Problem 54 - Calculus for Biology and Medicine

Question

Find the normal line, in standard form, to \(y=f(x)\) at the indicated point. $$ y=1-\pi x^{2}, \text { at } x=-1 $$

Step-by-Step Solution

Answer

The normal line is \(x + 2\pi y = 2\pi^2 - 2\pi + 1\) in standard form.

Step 1: Determine the derivative

To find the equation of the normal line, we first need the derivative of the function since the normal line is perpendicular to the tangent line at a point. The function given is \(y = 1 - \pi x^2\). The derivative is found using the power rule: \(f'(x) = -2\pi x\).

Step 2: Evaluate the derivative at the given point

Evaluate the derivative at the point \(x = -1\) to find the slope of the tangent line. Substitute \(-1\) into \(f'(x) = -2\pi x\): \[f'(-1) = -2\pi(-1) = 2\pi\]. Thus, the slope of the tangent line at \(x = -1\) is \(2\pi\).

Step 3: Calculate the slope of the normal line

The normal line is perpendicular to the tangent line, so we use the negative reciprocal of the tangent slope. The slope of the tangent line is \(2\pi\), so the slope of the normal line is \[-\frac{1}{2\pi}\].

Step 4: Find the point on the curve

Calculate the y-coordinate of the point on the curve when \(x = -1\). Substitute \(x = -1\) into the original equation: \[y = 1 - \pi(-1)^2 = 1 - \pi\]. Thus, the point is \((-1, 1 - \pi)\).

Step 5: Write the equation of the normal line in point-slope form

Use the point-slope form of a line's equation: \[y - y_1 = m(x - x_1)\]. Substitute \((x_1, y_1)\) = \((-1, 1 - \pi)\) and \(m = -\frac{1}{2\pi}\):\[y - (1 - \pi) = -\frac{1}{2\pi}(x + 1)\].

Step 6: Convert to standard form

Simplify and rearrange the equation from Step 5 to get it in the standard form \(Ax + By = C\): \\[y - 1 + \pi = -\frac{1}{2\pi}x - \frac{1}{2\pi}\]Multiply throughout by \(2\pi\) to clear the fraction: \[2\pi y - 2\pi + 2\pi^2 = -x - 1\]Rearrange terms: \[x + 2\pi y = 2\pi^2 - 2\pi + 1\].

Key Concepts

Understanding Derivatives

Derivatives are fundamental in calculus. They represent how a function changes at any given point. When you find the derivative of a function, you are essentially finding the slope of the tangent line to the graph of that function. This slope indicates the rate of change of the function's value with respect to a change in input value, \(x\).
For the function \(y = 1 - \pi x^2\), we used the power rule to find its derivative, \(f'(x) = -2\pi x\). This rule helps us quickly differentiate functions like \(x^n\), giving the result \(nx^{n-1}\).

• Key point: A derivative gives us the slope of the tangent line at a specific point on the curve!

The Tangent Line Explained

A tangent line is a straight line that touches a curve at just one point, reflecting the curve's slope at that point. It is crucial because it gives us a linear approximation of the curve near the point.
Once we have the derivative, we can find the tangent line by evaluating the derivative at the desired point. In our exercise, we determined the slope of the tangent line at \(x = -1\) was \(2\pi\).

• Remember: The tangent line shows the direction the curve is moving at a single point.

Normal Line and Its Perpendicular Nature

The normal line is perpendicular to the tangent line at a given point on the curve. This means it crosses the tangent line at a right angle (90 degrees). Its slope is the negative reciprocal of the tangent line's slope.
For instance, if the slope of the tangent line is \(2\pi\), then the slope of the normal line is \(-\frac{1}{2\pi}\). This is because perpendicular lines have slopes that multiply to \(-1\).

• Tip: Use the negative reciprocal to find the slope of the normal line easily!

Using the Power Rule

The power rule is a handy formula for speeding up the process of finding derivatives. It's especially useful for polynomial functions. If you have a function like \(x^n\), the power rule tells us the derivative is \(nx^{n-1}\).
This rule was applied to our function \(y = 1 - \pi x^2\) to find its derivative. Applying the rule, we found \(f'(x) = -2\pi x\). In essence, the power rule streamlines calculating derivatives for functions involving powers of \(x\).

• Practical advice: Master the power rule to solve derivatives much faster!