Problem 6 - APEX Calculus

Question

A function \(z=f(x, y),\) a vector \(\vec{v}\) and a point \(P\) are given. Give the parametric equations of the following directional tangent lines to \(f\) at \(P\) : (a) \(\ell_{x}(t)\) (b) \(\ell_{y}(t)\) (c) \(\ell_{\vec{u}}(t),\) where \(\vec{u}\) is the unit vector in the direction of \(\vec{v}\). $$ f(x, y)=3 \cos x \sin y, \vec{v}=\langle 1,2\rangle, P=(\pi / 3, \pi / 6) $$

Step-by-Step Solution

Answer

(a) \( \ell_x(t) = \left( \frac{\pi}{3} + t, \frac{\pi}{6}, \frac{3\sqrt{3}}{4} - \frac{3\sqrt{3}}{4} t \right) \), (b) \( \ell_y(t) = \left( \frac{\pi}{3}, \frac{\pi}{6} + t, \frac{3\sqrt{3}}{4} + \frac{9}{4} t \right) \), (c) \( \ell_{\vec{u}}(t) = \left( \frac{\pi}{3} + \frac{t}{\sqrt{5}}, \frac{\pi}{6} + \frac{2t}{\sqrt{5}}, \frac{3\sqrt{3}}{4} + t \times \text{expression} \right) \).

Step 1: Find the Partial Derivatives

We first find the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) of the given function \( z = f(x, y) = 3 \cos x \sin y \). The partial derivative with respect to \( x \) is:\[ \frac{\partial z}{\partial x} = -3 \sin x \sin y \]The partial derivative with respect to \( y \) is:\[ \frac{\partial z}{\partial y} = 3 \cos x \cos y \]

Step 2: Evaluate the Partial Derivatives at Point P

Now we evaluate the partial derivatives at the point \( P = \left( \frac{\pi}{3}, \frac{\pi}{6} \right) \).\[ \frac{\partial z}{\partial x} \bigg|_P = -3 \sin\left(\frac{\pi}{3}\right) \sin\left(\frac{\pi}{6}\right) = -\frac{3 \sqrt{3}}{4} \]\[ \frac{\partial z}{\partial y} \bigg|_P = 3 \cos\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{6}\right) = \frac{9}{4} \]

Step 3: Equation of Directional Tangent Line \( \ell_x(t) \)

The parametric equation for \( \ell_x(t) \) is given by:\[ \ell_x(t) = \left( x_0 + t, y_0, z_0 + t \cdot \frac{\partial z}{\partial x}\bigg|_P \right) \]Substituting the values, we have:\[ \ell_x(t) = \left( \frac{\pi}{3} + t, \frac{\pi}{6}, \frac{3\sqrt{3}}{4} - \frac{3\sqrt{3}}{4} t \right) \]

Step 4: Equation of Directional Tangent Line \( \ell_y(t) \)

The parametric equation for \( \ell_y(t) \) is:\[ \ell_y(t) = \left( x_0, y_0 + t, z_0 + t \cdot \frac{\partial z}{\partial y}\bigg|_P \right) \]Substituting the values, we have:\[ \ell_y(t) = \left( \frac{\pi}{3}, \frac{\pi}{6} + t, \frac{3\sqrt{3}}{4} + \frac{9}{4} t \right) \]

Step 5: Find the Unit Vector \( \vec{u} \)

The unit vector in the direction of \( \vec{v} = \langle 1, 2 \rangle \) is found by normalizing \( \vec{v} \):\[ \vec{u} = \frac{\langle 1, 2 \rangle}{\sqrt{1^2 + 2^2}} = \frac{1}{\sqrt{5}} \langle 1, 2 \rangle \]

Step 6: Equation of Directional Tangent Line \( \ell_{\vec{u}}(t) \)

