Problem 31

Question

Find the equation of the normal line to the curve $y=-3 x^{2}+x$ at the point $(-1,-4)$.

Step-by-Step Solution

Verified

Answer

The equation of the normal line is $ y = -\frac{1}{7}x - \frac{29}{7} $.

1Step 1: Find the derivative of the function

To find the slope of the tangent line, we first need the derivative of the function. The original function is given as $ y = -3x^2 + x $. Using basic differentiation rules, the derivative is $ y' = -6x + 1 $.

2Step 2: Calculate the slope at the given point

Substitute $ x = -1 $ into the derivative to find the slope of the tangent line at the point $(-1, -4)$. Thus, $-6(-1) + 1 = 6 + 1 = 7$, so the slope of the tangent line is 7.

3Step 3: Determine the slope of the normal line

The normal line is perpendicular to the tangent line. Therefore, its slope is the negative reciprocal of the tangent's slope. The slope of the tangent line is 7, so the slope of the normal line is $-\frac{1}{7}$.

4Step 4: Use the point-slope form to find the equation of the normal line

With a slope $-\frac{1}{7}$ and passing through the point $(-1, -4)$, use the point-slope form of the equation: $ y - y_1 = m(x - x_1) $. Substituting $ x_1 = -1 $, $ y_1 = -4 $, and $ m = -\frac{1}{7} $ gives $ y + 4 = -\frac{1}{7}(x + 1) $.

5Step 5: Simplify the equation

Simplify the normal line equation from Step 4. Multiply out the terms: $ y + 4 = -\frac{1}{7}x - \frac{1}{7} $, then isolate $ y $: $ y = -\frac{1}{7}x - \frac{1}{7} - 4 $, which simplifies to $ y = -\frac{1}{7}x - \frac{29}{7} $.

Key Concepts

Tangent LineDerivativePerpendicular SlopePoint-Slope Form

Tangent Line

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. This line is critical because it depicts the path or direction the curve is taking at exactly that point.

This tangent acts like a local linear approximation to the curve.
The slope of the tangent line at a specific point on the curve gives us the instantaneous rate of change of the function at that point.

In simpler terms, imagine the curve as a mountain. The tangent line is like a small, flat plank of wood resting on the side of the mountain, perfectly at the edge. It doesn't cut through but just fits snugly against the side. This concept of tangent lines is widespread, not only in mathematics but also in physics (for understanding velocities) and biology (for growth rates). The slope of this tangent is derived from the derivative of the function. This aspect seamlessly brings us to the next point.

Derivative

The derivative of a function provides a powerful tool for understanding the geometry of the curve represented by the function. Generally seen as a rate of change, the derivative helps in determining the slope of the tangent line.

For instance, the derivative of a function at a point gives the slope of the line that is tangent to the function's graph at that point.
In essence, it tells us how steep the curve is becoming.
Differentiation techniques include simple power rules, chain rule, and others, depending on the complexity of the function.

For example, consider the function given by the curve $ y = -3x^2 + x $. Its derivative, calculated using basic differentiation, is $ y' = -6x + 1 $. By substituting a specific value of $ x $, such as $ x = -1 $, we can determine the tangent's slope at that exact point. The result helps form the tangent line equation, connecting seamlessly to determining normal lines.

Perpendicular Slope

When you have two lines that are perpendicular to each other, the slopes of these lines are related in a specific way: they are negative reciprocals of each other. This is a critical concept when working with normal lines to a curve.

If a line has a slope of $ m $, the line perpendicular to it will have a slope of $-\frac{1}{m}$.
Ensuring this relationship maintains the exact right-angle (90 degrees) between the tangent and normal lines.

For instance, if the slope of the tangent line at a specific point is 7, then the slope of the normal line (perpendicular to the tangent) will be $-\frac{1}{7}$. This idea allows us to accurately define the unique line that perpendicularly intersects the curve precisely at that point of tangency.

Point-Slope Form

The point-slope form of a line is a practical tool for writing the equation of any straight line. It's especially handy when you have a point on the line and the slope. The general form is: $ y - y_1 = m(x - x_1) $.

Here, $ (x_1, y_1) $ is a point on the line.
$ m $ is the slope.

To illustrate, if we need the equation of the normal line that has a slope of $-\frac{1}{7}$ and passes through the point $(-1, -4)$, we utilize this form with $ x_1 = -1 $, $ y_1 = -4 $, and $ m = -\frac{1}{7} $. This gives us:$ y + 4 = -\frac{1}{7}(x + 1) $.Adjusting and simplifying this equation leads you to the proper line equation: $ y = -\frac{1}{7}x - \frac{29}{7} $.Again, this form neatly connects the core concepts of slope, points, and linear equations to form both the tangent and normal line equations.

Problem 31

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