Question

A function and an \(x-\) value \(C\) are given. (Note: these functions are the same as those given in Exercises 7 through 14.) (a) Give the equation of the tangent line at \(x=c .\) (b) Give the equation of the normal line at \(x=c\). $$f(x)=6,\( at \)x=-2$$

Short answer

The tangent line is \( y = 6 \); the normal line is \( x = -2 \).

Step-by-step solution

Understanding the Problem

We need to find the equations of both the tangent line and the normal line for the function \( f(x) = 6 \) at \( x = -2 \). This is a constant function.

Recognizing a Constant Function

Since \( f(x) = 6 \) is a constant function, its graph is a horizontal line at \( y = 6 \).

Finding the Derivative

For a constant function, the derivative \( f'(x) = 0 \). This represents the slope of the tangent line.

Equation of the Tangent Line

The tangent line will have the same value as the function itself at every point, thus it is also \( y = 6 \) for all \( x \). In slope-intercept form, it's \( y = 0(x + 2) + 6 = 6 \).

Equation of the Normal Line

The slope of the normal line is the negative reciprocal of the tangent line's slope. Since the tangent line's slope is 0, the slope of the normal line is undefined, meaning it is a vertical line at \( x = -2 \).

Conclusion

The equation of the tangent line is \( y = 6 \) and the equation of the normal line is \( x = -2 \). These lines reflect the properties of horizontal and vertical lines.

Key concepts

Tangent Lines

A tangent line to a curve at a particular point is a line that just touches the curve at that point. It reflects the immediate direction of the curve there. For a constant function like \(f(x) = 6\), the graph is a horizontal line. Thus, the tangent line is simply the line itself: horizontal at every point. For the given function, the tangent line at any point including \(x = -2\) is \(y = 6\).
This is because the slope of a horizontal line remains zero and does not change. A tangent line always shares the same slope as the function at the point of tangency, which in this case is 0. Hence, the equation of the tangent line remains \(y = 6\).

Normal Lines

Normal lines are perpendicular to the tangent lines at the point of tangency. To find the normal line, you take the negative reciprocal of the tangent line's slope. When dealing with a constant function, the slope of the tangent is zero, making the normal line undefined, as its slope would be the reciprocal of zero (infinity).
Thus, in such a scenario, the normal line becomes a vertical line. For \(f(x) = 6\) at \(x = -2\), the normal line's equation is simply \(x = -2\). It shows us that the normal line doesn't move horizontally and remains fixed at \(x = -2\).

Constant Functions

Constant functions such as \(f(x) = 6\) are functions that do not change with different values of \(x\). The output remains the same for all inputs, thus creating a horizontal line graph. This constancy makes determining slopes straightforward since the derivative will always be zero.
These functions exemplify the simplest slope context as the tangent and the function line coincide entirely. For a constant function, the tangent line is the function itself, creating scenarios where normal lines turn into vertical lines, emphasizing the unique geometry of constant functions.

Derivatives

In calculus, derivatives represent the rate of change of a function. For a constant function like \(f(x) = 6\), the derivative is zero, indicating no change regardless of \(x\). This zero slope means a horizontal tangent line.
Derivatives help in understanding the behavior of functions at specific points. By using derivatives, you can easily determine tangent line equations and understand the nature of functions. In our function \(f'(x) = 0\), which reinforces the idea of the tangent line being the same horizontal line \(y = 6\). Understanding derivatives is crucial as it provides insights into slopes and how curves behave at different points.

Slope of Lines

The slope of a line measures its steepness and is defined as the rise over run. For any horizontal line, like our constant function, the slope is zero, resulting in a flat line. The slope also decides how tangent and normal lines are aligned.
When the tangent slope is zero, the normal line inherits an undefined slope, leading to a vertical line. Understanding slopes is essential when interpreting how lines interact with curves. It allows us to simplify complex equations into straightforward linear ones, assisting in finding equations of tangent and normal lines comfortably. In this exercise, the concept of slope shows us both the unchanging nature of the constant function and the peculiar case of the normal line being vertical.