Problem 4 - 5 Steps to A 5: AP Calculus AB 2021

Question

Find the point on the graph of \(y=|x|^{3}\) such that the tangent at the point is parallel to the line \(y-12 x=3\).

Step-by-Step Solution

Answer

The point on the graph of \(y = |x|^3\) with a tangent parallel to the line \(y - 12x = 3\) is \((2, 8)\).

Step 1: Find the derivative of \(y=f(x) = |x|^3\)

Since \(y = |x|^3\), we need to treat this function as a piecewise function to find the derivative. $$ f(x) = \begin{cases} x^3 & \text{if } x \geq 0 \\ -x^3 & \text{if } x < 0 \end{cases} $$ Now find the derivative for both cases. $$ f'(x) = \begin{cases} 3x^2 & \text{if } x \geq 0 \\ -3x^2 & \text{if } x < 0 \end{cases} $$

Step 2: Find the slope of the given line

The given line equation is \(y - 12x = 3\). In order to find the slope, we need to rewrite this equation in the slope-intercept form (\(y = mx+b\)) where m is the slope. $$ y = 12x + 3 $$ Now we can easily see that the slope (m) of this line is 12.

Step 3: Set the derivative equal to the slope of the given line and solve for x

Set the derivative equal to the slope of the given line (12). $$ f'(x) = 12 $$ Solve the equation for both positive and negative cases. For the positive case (\(3x^2 = 12\)): $$ 3x^2 = 12 \Rightarrow x^2 = 4 \Rightarrow x = \pm2 $$ For the negative case (\(-3x^2 = 12\)): There is no solution for the negative case, as it represents a downward slope, and our given line has an upward slope. Thus, \(x = 2\) is the x-coordinate of the desired point.

Step 4: Plug the x-coordinate back into the original function to find the y-coordinate

Plug \(x = 2\) into the original function \(y = |x|^3\) to find the corresponding y-coordinate. $$ y = |2|^3 = 2^3 = 8 $$ The y-coordinate of the desired point is 8.

Step 5: Final Answer

The point on the graph of \(y = |x|^3\) such that the tangent at that point is parallel to the line \(y - 12x = 3\) is \((2, 8)\).

Key Concepts

Derivatives

In calculus, derivatives represent the rate at which a function is changing at any given point. They are essentially a way to calculate the slope of the tangent line to the function graph at a specific point. To find the derivatives, especially for functions like \(y=|x|^3\), it is essential to understand how they can behave differently over various intervals. For functions involving the absolute value, you often deal with piecewise functions.

• The derivative tells us how steep a slope of a graph is at any instant.
• In finding derivatives of \(y=|x|^3\), we need to break it into two parts based on the piecewise definition.
• Derivatives can indicate increasing or decreasing nature of functions over intervals.

For this function, \(f(x)\) is equal to \(x^3\) when \(x \geq 0\) and \(-x^3\) when \(x < 0\). Calculating the derivative in both cases gives \(3x^2\) and \(-3x^2\), respectively. Knowing how to differentiate piecewise functions is crucial for such problems.

Tangent Line

A tangent line to a curve at a given point is a straight line that "just touches" the curve at that point. It has the same slope as the curve at that location.

Calculating the tangent line involves both finding the derivative and understanding the relationship this line has with the curve. Once you solve for the derivative, the value indicates the slope of this tangent.

• Tangent lines are vital for assessing how quickly a graph is climbing or descending at any point.
• To find where a tangent line is parallel to a given line, set the slope equal to the derivative’s slope.

In the exercise, we find the tangent line by setting the derived gradient of our function equal to the slope of the line \(y - 12x = 3\). This gives us understanding of tangency and parallel slopes.

Slope

The slope of a line is a measure of its steepness. Expressed as rise over run, the slope essentially tells us how much the line goes up (or down) for each unit it moves horizontally.
For a line equation like \(y - 12x = 3\), converting it into slope-intercept form \(y = mx + b\) reveals that the slope \((m) = 12\).

• A positive slope indicates an upward direction, while a negative one points downward.
• For a tangent line to be parallel to another, their slopes must be equal.

In the exercise, this slope was used to find where on the \(y = |x|^3\) curve the tangent is exactly parallel to this line. Understanding this helps identify the point of tangency, ensuring correct position on the graph.

Piecewise Functions

Piecewise functions are those that have different rules or expressions for different intervals of the independent variable. For \(y = |x|^3\), the function changes depending on whether \(x\) is positive or negative.

• Piecewise functions allow us to include absolute values and other conditional calculations smoothly.
• They enable us to handle functions that may behave differently across different portions of their domains.

In calculus, when determining derivatives of piecewise functions like \(y = |x|^3\), distinct intervals for \(x\) require individual treatment. The key is understanding how each part of the piece function relates to derivatives and graph behavior. This kind of function greatly aids in modeling real-world scenarios that involve changing conditions.