Problem 63 - Calculus and its applications

Question

For each function, find the points on the graph at which the tangent line is horizontal. If none exist, state that fact. $$ y=x^{3}-2 $$

Step-by-Step Solution

Answer

The tangent line is horizontal at the point (0, -2).

Step 1: Understanding the Problem

We are given the function \( y = x^3 - 2 \) and need to find where the tangent line to the graph is horizontal. A tangent line is horizontal where its slope is zero.

Step 2: Derive the Function

To find the slope of the tangent line, first compute the derivative of the function. Differentiate \( y = x^3 - 2 \) with respect to \( x \). \[ \frac{dy}{dx} = 3x^2 \]

Step 3: Set the Derivative to Zero

Horizontal tangents occur where the derivative equals zero. Set the derivative \( 3x^2 \) equal to zero and solve for \( x \).\[ 3x^2 = 0 \]

Step 4: Solve for \( x \)

Solve \( 3x^2 = 0 \): - Divide both sides by 3 to get \( x^2 = 0 \).- Take the square root of both sides to find \( x = 0 \).

Step 5: Find the Corresponding \( y \)-value

To find the point on the graph, substitute \( x = 0 \) back into the original equation \( y = x^3 - 2 \) to get the corresponding \( y \)-value. \[ y = 0^3 - 2 = -2 \]

Step 6: State the Result

The point on the graph where the tangent is horizontal is \((0, -2)\).

Key Concepts

Slope of a Tangent Line

The whole idea behind the tangent line is its relationship to a curve. A tangent line touches a curve at one unique point, and the slope of this line at that point represents the instantaneous rate of change of the function. In simpler terms, it's like peeking into the curve's behavior just at that specific point.

• A horizontal tangent line means that the slope at that point equals zero. Think of horizontal lines that run flat, neither rising nor falling.
• If the slope is zero, the curve right there is flattening out briefly, if just for an instant.

One way to visualize this is by imagining a hill. If you're standing at the very top, the ground beneath you is flat for a moment—your `slope` would be zero, just like a horizontal tangent line on a graph.

Derivative of a Function

Calculating the derivative of a function is like finding a special formula that tells us the slope of the tangent line at any given point on the curve. This formula changes depending on the point we're interested in.

• For the function in our example, the derivative is found using a straightforward method of differentiation. Given the function \( y = x^3 - 2 \), we differentiate with respect to \( x \).
• The power rule, one of the basic rules of differentiation, gives us \( \frac{dy}{dx} = 3x^2 \). It's really that simple. Just multiply by the power and reduce the power by one.
• This derivative, \( 3x^2 \), is our magical formula that reveals the slope of the tangent line for any \( x \).

It's powerful because, at a glance, it tells us how steep the curve is or if it's turning up or down.

Solving Equations

Once we've got the derivative, finding horizontal tangent lines boils down to a bit of simple equation solving.

• Set the derivative equal to zero because, remember, a horizontal line has a slope of zero.
• In our exercise, the derivative \( 3x^2 = 0 \) tells us exactly where the slope is zero. Solving this equation is straightforward; first, divide by 3 (which doesn't change anything significant) to get \( x^2 = 0 \).
• Next, take the square root of both sides, finding \( x = 0 \). What you've done is find the specific point or points where our curve flattens.

By substituting back into the original equation, we can easily find the full coordinates of that special point, revealing where exactly the curve has a horizontal tangent.