Problem 79 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

The curve \(y^{2}(2-x)=x^{3}\) is called a cissoid. a. Use a graphing utility to sketch the curve. b. Find an equation for the tangent line to the curve at all points where \(x=1\). c. What happens to the curve as \(x\) approaches 2 from the left? d. Does the curve have a tangent line at the origin? If so, what is its equation?

Step-by-Step Solution

Answer

a. Graph using a utility.b. Tangent lines at x=1: y=2x-1 and y=-2x+1.c. As x approaches 2 from the left, y^2 approaches -∞.d. No tangent line at the origin.

Step 1: Sketch the curve

Using a graphing utility, plot the given curve defined by the equation \(y^{2}(2-x)=x^{3}\). This will help visualize the behavior of the curve.

Step 2: Find the tangent line at x=1

Differentiate the equation implicitly with respect to \(x\). First, rewrite it as \(y^2(2-x) - x^3 = 0\). Apply implicit differentiation to get an expression for \(dy/dx\). Then, substitute \(x=1\) to find the slope of the tangent line at that point. Finally, use the point-slope form to find the equation of the tangent line.

Step 3: Differentiate implicitly

Differentiate both sides of \(y^2(2-x) = x^3\) with respect to \(x\). We get: \(2y\frac{dy}{dx}(2-x) + y^2(-1) = 3x^2\). Solving for \(\frac{dy}{dx}\), we obtain: \[ \frac{dy}{dx} = \frac{3x^2 + y^2}{2y(2-x)} \]

Step 4: Substitute x=1 to find slope

Substitute \(x=1\) into the original equation to solve for \(y\): \(y^2(2-1) = 1^3\), yielding \(y^2 = 1\) so \(y = \pm 1\). Then, substitute \(x=1\) and \(y=\pm 1\) into the derivative to find the slope: \(\frac{dy}{dx} = \frac{3(1)^2 + (\pm 1)^2}{2(\pm 1)(2-1)} = \pm 2\). Thus, the slopes are 2 and -2.

Step 5: Find equations for the tangent lines

Using the point-slope form \(y-y_1=m(x-x_1)\), plug in the points and slopes: For \( (1,1) \) with slope 2: \( y-1=2(x-1) \Rightarrow y=2x-1 \). For \( (1,-1) \) with slope -2: \( y+1=-2(x-1) \Rightarrow y=-2x+1 \).

Step 6: Behavior as x approaches 2 from the left

Examine the original equation \(y^2(2-x)=x^3\) as \(x\rightarrow 2^{-}\). As \( x \) gets closer to 2, \( 2-x \) approaches 0, causing \(y^2 \) to grow very large (as \(x^3eq 0 \)). Thus, \( y^2 \rightarrow \, - \infty\).

Step 7: Tangent line at the origin

Examine the curve at the origin \((0,0)\). Differentiate the original equation using implicit differentiation: No real tangent exists at the origin since the derivative does not exist.

Key Concepts

Tangent Line Equations

In calculus, a tangent line to a curve at a given point is a straight line that just 'touches' the curve at that point. To find the equation of a tangent line, we need two things: the slope of the tangent line at the point and the coordinates of the point itself. For the given curve, we differentiated implicitly to find \(\frac{dy}{dx} = \frac{3x^2 + y^2}{2y(2-x)} \). Plugging in the point where x=1, we calculated the slopes as 2 and -2. Using the point-slope form equation, we found two tangent lines: \(y=2x-1\) and \(y=-2x+1 \). These lines describe how the curve behaves and slopes at those specific points.

Behavior of Functions Near Limits

Understanding how curves behave near certain values of x helps us grasp the bigger picture. For our cissoid curve, we examined its behavior as x approaches 2 from the left. In our cissoid equation \(y^2(2-x)=x^3\), as x gets closer to 2, the term \(2-x\) gets smaller and smaller. This causes \(y^2\) to become very large since x\textsuperscript{3} is still non-zero. Practically, this means y grows towards infinity as x nears 2, better illustrating how the curve 'stretches' vertically at x=2.

Graphing Implicit Functions

Implicit functions like the given \(y^2(2-x)=x^3\) are not always easy to solve for y explicitly. When graphing these, it's helpful to use graphing utilities. Implicit differentiation allows us to understand the slope and tangent lines without having to solve for y. To graph implicitly defined curves:

• Begin by re-writing the equation in a more manageable form, if needed.
• Use graphing utilities to visualize the curve.
• Identify key points and behaviors, such as where the curve intersects the axes.
• Use derivatives to understand slopes and tangent lines.

This gives us deeper insight into both the graphical and analytical properties of the function.

Cissoid Curves

A cissoid is a type of curve defined implicitly, such as our example: \(y^2(2-x)=x^3\). These curves have unique features and characteristics worth studying. In our case:

• The curve originates from a classic Greek mathematical problem.
• It displays interesting properties near specific points, like sharp increases or decreases.
• We can analyze their behavior through techniques like implicit differentiation and limit evaluation.

This particular cissoid exhibits a dramatic rise in y-values as \(x \) nears 2, providing a visual example of how complex and fascinating these curves can be.