Problem 61 - Calculus Single Variable

Question

Find the equation of the line that is tangent to the graph of \(f(x)=3 x^{3}+12\) and that passes through the origin.

Step-by-Step Solution

Answer

The tangent line is \(y = 9\sqrt[3]{4} x\).

Step 1: Find the Derivative

Start by differentiating the function \(f(x) = 3x^3 + 12\) to find \(f'(x)\), which represents the slope of the tangent line. Using the power rule, you get \(f'(x) = 9x^2\).

Step 2: Tangent Line Slope at a Specific Point

For the tangent line passing through the origin, the equation will be of the form \(y = mx\). Here, \(m\) is the slope that matches the derivative at some point \(x = a\). So, \(m = 9a^2\).

Step 3: Tangent Line Passes Through the Origin

Since the tangent line passes through the origin \((0, 0)\), substitute the point \((a, f(a))\) on the curve into the line equation \(y = 9a^2x\) to get \(3a^3 + 12 = 9a^2 \cdot a\). This simplifies to \(3a^3 + 12 = 9a^3\).

Step 4: Solving the Equation for \(a\)

Solve for \(a\) by rearranging and simplifying the equation from Step 3: \(3a^3 + 12 = 9a^3\) becomes \(12 = 6a^3\), which simplifies to \(a^3 = 2\). Then \(a = \sqrt[3]{2}\).

Step 5: Determine the Equation of the Tangent Line

The slope of the tangent line at \(x = \sqrt[3]{2}\) is \(m = 9 \times (\sqrt[3]{2})^2 = 9(\sqrt[3]{4})\). Hence, the equation of the tangent line is \(y = 9 \sqrt[3]{4} x\).

Key Concepts

Tangent Line

A tangent line is a straight line that just touches a curve at a specific point. It does not cross the curve at this point, but only "grazes" it.
The key purpose of a tangent line is to show the direction the curve is heading at that specific point.
For any given curve described by a function, the tangent line at a point can be thought of as the immediate 'linearity' the curve is exhibiting at that exact location.

To find this line, one identifies the slope of the curve at that point and then determines the equation of the line with that slope passing through the designated point on the curve. In basic terms:

• The tangent line is an accurate approximation to the function's behavior near the point.
• It is useful in various applications, such as optimization and curve sketching.

When you see a curve like a hilly road, the tangent line is like laying a flat plank on the road surface, just touching it without crossing.

Power Rule

The power rule is a basic yet powerful principle in calculus used to differentiate functions. If you have a function of the form \(x^n\), the power rule states that its derivative is \(nx^{n-1}\).
Leveraging this rule simplifies differentiation tremendously, providing a quick way to find the slope of a curve at any point for polynomial functions.

Let's highlight its usage in a practical setting:

• If \(f(x) = x^2\), the derivative \(f'(x) = 2x\).
• Similarly, if \(g(x) = x^5\), then \(g'(x) = 5x^4\).

The power rule enables easy computation of the rate of change for functions expressed as polynomials. In our exercise problem, we apply this to find the derivative of a cubic function, which then helps us to calculate the specific slope of the curve at a point.

Slope of a Curve

The slope of a curve at a specific point expresses how steep or flat the curve is at that exact spot.
Mathematically, this slope is obtained by taking the derivative of a function at a particular point.

The concept of slope for curves is a natural extension of the idea of slope for straight lines, which is typically calculated by "rise over run."
With curves, we look at infinitesimally small sections to determine this slope.

• If the slope is positive, the curve ascends as you move from left to right.
• If the slope is negative, the curve descends.
• If the slope is zero, we have a flat or horizontal spot on the curve – like the top of a hill or the bottom of a valley.

In the context of our problem, the derivative \(f'(x)\) tells us the slope of the cubic curve at various points, which is crucial for finding where a tangent could effectively model the curve.

Cubic Function

Cubic functions represent equations of the form \(ax^3 + bx^2 + cx + d\). These are polynomials of degree three and are characterized by their S-shaped curve.
They may have one or two turns, which are referred to as critical points where the direction of the curve changes.

This type of function is especially interesting due to its behavior and diversity in shapes:

• Cubic functions can have up to three real roots, signifying where they intersect the x-axis.
• Their dynamic nature makes them useful in physics and engineering, modeling phenomena like population growth or financial projections.

In the exercise, the given function \(f(x) = 3x^3 + 12\) is cubic, and differentiating it helps unveil just how swiftly its slope changes at any point, allowing us to determine tangents.