Problem 16 - Active Calculus Multivariable

Question

Suppose that a function \(f=f(x, y)\) is differentiable at a point \(\left(x_{0}, y_{0}\right)\). Let \(L=L(x, y)=f\left(x_{0}, y_{0}\right)+m\left(x-x_{0}\right)+n\left(y-y_{0}\right)\) as in the conditions of Definition 10.4.14. Show that \(m=f_{x}\left(x_{0}, y_{0}\right)\) and \(n=f_{y}\left(x_{0}, y_{0}\right) .\) (Hint: Calculate the limits of the relative errors when \(h=0\) and \(k=0 .\) )

Step-by-Step Solution

Answer

In conclusion, we have shown that the coefficients \(m\) and \(n\) in the linear approximation function \(L(x, y)\) of a differentiable function \(f(x, y)\) at a point \((x_0, y_0)\) are equal to the partial derivatives \( f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\), respectively. We compute the relative errors and find their limits as \(h \to 0\) and \(k \to 0\), which are equal to the partial derivatives, confirming the relationship between the linear approximation function and the partial derivatives of the function.

Step 1: Review the given function L(x, y) and Definitions

The given function is: \[L(x, y) = f(x_0, y_0) + m(x - x_0) + n(y - y_0)\] According to the definition of differentiability, the tangent plane approximation at \((x_0, y_0)\) can be given in terms of the partial derivatives of the function. Our task is to prove that the coefficients \(m\) and \(n\) of the linear function given are equal to the partial derivatives \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\), respectively.

Step 2: Find relative errors

To prove the relationship between the coefficients and the partial derivatives, we need to compute the relative errors for the functions. The relative errors are given as: \[\frac{1}{h}(f(x_0 + h, y_0) - f(x_0, y_0) - m h)\] \[\frac{1}{k}(f(x_0, y_0 + k) - f(x_0, y_0) - n k)\]

Step 3: Take limits h -> 0 and k -> 0

Now, we need to evaluate the limits of the relative errors as \(h\) approaches 0 and \(k\) approaches 0. \(\lim_{h \to 0} \frac{1}{h}(f(x_0 + h, y_0) - f(x_0, y_0) - m h) = f_x(x_0, y_0)\) \(\lim_{k \to 0} \frac{1}{k}(f(x_0, y_0 + k) - f(x_0, y_0) - n k) = f_y(x_0, y_0)\) We have applied the basic definitions of partial derivatives to find the limits.

Step 4: Compare limits with coefficients m and n

Based on the results in the previous step, we can compare the limits with the coefficients \(m\) and \(n\) in the linear function \(L(x, y)\). From the first limit: \[\lim_{h \to 0} \frac{1}{h}(f(x_0 + h, y_0) - f(x_0, y_0) - m h) = f_x(x_0, y_0)\] Since \(m\) is the coefficient in front of the term \((x - x_0)\), we can conclude that: \[m = f_x(x_0, y_0)\] Similarly, from the second limit: \[\lim_{k \to 0} \frac{1}{k}(f(x_0, y_0 + k) - f(x_0, y_0) - n k) = f_y(x_0, y_0)\] Since \(n\) is the coefficient in front of the term \((y - y_0)\), we can conclude that: \[n = f_y(x_0, y_0)\]

Step 5: Conclusion

In conclusion, we have shown that the coefficients \(m\) and \(n\) in the linear approximation function \(L(x, y)\) of a differentiable function \(f(x, y)\) at a point \((x_0, y_0)\) are equal to the partial derivatives \( f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\) respectively. This confirms the relationship between the linear approximation function and the partial derivatives of the function.

Key Concepts

Differentiable Functions

Differentiable functions are a key concept in calculus, particularly when discussing multivariable functions like those of the form \(f(x, y)\). Such a function is considered differentiable at a certain point \((x_0, y_0)\) if it can be well-approximated by a linear function at that point.
This means that around \((x_0, y_0)\), the function's behavior is similar to a plane rather than a curved surface. In other words, a small change in \(x\) or \(y\) will result in a predictable, "linear-like" change in the value of \(f\).
This is crucial for analyzing complex functions as it simplifies computations by letting us use straight-line approximations within small ranges. The ability to approximate the function by a plane is closely tied to the concepts of partial derivatives, which measure how the function changes as \(x\) or \(y\) change independently.

Tangent Plane

In multivariable calculus, the tangent plane is an extension of the tangent line concept to functions of two variables. If you've ever visualized a 3D surface, imagine a flat plane that just "grazes" the surface we're studying at a particular point, \((x_0, y_0)\).
This plane doesn't contain the entirety of the surface but provides sufficient information about the surface's inclination near that point.
The tangent plane equation takes the form:

• \(L(x, y) = f(x_0, y_0) + m(x - x_0) + n(y - y_0)\)

Here, \(m\) and \(n\) represent the slope of the tangent plane in the \(x\) and \(y\) directions, respectively. These slopes correspond to the partial derivatives \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\), which show how steeply the function changes with respect to each variable.

Linear Approximation

Linear approximation is a handy method to estimate the value of complex functions by using simpler linear functions. For a differentiable function, the best linear approximation near a point \((x_0, y_0)\) is its tangent plane. This approximation hinges on finding the partial derivatives precisely at that point.
In essence, linear approximation uses the tangent plane formula discussed earlier, where the partial derivatives determine the slope and hence the direction of the plane.
This is particularly useful:

• In physics and engineering problems where exact solutions might be cumbersome.
• In financial modeling where predictions based on constant rates of change are needed.

While it is incredibly effective for small ranges around \((x_0, y_0)\), the approximation's accuracy can wane as we move further from that point.

Limits in Calculus

Limits are foundational to calculus, bridging the gap between algebraic functions and their derivatives. When we examine something approaching a particular value, the limit lets us talk about "approaching zero" in concrete terms.
In this exercise, limits help verify the relationship between coefficients of the tangent plane and partial derivatives. When we take the limit
\(\lim_{h \to 0}\) for the expression \(\frac{1}{h}(f(x_0 + h, y_0) - f(x_0, y_0) - m h)\), it simplifies to the partial derivative \(f_x(x_0, y_0)\).
This shows that as changes become infinitesimally small, the prediction from the tangent becomes exact. Similarly, \(\lim_{k \to 0}\) for the expression \(\frac{1}{k}(f(x_0, y_0 + k) - f(x_0, y_0) - n k)\) confirms that \(n = f_y(x_0, y_0)\). Through limits, we understand how locally the function resembles its tangent plane and underpins the validity of linear approximation at \((x_0, y_0)\).