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Chapter 12: Problem 1

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

Expert verified
Answer: The partial derivatives \(f_x(a, b)\) and \(f_y(a, b)\) represent the rates of change of the function \(f(x, y)\) in the \(x\) and \(y\) directions, respectively, at the point \((a, b, f(a, b))\). They can also be interpreted as slopes of the tangent plane to the surface \(z = f(x, y)\) at the point \((a, b, f(a, b))\) in the direction of the \(x\)- and \(y\)-axis, respectively.

Step by step solution

01

Define the partial derivatives

Partial derivatives are the derivatives of a multivariable function with respect to only one of its variables, keeping the other variables constant. For this exercise, we're focusing on the partial derivatives \(f_x(a, b)\) and \(f_y(a, b)\). The notation \(f_x\) indicates taking the partial derivative of the function \(f(x, y)\) with respect to \(x\), and likewise, the notation \(f_y\) indicates taking the partial derivative of the function with respect to \(y\).
02

Interpret \(f_x(a, b)\) as a rate of change or slope in the x-direction

The partial derivative \(f_x(a, b)\) represents the rate of change of the function \(f(x, y)\) in the \(x\) direction at the point \((a, b, f(a, b))\). More specifically, it tells us how much the function \(f(x, y)\) is changing with respect to \(x\) when \(x\) changes but \(y\) remains constant at \(b\). In terms of slope, \(f_x(a, b)\) can be interpreted as the slope of the tangent plane to the surface \(z = f(x, y)\) at the point \((a, b, f(a, b))\) in the direction of the \(x\)-axis.
03

Interpret \(f_y(a, b)\) as a rate of change or slope in the y-direction

Similarly, the partial derivative \(f_y(a, b)\) represents the rate of change of the function \(f(x, y)\) in the \(y\) direction at the point \((a, b, f(a, b))\). This means that \(f_y(a, b)\) tells us how much the function \(f(x, y)\) is changing with respect to \(y\) when \(y\) changes but \(x\) remains constant at \(a\). In terms of slope, \(f_y(a, b)\) can be interpreted as the slope of the tangent plane to the surface \(z = f(x, y)\) at the point \((a, b, f(a, b))\) in the direction of the \(y\)-axis.

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