Question

Find the slope of the tangent to the curve of intersection of the surface \(36 z=4 x^{2}+9 y^{2}\) and the plane \(x=3\) at the point \((3,2,2)\)

Short answer

The slope of the tangent is 1.

Step-by-step solution

Understand the Surface Equation

The surface equation is given by \( 36z = 4x^2 + 9y^2 \). This can be rewritten as \( z = \frac{4}{36}x^2 + \frac{9}{36}y^2 \) or \( z = \frac{1}{9}x^2 + \frac{1}{4}y^2 \). This represents an elliptic paraboloid.

Determine the Intersection Curve

Set \( x = 3 \) in the surface equation \( z = \frac{1}{9}x^2 + \frac{1}{4}y^2 \). Substitute \( x = 3 \) to get \( z = \frac{1}{9}(3)^2 + \frac{1}{4}y^2 = \frac{1}{9} \times 9 + \frac{1}{4}y^2 = 1 + \frac{1}{4}y^2 \). Thus, the equation of the curve of intersection is \( z = 1 + \frac{1}{4}y^2 \).

Derive z with Respect to y

We need to find the slope of the tangent, which is \( \frac{dz}{dy} \). Differentiate \( z = 1 + \frac{1}{4}y^2 \) with respect to \( y \). This gives \( \frac{dz}{dy} = \frac{1}{2}y \), where \( \frac{1}{4}y^2 \) differentiates to \( \frac{1}{2}y \).

Evaluate the Derivative at the Given Point

At the point \((3, 2, 2)\), the \( y \)-value is 2. Substitute \( y = 2 \) into \( \frac{dz}{dy} = \frac{1}{2}y \) to get \( \frac{dz}{dy} = \frac{1}{2} \times 2 = 1 \). Thus, the slope of the tangent line at \((3, 2, 2)\) is 1.

Key concepts

Surface Theory

In mathematics, surface theory is the study of surfaces in three-dimensional spaces. Surfaces are two-dimensional objects, meaning every point on a surface can be defined by two parameters. This concept is used in geometry to explore different shapes and structures.
In calculus, we're interested in the relationships and interactions between surfaces, curves, and other geometric entities.
An essential part of surface theory is understanding how these surfaces can be mathematically represented through equations, such as those describing planes, spheres, and paraboloids. The general equation of a surface often takes functions of the form \(z = f(x, y)\), where \(z\) represents the height in a three-dimensional coordinate system.
By understanding the surface equation, we can explore how different surfaces interact or intersect with one another, which helps to solve complex geometry and calculus problems.

Tangent Lines

Tangent lines are straight lines that touch a curve at one specific point without crossing it. They can be thought of as the best linear approximation of the curve at that point.
When dealing with surfaces and curves, tangent lines play a significant role as they give us the slope or the rate of change of the curve at a particular point.
To find the tangent line, we use derivatives, which allow us to calculate the slope. The slope can tell us how steep the tangent line is at the point of tangency.

• If the curve represents a physical path, the slope gives us the inclination of the path at that point.
• When the slope is zero, the tangent line is horizontal, indicating a potential maximum, minimum, or inflection point on the curve.

Understanding how tangent lines interact with surfaces and curves is pivotal in bringing clarity to the behavior of complex geometrical shapes.

Elliptic Paraboloid

An elliptic paraboloid is a type of three-dimensional surface that looks like an elongated bowl. Its defining feature is its cross-sections:

• If you cut across it horizontally, you'll see ellipses.
• If you cut it vertically, you may get parabolas.

The equation \( z = \frac{1}{9}x^2 + \frac{1}{4}y^2 \) is an example of an elliptic paraboloid, where the coefficients dictate the 'width' and 'height' of the paraboloid in the \(x\) and \(y\) direction.
The elliptic paraboloid appears frequently in physics and engineering, often modeling scenarios like projectile trajectories or satellite dishes. Understanding its structure helps us to predict and analyze the path of physical systems that display this shape. Exploring this surface requires integrating knowledge from both calculus and geometry.

Derivative

In calculus, the derivative of a function measures how the function's output value changes as its input changes. It's essentially the "rate of change" or "slope" of the function.
Derivatives are crucial in understanding how functions behave, especially how surfaces and curves rise and fall.
To compute derivatives, we use various rules like the power rule, product rule, or chain rule. For example, when we found \( \frac{dz}{dy} = \frac{1}{2}y \), we used the power rule to differentiate \( \frac{1}{4}y^2 \).
Calculating derivatives at a point, as we did in the exercise, helps determine the slope of the tangent to the curve—connecting the mathematics to tangible concepts like velocity or acceleration in real-world applications.

• Derivatives can be used to find maxima or minima of functions, indicating the highest or lowest points on a curve.
• They also help us to assess concavity, understanding whether the curve opens upwards or downwards.

Mastering derivatives unlocks a deeper understanding of dynamic and changing systems in mathematics.