The parametric equation \( \ell_{\vec{u}}(t) \) is:\[ \ell_{\vec{u}}(t) = \left( x_0 + tu_1, y_0 + tu_2, z_0 + t(u_1 \cdot \frac{\partial z}{\partial x} \bigg|_P + u_2 \cdot \frac{\partial z}{\partial y} \bigg|_P) \right) \]Substituting the values, we have:\[ \ell_{\vec{u}}(t) = \left( \frac{\pi}{3} + \frac{t}{\sqrt{5}}, \frac{\pi}{6} + \frac{2t}{\sqrt{5}}, \frac{3\sqrt{3}}{4} + t \left( \frac{1}{\sqrt{5}} \left( -\frac{3\sqrt{3}}{4} \right) + \frac{2}{\sqrt{5}} \left( \frac{9}{4} \right) \right) \right) \]

Key Concepts

Partial Derivatives

Understanding partial derivatives is key to dealing with functions of multiple variables like our given function, \(z = f(x, y) = 3 \cos x \sin y\). Partial derivatives measure how the function changes as one specific variable changes while the others are kept constant. In simpler terms, they're like taking a regular derivative, but just focusing on one direction. Let's break it down:

• With respect to \(x\): The notation \( \frac{\partial z}{\partial x} \) refers to changing \(x\) while keeping \(y\) constant. Here we find it as \( -3 \sin x \sin y \).
• With respect to \(y\): The notation \( \frac{\partial z}{\partial y} \) refers to changing \(y\) while keeping \(x\) constant. This derivative is \( 3 \cos x \cos y \).

Evaluating these at the point \( P = \left( \frac{\pi}{3}, \frac{\pi}{6} \right)\), gives the rate of change of \(z\) at that particular point. By plugging in these values, the partial derivatives provide the steepness of the tangent lines in the direction of \(x\) and \(y\).

Parametric Equations

Parametric equations are utilized to express geometric features like lines or curves where one or more variables are expressed as functions of a parameter, usually \(t\). In the directional derivative context, it lets us neatly describe the path of a tangent line. Here's how it applies:

• For the line \( \ell_x(t) \), we express changes in the \(x\)-direction.
• For \( \ell_y(t) \), we illustrate movements in the \(y\)-direction.
• \( \ell_\vec{u}(t) \) gives the line in a custom direction, described by our unit vector \(\vec{u}\).

This form allows us to substitute known values and abstractly visualize the change along a curve. This is particularly effective in multi-dimensional calculus for describing how functions behave around specific points using tangent constructions.

Unit Vector

A unit vector plays a crucial role when determining directional derivatives in any specified direction. It is derived from a given vector by dividing each component by the vector's magnitude, thus giving it a length of one while maintaining its direction. The process is straightforward:

• Calculate the magnitude of the vector \( \vec{v} = \langle 1, 2 \rangle \) as \( \sqrt{1^2 + 2^2} = \sqrt{5} \).
• Normalize \( \vec{v} \) to obtain \( \vec{u} = \frac{1}{\sqrt{5}} \langle 1, 2 \rangle \).

This normalization process ensures that the direction is preserved while enabling the vector to point along a path with unit scale. Using unit vectors is beneficial in forming directional derivatives, amplifying the effect of change precisely without the influence of the magnitude of the original vector.

Tangent Lines

Tangent lines are straight lines that just "touch" a curve at a given point, without crossing over. They give an excellent linear approximation of the function near that point. In multivariable calculus, tangent lines at a point provide insight into the directional behavior of a surface. For our function \(z = f(x, y) = 3 \cos x \sin y\), tangent lines in specified directions are described with their parametric equations.

• \( \ell_x(t) \): Represents tangent in the \(x\)-direction.
• \( \ell_y(t) \): Portrays tangent in the \(y\)-direction.
• \( \ell_\vec{u}(t) \): Specifies tangent in the direction of unit vector \( \vec{u} \).

By using partial derivatives evaluated at a point, each tangent line equation encapsulates the directional slope, offering a linear perspective of the surface's geometry. This allows a local understanding of function behavior by observing how the function can be linearly approximated around the given point \(P\).