1. What is a Vertical Tangent Line?

A vertical tangent line occurs at points where the derivative of a function becomes infinite (undefined). This typically happens when the curve has a vertical slope at that point.

2. How Does the Calculator Work?

The calculator finds points where:

Which corresponds to vertical tangent lines at:

• \( x = x_0 \)

Explanation: The calculator numerically evaluates the derivative and identifies points where it approaches infinity.

3. Mathematical Explanation

Details: Vertical tangents occur when the denominator of dy/dx is zero while the numerator is non-zero. Common examples include functions with fractional exponents like \( y = x^{1/3} \) at x=0.

4. Using the Calculator

Tips: Enter a function of x (like "x^(1/3)" or "sqrt(x)"). The calculator will identify all x-values where vertical tangents occur.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between vertical tangent and vertical asymptote?
A: A vertical tangent occurs when the function is defined but has infinite slope. A vertical asymptote occurs when the function approaches infinity.

Q2: Can a function have multiple vertical tangents?
A: Yes, functions like \( y = x^{1/3}(x-1)^{1/3} \) can have vertical tangents at multiple points.

Q3: How is this different from a cusp?
A: At a cusp, both left and right derivatives approach ±∞. For a vertical tangent, both sides approach the same infinite slope.

Q4: What functions commonly have vertical tangents?
A: Functions with fractional exponents (especially denominators of 3, 5, etc.), certain parametric equations, and implicitly defined functions.

Q5: Can vertical tangents occur at endpoints?
A: Yes, if the function is defined at the endpoint and the one-sided derivative is infinite.