Question

Vertical tangent lines a. Determine the points at which the curve \(x+y^{3}-y=1\) has a vertical tangent line (see Exercise 52 ). b. Does the curve have any horizontal tangent lines? Explain.

Short answer

If so, determine the points at which these tangents occur. Answer: The curve \(x+y^{3}-y=1\) does not have any vertical or horizontal tangent lines.

Step-by-step solution

Re-write the equation in parametric form

Rewrite the curve equation as a parametric equation using a parameter t: $$x(t) = x$$ $$y(t) = y$$.

Calculate dx/dt and dy/dt

Take the derivatives of x and y with respect to the parameter t: $$\frac{dx}{dt} = 1$$ $$\frac{dy}{dt} = \frac{dy}{dy}=1$$.

Substitute the parametric equations into the original curve equation

Now we need to eliminate the parameter t, so plug x(t) and y(t) back into the curve equation: $$x+y^{3}-y=1$$ Substituting the parameterized expressions for x and y, we have: $$x+y^{3}-y=1$$.

Determine where the slope is undefined (vertical tangent)

Now we find the points where the slope is undefined, so we need to find where the derivative \(\frac{dy}{dx}\) is undefined. To do this, we first find \(\frac{dy}{dx}\): $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{1}{1} = 1$$ Since the derivative is always defined, there are no vertical tangent lines for this curve. #b. Does the curve have any horizontal tangent lines? Explain.#

Determine where the slope is 0 (horizontal tangent)

Now we find the points where the slope is horizontal or the derivative \(\frac{dy}{dx}\) is equal to 0: Since we already found that the \(\frac{dy}{dx}=1\), there are no points where the derivative is equal to 0. Therefore, the curve has no horizontal tangent lines.

Key concepts

Vertical Tangent Line

A vertical tangent line occurs at a point on a curve where the tangent is perfectly vertical. This means that the slope of the tangent is undefined because the change in the y-values is happening infinitely faster than the change in x-values.

• In mathematical terms, this happens when the derivative \(\frac{dy}{dx}\) is undefined.
• If you have a parametric equation for the curve, this condition is equivalent to \(\frac{dx}{dt} = 0\) while \(\frac{dy}{dt}\) is not zero.

In the given exercise, the attempt to find a vertical tangent line reveals that \(\frac{dy}{dx}\) is never undefined. Therefore, there are no vertical tangent lines for this curve.

Parametric Equations

Parametric equations allow us to describe a curve by expressing the coordinates \(x\) and \(y\) as functions of a third parameter, typically \(t\).

• They are incredibly useful for tracing the paths of objects or describing curves in two-dimensional space.
• Unlike standard equations, parametric ones can represent curves that loop or have sharp corners without vertical tangent line issues.

In the exercise, the curve was placed in the parametric form, which simply expressed \(x\) and \(y\) as functions of themselves: \(x(t) = x\) and \(y(t) = y\). This was a straightforward transformation here to aid further analysis.

Derivative

The derivative \(\frac{dy}{dx}\) is a key concept in calculus that represents the slope of the tangent line to a curve at a point. It tells us how the y-values of the function change with respect to changes in x-values.

• If \(\frac{dy}{dx} > 0\), the curve is rising at that point.
• If \(\frac{dy}{dx} < 0\), the curve is falling.
• If \(\frac{dy}{dx} = 0\), the curve is flat - meaning it could potentially have a horizontal tangent.

In the exercise, calculations showed \(\frac{dy}{dx} = 1\), confirming a consistently rising curve with no points where \(\frac{dy}{dx}\) is undefined or zero.

Horizontal Tangent Line

A horizontal tangent line occurs at points on a curve where the slope is zero, meaning that the change in x-values happens without any change in y-values.

• This implies that \(\frac{dy}{dx} = 0\).
• For parametric equations, it means both \(\frac{dy}{dt} = 0\), while \(\frac{dx}{dt}\) must not be zero.

In the provided problem, it was discovered that \(\frac{dy}{dx} = 1\) consistently. Thus, there are no horizontal tangent lines for this curve since the slope is never zero.