Problem 14 - Calculus for Biology and Medicine

Question

(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

Answer

The derivative \( \frac{dy}{dx} \) at \((1, 3)\) is \( \frac{19}{3} \). Graph by plotting the upper \( y = \sqrt{10x^4 - x^2} \) and lower halves \( y = -\sqrt{10x^4 - x^2} \) separately.

Step 1: Differentiate Implicitly

To differentiate the equation \( y^2 = 10x^4 - x^2 \) implicitly, first apply the derivative to both sides with respect to \( x \). The left side becomes \( 2y \frac{dy}{dx} \) and the right side becomes \( 40x^3 - 2x \). The equation is:\[2y \frac{dy}{dx} = 40x^3 - 2x\]

Step 2: Solve for \( \frac{dy}{dx} \)

From the differentiated equation, solve for \( \frac{dy}{dx} \) by isolating it on one side:\[\frac{dy}{dx} = \frac{40x^3 - 2x}{2y}\]

Step 3: Evaluate at the Given Point

Now, substitute \( x = 1 \) and \( y = 3 \) into the equation for \( \frac{dy}{dx} \):\[\frac{dy}{dx} = \frac{40(1)^3 - 2(1)}{2(3)} = \frac{40 - 2}{6} = \frac{38}{6} = \frac{19}{3}\]

Step 4: Graph the Curve

Since a graphing calculator may not graph implicit functions directly, graph the upper and lower halves separately. For the upper half, use \( y = \sqrt{10x^4 - x^2} \) and for the lower half, use \( y = -\sqrt{10x^4 - x^2} \). Set the viewing window to \(-3 \leq x \leq 3\) and \(-10 \leq y \leq 10\).

Key Concepts

Graphing Calculator

Graphing calculators are powerful tools for visualizing complex mathematical equations. They allow you to see the shape and behavior of functions effectively. However, when dealing with implicit functions, graphing calculators can sometimes pose a challenge. Implicit functions are equations where both variables, usually \(x\) and \(y\), are interdependent on each other without a clear, explicit solution for one variable in terms of the other.

• In the case of implicit functions, simple graphing calculators might only support graphs of explicit equations, like \(y = f(x)\).
• To handle implicit equations such as \(y^2 = 10x^4 - x^2\), an effective approach is to graph the two parts separately.
• The equation can be split into an upper portion, \(y = \sqrt{10x^4 - x^2}\), and a lower portion, \(y = -\sqrt{10x^4 - x^2}\).

Using this method, you can accurately represent the entire curve by graphing each part individually. Set your graphing window appropriately (e.g., \(-3 \leq x \leq 3\) and \(-10 \leq y \leq 10\)) to ensure both halves of the curve are clearly visible. This approach ensures a precise visualization without the need for advanced graphing tools.

Implicit Functions

Implicit functions are equations where variables cannot easily be separated or solved for in terms of one another. This means neither variable is solely dependent on the other in an explicit form, like \(y = mx + b\). An example of this is the equation from the exercise, \(y^2 = 10x^4 - x^2\).

• To work with implicit functions, you often need to use implicit differentiation.
• In implicit differentiation, you differentiate both sides of the equation with respect to \(x\), treating \(y\) as a function of \(x\).
• For \(y^2 = 10x^4 - x^2\), differentiating both sides gives \(2y\frac{dy}{dx} = 40x^3 - 2x\).

By using implicit differentiation, you can find the derivative \(\frac{dy}{dx}\), which represents how \(y\) changes concerning \(x\). This process is beneficial in calculus when solving implicit equations, making it possible to analyze how variables interact within such functions.

Differential Calculus

Differential calculus focuses on the concept of the derivative, which measures how a function changes as its input changes. Implicit differentiation is a key technique in differential calculus used when dealing with implicit functions. By finding derivatives, you are determining the rate of change of one variable with respect to another.

• The derivative \(\frac{dy}{dx}\) can indicate the slope of the tangent line to a curve at a given point.
• In our example, once the implicit equation is differentiated, solving for \(\frac{dy}{dx}\) gives \(\frac{dy}{dx} = \frac{40x^3 - 2x}{2y}\).
• Substituting specific values of \(x\) and \(y\), such as \((1,3)\), lets you find the slope at that point, \(\frac{19}{3}\).

Understanding this slope is crucial as it provides insight into the behavior and direction of the curve at any given point. This application of differential calculus is fundamental in analyzing and comprehensively understanding the intricacies of curves and surfaces described by mathematical functions.