Question

Standardized Test Questions You should solve the following problems without using a graphing calculator. True or False If the graph of a function has a tangent line at \(x=a,\) then the graph also has a normal line at \(x=a\) . Justify your answer.

Short answer

True. If the graph of a function has a tangent line at \(x=a\), then the graph also has a normal line at \(x=a\). This is because every line has an infinite number of lines perpendicular to it, including the tangent line to a function's graph at any given point.

Step-by-step solution

Understanding Terms

First, recall the definitions of tangent and normal lines in the context of a function. A tangent line is a line that 'just touches' the curve at a given point, while a normal line is a line perpendicular to the tangent line at the point of tangency.

Analysing the Relationship

Knowing the definitions, think about the relationship between the tangent line and the normal line. By definition, if a function is differentiable (has a derivative) at a point, it has a tangent line at that point. Because a normal line is simply the line perpendicular to the tangent, any function that has a tangent line also has a normal line.

Concluding the Response

Based on the relationship between tangent and normal line, we can conclude and justify that if a function has a tangent line at \(x=a\), then it also has a normal line at that point. This is because any line has an infinite number of lines perpendicular to it at any point on the line, thus there must be a line that is perpendicular to the tangent line at \(x=a\) which we can consider as the normal line at \(x=a\).

Key concepts

Tangent Line Definition

In calculus, a tangent line refers to a straight line that touches a curve at exactly one point without crossing it. This line represents the instantaneous direction the curve is heading at that point, which is essentially the same as the direction of the function's slope at that point.

The formal mathematical way to describe a tangent line at a point \(x = a\) for a function \(f(x)\) is to find where the function's derivative \(f'(a)\) matches the slope of the tangent line. This slope is a real number that gives us the rate of change of the function at \(x = a\).

To improve understanding, it is beneficial to visualize the situation: imagine your curve as a road, and the tangent line as the trajectory a car would follow if it had to leave the road at point \(a\). The car would not turn immediately but would follow the path of the tangent line, reflecting the direction of the road at the very moment the car leaves it.

Normal Line Definition

Opposite in concept to the tangent line is the normal line. This is a line that passes through the point of tangency and is perpendicular to the tangent line at that point. In the Cartesian plane, if a tangent line has a slope of \(m\), then the normal line at that point will have a slope of \(\frac{-1}{m}\), provided \(m\) is not zero, since perpendicular lines have slopes that are negative reciprocals of each other.

An easy way to remember the normal line is to think of it as 'standing up' against the tangent. If the tangent line represents the path you would continue on, the normal represents a path you would take if you wanted to move directly away from the curve at the point of tangency, in the steepest possible ascent or descent.

When visualizing the normal line, picture a ladder leaning against a wall, where the wall is your curve. The ladder would be perpendicular to the wall representing the normal line where it touches the wall or curve.

Differentiability of Functions

A core concept in calculus is the differentiability of functions. A function \(f(x)\) is said to be differentiable at a point \(x = a\) if it has a derivative there. This implies that \(f(x)\) is smooth and continuous at \(x = a\), with no sharp corners or discontinuities.

Differentiability guarantees that there is a tangent line at the point \(x = a\), which also confirms the existence of a normal line. To determine differentiability, we look for the limit of the difference quotient as \(x\) approaches \(a\). If this limit exists, we have found \(f'(a)\), and hence, the function is differentiable at \(a\).

Understanding differentiability is crucial because it is the gateway to finding both tangent and normal lines to functions. Whenever you're asked if a function is differentiable at a point, you’re essentially being asked whether the function behaves nicely enough at that point for us to talk sensibly about its tangent and normal lines.