Problem 60
Question
Kampyle of Eudoxus (a) Consider the curve with equation \(y^{2}=10 x^{4}-x^{2}\). Find \(\frac{d y}{d x}\) at \((1,3)\) (b) Use a graphing calculator to graph the curve in (a). If the calculator cannot graph implicit functions, graph the upper and the lower halves of the curve separately. Choose the viewing rectangle \(-3 \leq x \leq 3,-10 \leq y \leq 10\)
Step-by-Step Solution
Verified Answer
\(\frac{dy}{dx}\) at (1,3) is \(\frac{19}{3}\); graph both halves separately.
1Step 1: Differentiate Implicitly
Given the equation of the curve: \[ y^2 = 10x^4 - x^2 \]Take the derivative of both sides with respect to \(x\). The left side differentiates to \(2y \frac{dy}{dx}\) using the chain rule. The right side differentiates to \(40x^3 - 2x\). Equating both derivatives gives:\[ 2y \frac{dy}{dx} = 40x^3 - 2x \]
2Step 2: Solve for \(\frac{dy}{dx}\)
Solve the equation from Step 1 for \(\frac{dy}{dx}\):\[ \frac{dy}{dx} = \frac{40x^3 - 2x}{2y} \]
3Step 3: Substitute Point (1,3) into Derivative
Substitute \(x = 1\) and \(y = 3\) into the expression for \(\frac{dy}{dx}\):\[ \frac{dy}{dx} = \frac{40(1)^3 - 2(1)}{2(3)} = \frac{40 - 2}{6} = \frac{38}{6} = \frac{19}{3} \]
4Step 4: Graph the Upper Half of the Curve
Solve for \(y\) in terms of \(x\) for the upper half of the curve:\[ y = \sqrt{10x^4 - x^2} \]Graph this equation from \(-3\) to \(3\) for \(x\), ensuring \(-10 \leq y \leq 10\). This represents the upper half of the curve.
5Step 5: Graph the Lower Half of the Curve
For the lower half of the curve, solve for \(y\) as:\[ y = -\sqrt{10x^4 - x^2} \]Graph this equation similarly from \(-3\) to \(3\) for \(x\), within the same viewing window \(-10 \leq y \leq 10\).
Key Concepts
Implicit DifferentiationDerivative CalculationGraphing Implicit Functions
Implicit Differentiation
Implicit differentiation is a crucial technique in calculus used when dealing with equations that define curves indirectly. These equations contain multiple variables in such a way that one variable is not easily solved in terms of another. In the context of the Kampyle of Eudoxus, we have \[ y^2 = 10x^4 - x^2 \].
To find the derivative of this equation with respect to \( x \), we need to differentiate both sides as they stand.
This involves treating \( y \) as an implicit function of \( x \). Thus, when differentiating \( y^2 \), we apply the chain rule: the derivative becomes \( 2y \frac{dy}{dx} \). For the right side, treating each term separately yields \( 40x^3 - 2x \).
By equating these derivatives, we form an equation involving \( \frac{dy}{dx} \) that we can solve for this derivative: \[ 2y \frac{dy}{dx} = 40x^3 - 2x \]. Learning this technique enables us to understand how changes in \( x \) or \( y \) impact one another, even when they are interlinked in complicated ways.
To find the derivative of this equation with respect to \( x \), we need to differentiate both sides as they stand.
This involves treating \( y \) as an implicit function of \( x \). Thus, when differentiating \( y^2 \), we apply the chain rule: the derivative becomes \( 2y \frac{dy}{dx} \). For the right side, treating each term separately yields \( 40x^3 - 2x \).
By equating these derivatives, we form an equation involving \( \frac{dy}{dx} \) that we can solve for this derivative: \[ 2y \frac{dy}{dx} = 40x^3 - 2x \]. Learning this technique enables us to understand how changes in \( x \) or \( y \) impact one another, even when they are interlinked in complicated ways.
Derivative Calculation
Once you have the implicit derivative equation \[ 2y \frac{dy}{dx} = 40x^3 - 2x \], the next step is to isolate \( \frac{dy}{dx} \).
This means rearranging this equation by dividing both sides by \( 2y \). Once simplified, it becomes: \[ \frac{dy}{dx} = \frac{40x^3 - 2x}{2y} \].
To find the specific slope at a given point, in this case at \( (1,3) \), plug these values into the derivative expression. You substitute \( x = 1 \) and \( y = 3 \) resulting in: \[ \frac{dy}{dx} = \frac{40(1)^3 - 2(1)}{2(3)} = \frac{38}{6} = \frac{19}{3} \].
This step gives you the slope of the tangent to the curve at the specified point. Calculating derivatives at specific points like this allows you to understand how the curve behaves at certain locations, providing insights into rates of change.
This means rearranging this equation by dividing both sides by \( 2y \). Once simplified, it becomes: \[ \frac{dy}{dx} = \frac{40x^3 - 2x}{2y} \].
To find the specific slope at a given point, in this case at \( (1,3) \), plug these values into the derivative expression. You substitute \( x = 1 \) and \( y = 3 \) resulting in: \[ \frac{dy}{dx} = \frac{40(1)^3 - 2(1)}{2(3)} = \frac{38}{6} = \frac{19}{3} \].
This step gives you the slope of the tangent to the curve at the specified point. Calculating derivatives at specific points like this allows you to understand how the curve behaves at certain locations, providing insights into rates of change.
Graphing Implicit Functions
Graphing implicit functions can be challenging, particularly when your calculator or software doesn't support implicit plotting directly. For the Kampyle of Eudoxus, we need to approach this by separating the equation into its two symmetrical parts.
For the upper half of the curve, solve for \( y \) by taking the positive square root: \[ y = \sqrt{10x^4 - x^2} \]. This gives the positive values of \( y \) over the domain where the equation is valid.
Likewise, the lower half is obtained via the negative square root form: \[ y = -\sqrt{10x^4 - x^2} \].
You graph these individually over a defined interval, here from \( x = -3 \) to \( x = 3 \), with \( y \) extending from \( -10 \) to \( 10 \). Each half represents different sections of the implicit curve's geometry. Using visualization to represent the function helps in understanding its symmetry and behavior across its domain, enhancing comprehension of both the equation's properties and mathematical aesthetics.
For the upper half of the curve, solve for \( y \) by taking the positive square root: \[ y = \sqrt{10x^4 - x^2} \]. This gives the positive values of \( y \) over the domain where the equation is valid.
Likewise, the lower half is obtained via the negative square root form: \[ y = -\sqrt{10x^4 - x^2} \].
You graph these individually over a defined interval, here from \( x = -3 \) to \( x = 3 \), with \( y \) extending from \( -10 \) to \( 10 \). Each half represents different sections of the implicit curve's geometry. Using visualization to represent the function helps in understanding its symmetry and behavior across its domain, enhancing comprehension of both the equation's properties and mathematical aesthetics.
Other exercises in this chapter
Problem 59
Differentiate the functions with respect to the independent variable. $$ f(u)=\log _{3}\left(3+u^{4}\right) $$
View solution Problem 60
Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=\frac{1}{2+x} $$
View solution Problem 60
Find the points on the curve \(y=\cos ^{2} x\) that have a horizontal tangent.
View solution Problem 60
Population Growth Suppose that the population size at time is $$ N(t)=N_{0} e^{r t}, \quad t \geq 0 $$ where \(N_{0}\) is a positive constant and \(r\) is a rea
View solution