Problem 35 - Calculus of a Single Variable

Question

Finding an Equation of a Tangent Line In Exercises \(33-38,\) find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. $$\begin{array}{ll}{\text { Function }} & {\text { Line }} \\ {f(x)=x^{3}} & {3 x-y+1=0}\end{array}$$

Step-by-Step Solution

Answer

The equations of the lines that are tangent to the graph of \(f(x) = x^3\) and parallel to the line \(3x-y+1=0\), are \(y=3x+4\) and \(y=3x-2\).

Step 1: Derive the function

Before a tangent line to \(f(x)=x^{3}\) can be found, we need to find its derivative. The derivative of \(f(x)=x^{3}\) is \(f'(x) = 3x^2\).

Step 2: Determine the slope of the parallel line

The line parallel to \(3x-y+1=0\) would have the same slope. We can solve for the slope by rearranging this equation into slope-intercept form (i.e., \(y = mx + b\)). The equation \(3x-y+1=0\) can be written as \(y=3x+1\), so the slope, \(m\), is 3.

Step 3: Equate the derivative to the slope of the line

We want to find the `x` value where the slope of the tangent to the curve \(f(x)=x^3\) is the same as the slope of the parallel line (\(m=3\)). That means we set the derivative equal to the slope: \(3x^2=3\). This leads to \(x^2=1\), and thus \(x=-1\) and \(x=1\).

Step 4: Find the \(y\)-coordinates

Substitute these x-values into the original function \(f(x)=x^{3}\) to find the corresponding \(y\)-coordinates. Hence, when \(x=-1, y=-1*(-1)^3=-1\) and when \(x=1, y=1*1^3=1\).

Step 5: Use the point-slope form to get the line equations

Use the point-slope form, \(y-y_1=m(x-x_1)\), to find the equations of the tangent lines. Substituting \(m=3\) and the points \((-1,-1)\) and \((1,1)\), you get the tangent lines \(y+1=3(x+1)\) and \(y-1=3(x-1)\), i.e. \(y=3x+4\) and \(y=3x-2\).

Key Concepts

Derivative of a Function

Understanding the derivative of a function is crucial when finding a tangent line. A derivative represents the function's instantaneous rate of change, which is mathematically expressed as the slope at any given point on the function's graph.
For a cubic function like \( f(x) = x^3 \), we find its derivative by applying the power rule. The power rule states that if \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \).
Hence, for \( f(x) = x^3 \), the derivative is \( f'(x) = 3x^2 \).

• This derivative tells us how steep the graph is at any point \( x \).
• It is a necessary element to determine where the tangent line will touch the curve.

Slope of a Line

The slope of a line indicates how steep the line is. It's often represented by the letter \( m \). In the problem, we're dealing with a line in the equation \( 3x - y + 1 = 0 \). To find the slope of this line, we need to rearrange it into the slope-intercept form, which is \( y = mx + b \). Transitioning it gives us \( y = 3x + 1 \), where the slope \( m = 3 \).

• The slope is a crucial component when dealing with parallel lines.
• Parallel lines have identical slopes, which will be crucial for our tangent line calculation.

Point-Slope Form

The point-slope form is an equation format that uses a point on a line and the slope to define the line. It is written as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) are the coordinates of a point on the line.
In our task, once we determine the points \( (-1, -1) \) and \( (1, 1) \) on the function and know the line's slope is 3, we can use this form.

• Insert the slope and the points into the equation.
• This method directly provides the equation of the tangent lines.

For point \( (-1, -1) \), it becomes \( y + 1 = 3(x + 1) \). For point \( (1, 1) \), it's \( y - 1 = 3(x - 1) \). After simplifying, the equations are \( y = 3x + 4 \) and \( y = 3x - 2 \).

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the exact same slope, which is a key feature when determining parallelism.
In the context of finding a tangent line that is also parallel, we need our tangent line's slope to match the given line's slope. In this scenario, both tangent lines derived should have a slope of 3, matching that of the parallel line equation \( y = 3x + 1 \).

• This ensures the tangent line and the given line are parallel.
• The concept of parallelism is integral for constructing geometric relationships on graphs.

Ensuring the slopes match while defining tangent lines guarantees that they run parallel, adding symmetrical and aesthetic structural integrity to the graph.