on two variables, \(x\) and \(y\). The rate of change describes how \(z = f(x, y)\) varies as one or both variables are modified.

For instance, in partial derivatives, the rate of change is measured separately for each variable. \(f_x(a, b)\), the partial derivative with respect to \(x\), shows how \(z\) changes as \(x\) changes, keeping \(y\) fixed. Similarly, \(f_y(a, b)\) shows the change when \(y\) alters, while \(x\) remains constant.

Think of it as hiking on a hill. If you turn left or right (change \(x\)), you face a certain slope, and if you go forward or backward (change \(y\)), there might be a different slope. Both directions have specific rates of change which are indicated by the slopes.

Slopes

Slopes in the context of a surface like \(z = f(x, y)\) offer insight into how steep a function is in particular directions. Imagine standing on a hill – the steepness felt while moving directly east or west might differ from the steepness in a north-south direction.

The partial derivatives \(f_x(a, b)\) and \(f_y(a, b)\) represent these "directional slopes."

• \(f_x(a, b)\) captures the slope along the \(x\)-axis, signals how \(z\) feels as it shifts left or right.
• \(f_y(a, b)\) captures the slope along the \(y\)-axis, showing how \(z\) changes moving forward or back.

In summary, the slopes given by the partial derivatives tell you about the steepness of the surface at a specific point which determines how you ascend or descend a curved plane.

Function of Two Variables

A function of two variables, often written as \(z = f(x, y)\), represents a surface in three-dimensional space. Think of it as a landscape made by altering \(x\) and \(y\), with \(z\) showing the height at each point.

When analyzing such functions, one can study how the value of \(z\) changes when \(x\) or \(y\) is tweaked. This leads to the idea of partial derivatives, which are tools to examine how surfaces behave when there's a change in just one of the variables (while keeping the other constant).

Imagine adjusting \(x\) while holding \(y\) steady – you track the change in \(z\) specifically because of \(x\). The analysis is mirrored if you hold \(x\) constant and vary \(y\).

• For complex models, understanding these changes helps in optimization and forecasting outcomes.

Overall, a function of two variables forms the groundwork for modeling and plotting surfaces on graphs.

Tangent Plane

A tangent plane is akin to a flat surface touching a curved surface at a singular point. This idea extends the concept of a tangent line to three-dimensional surfaces.

Imagine a point on a hill: the tangent plane here would be the flat surface that perfectly matches the hill's slope at that point. This abstraction speaks to how the surface might look very close up.

When the partial derivatives \(f_x(a, b)\) and \(f_y(a, b)\) are calculated, they provide the slopes in the \(x\)- and \(y\)-directions at the point \(a, b\). These slopes help construct the equation of the tangent plane.

• Having a tangent plane offers a local linear approximation of the surface.
• This approximation is crucial for solving problems that rely on linear assumptions.

Thus, the tangent plane gives us insights into how a surface behaves around a point, simplifying complex surfaces into linear forms for practical analysis.