Problem 11 - Calculus Early Transcendentals

Question

If \(f(x, y)=16-4 x^{2}-y^{2},\) find \(f_{x}(1,2)\) and \(f_{y}(1,2)\) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.

Step-by-Step Solution

Answer

The slopes are \(-8\) in the \(x\)-direction and \(-4\) in the \(y\)-direction at \((1, 2)\).

Step 1: Understand the Objective

We need to find the partial derivatives \(f_x(1,2)\) and \(f_y(1,2)\) of the function \(f(x, y) = 16 - 4x^2 - y^2\). These partial derivatives represent the slopes of the tangent lines to the surface \(z = f(x, y)\) at the point \((1, 2)\) along the \(x\)-axis and \(y\)-axis, respectively.

Step 2: Compute the Partial Derivative with respect to x

To compute \(f_x(x, y)\), we differentiate \(f(x, y)\) partially with respect to \(x\), treating \(y\) as a constant. The derivative \(f_x(x, y)\) is given by \(f_x(x, y) = \frac{\partial}{\partial x}(16 - 4x^2 - y^2) = -8x\).

Step 3: Evaluate f_x at (1, 2)

Substitute \(x = 1\) and \(y = 2\) into \(f_x(x, y) = -8x\). This yields \(f_x(1, 2) = -8 \times 1 = -8\). Thus, the slope of the tangent line in the \(x\)-direction at \((1, 2)\) is \(-8\).

Step 4: Compute the Partial Derivative with respect to y

To compute \(f_y(x, y)\), differentiate \(f(x, y)\) partially with respect to \(y\), treating \(x\) as constant. The derivative \(f_y(x, y)\) is given by \(f_y(x, y) = \frac{\partial}{\partial y}(16 - 4x^2 - y^2) = -2y\).

Step 5: Evaluate f_y at (1, 2)

Substitute \(x = 1\) and \(y = 2\) into \(f_y(x, y) = -2y\). This gives \(f_y(1, 2) = -2 \times 2 = -4\). Thus, the slope of the tangent line in the \(y\)-direction at \((1, 2)\) is \(-4\).

Step 6: Interpretation and Illustration

The partial derivative \(f_x(1,2) = -8\) indicates that for a small increase in \(x\) while keeping \(y\) constant, the value of \(f(x,y)\) decreases at a rate of 8 units per unit increase in \(x\). Similarly, \(f_y(1,2) = -4\) indicates that for a small increase in \(y\) while keeping \(x\) constant, \(f(x, y)\) decreases at a rate of 4 units per unit increase in \(y\). These slopes can be visualized on a 3D surface plot, showing steeper decline in the \(x\) direction compared to the \(y\) direction at the point \((1, 2)\).

Key Concepts

Tangent Plane

In multivariable calculus, the tangent plane plays a similar role to the tangent line in single-variable calculus. Just like a tangent line touches a curve at a point and approximates the curve's slope at that point, a tangent plane touches a surface at a point and approximates the surface's slopes in all directions locally around that point.

For a function of two variables, say \(f(x, y)\), the tangent plane at a point \((a, b)\) can be expressed by the equation: \[ z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) \] where \(f_x(a, b)\) and \(f_y(a, b)\) are the partial derivatives of the function \(f\) at \((a, b)\).

These partial derivatives represent how the function \(f(x, y)\) changes as you move in the \(x\) and \(y\) directions from \((a, b)\). Thus, they are key components for defining the tangent plane, giving us both the direction and rate of change.

• It provides a linear approximation for the surface \(z = f(x, y)\) close to \((a, b)\).
• The tangent plane is crucial for visualizing the behavior of multivariable functions graphically.

Slope Interpretation

In the context of multivariable calculus, understanding slope involves interpreting the rate at which a surface changes when we alter one of the input variables while keeping the other constant.

For the function \(f(x, y) = 16 - 4x^2 - y^2\), the partial derivatives \(f_x(1,2)\) and \(f_y(1,2)\) give us the slopes of tangent lines to the surface along the \(x\)-axis and \(y\)-axis, respectively, at the point \((1,2)\).

The value \(f_x(1,2) = -8\) suggests a slope of -8 along the \(x\)-axis. This means that a small increase in \(x\) results in the function value decreasing by approximately 8 units for each unit increase in \(x\).

Similarly, \(f_y(1,2) = -4\) indicates a slope of -4 along the \(y\)-axis. Thus, increasing \(y\) by a small amount leads to the function value decreasing by approximately 4 units for each unit increase in \(y\).

• These slopes help us picture how the surface \(z = f(x, y)\) behaves at a specific point.
• Higher absolute values of slopes denote steeper surfaces in that direction.

Multivariable Calculus

The fascinating world of multivariable calculus extends the concepts of calculus from single-variable to functions with more than one variable, adding depth to our analytical toolkit.

At its core, multivariable calculus helps us explore and understand how quantities change when influenced by multiple factors. For functions like \(f(x, y)\), this branch of math allows us to:

• Define and evaluate how the function behaves locally by using partial derivatives.
• Visualize the function's surface with 3D graphs, observing slopes and contours.

Partial derivatives \(f_x\) and \(f_y\) serve as the building blocks for such analysis, breaking down complex multi-dimensional changes into understandable segments, each addressing one independent variable's influence.

Whether you’re analyzing geographic models, optimizing functions in economics, or controlling systems in physics, multivariable calculus equips you with the tools needed to dissect and interpret the intricate interactions between multiple changing quantities.