Vertical tangent lines If a function f is continuous at a and lim x→a| f′(x)|=∞, then the curve y=f(x) has a vertical tangent line at a, and the equation of the tangent line is x=a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 71–72) is used. Use this information to answer the following questions.
73. {Use of Tech} Graph the following functions and determine the location of the vertical tangent lines.
c. f(x) = √|x-4|

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the location of vertical tangent lines for the function f(x) = \sqrt{|x-4|}. A vertical tangent line occurs where the derivative of the function approaches infinity.
2. Step 2: Analyze the function. The function f(x) = \sqrt{|x-4|} is continuous everywhere in its domain, which is all real numbers. However, the expression inside the square root, |x-4|, changes behavior at x = 4.
3. Step 3: Find the derivative of the function. To find where the vertical tangent line occurs, we need to compute the derivative f'(x). Start by rewriting the function as f(x) = (|x-4|)^{1/2}. Use the chain rule and the derivative of the absolute value function to find f'(x).
4. Step 4: Evaluate the behavior of the derivative at x = 4. Since the absolute value function has a corner at x = 4, check the limit of the derivative as x approaches 4 from both sides. Specifically, calculate \lim_{x \to 4^-} f'(x) and \lim_{x \to 4^+} f'(x).
5. Step 5: Determine the location of the vertical tangent line. If either of the one-sided limits from Step 4 approaches infinity, then there is a vertical tangent line at x = 4. The equation of this vertical tangent line is x = 4.

Key Concepts

Vertical Tangent Lines

A vertical tangent line occurs at a point on a curve where the slope of the tangent approaches infinity. This typically indicates that the derivative of the function is undefined or infinite at that point. In the context of calculus, if the limit of the absolute value of the derivative approaches infinity as x approaches a certain value, the function has a vertical tangent line at that point.

Continuity of Functions

A function is continuous at a point if there are no breaks, jumps, or holes in the graph at that point. For a function to have a vertical tangent line, it must first be continuous at the point of interest. This means that the function's value at that point is well-defined, and the behavior of the function around that point can be analyzed using limits.

One-Sided Derivatives

One-sided derivatives are used to analyze the behavior of a function at endpoints or points where the function may not be differentiable in the traditional sense. The left-hand derivative considers the slope of the tangent as you approach the point from the left, while the right-hand derivative does so from the right. In cases where a function is defined only on one side of a point, one-sided derivatives provide crucial information about the function's behavior at that point.