Problem 83 - Calculus and its applications

Question

For each function, find the points on the graph at which the tangent line has slope 1 . $$ y=\frac{1}{3} x^{3}-x^{2}-4 x+1 $$

Step-by-Step Solution

Answer

The tangent lines with slope 1 are at the points \((1 + \sqrt{6}, y_1)\) and \((1 - \sqrt{6}, y_2)\) on the graph.

Step 1: Find the Derivative

The derivative of a function gives the slope of the tangent line at any given point. For the function \[ y = \frac{1}{3}x^3 - x^2 - 4x + 1, \]we need to find its derivative, which is calculated by differentiating each term individually. The derivative is:\[ y' = \frac{d}{dx}\left( \frac{1}{3}x^3 \right) - \frac{d}{dx}(x^2) - \frac{d}{dx}(4x) + \frac{d}{dx}(1). \]Therefore, \[ y' = x^2 - 2x - 4. \]

Step 2: Set Derivative Equal to 1

We want to find the points where the slope of the tangent line is 1. This means solving for:\[ y' = 1. \]Substitute the derivative found in Step 1:\[ x^2 - 2x - 4 = 1. \]

Step 3: Solve the Equation

Rearrange the equation to solve for x:\[ x^2 - 2x - 4 - 1 = 0 \]\[ x^2 - 2x - 5 = 0. \]Use the quadratic formula\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 1\), \(b = -2\), and \(c = -5\). Plug these values:\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1}. \]This simplifies to:\[ x = \frac{2 \pm \sqrt{4 + 20}}{2} \]\[ x = \frac{2 \pm \sqrt{24}}{2} \]\[ x = \frac{2 \pm 2\sqrt{6}}{2} \]\[ x = 1 \pm \sqrt{6}. \]

Step 4: Find Corresponding Y-Values

Using the x-values found in Step 3, substitute back into the original function to find the y-values. For \(x = 1 + \sqrt{6}\):\[ y = \frac{1}{3}(1 + \sqrt{6})^3 - (1 + \sqrt{6})^2 - 4(1 + \sqrt{6}) + 1. \]And for \(x = 1 - \sqrt{6}\):\[ y = \frac{1}{3}(1 - \sqrt{6})^3 - (1 - \sqrt{6})^2 - 4(1 - \sqrt{6}) + 1. \]Calculate these to find the corresponding y-values.

Key Concepts

Derivative

The concept of a derivative is central in calculus, representing the rate at which a function changes. In simpler terms, it gives us the slope of the tangent line to the graph of the function at any point.

• To find the derivative, you differentiate each term of the function separately.
• The derivative of a constant is 0 because constants do not change.
• The power rule is often used where the derivative of \( x^n \) is \( nx^{n-1} \).

For the function \( y = \frac{1}{3}x^3 - x^2 - 4x + 1 \), the derivative simplifies to \( y' = x^2 - 2x - 4 \) using these rules.
Understanding derivatives allows us to analyze how functions behave and change over time, making it a powerful tool in mathematics and various applications.

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it at that point. The slope of this tangent line is the derivative of the function at that specific point.

• For example, in physics, a tangent line can represent an object's instantaneous velocity at a certain time.
• Graphically, it gives us a linear approximation of the function at that point.

To find a tangent line with a specific slope, say 1, set the derivative equal to that slope: \( y' = 1 \). Solving this helps identify points on the graph where the tangent line has that specific slope.
Recognizing the role of tangent lines bridges our understanding between linear and non-linear functions.

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). When unable to easily factor or when the quadratic does not factor neatly, the quadratic formula offers a systematic approach:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation.
• This formula helps find the \( x \)-values where the function takes specific slopes, which is essential in determining points with a particular tangent line slope.

For instance, solving \( x^2 - 2x - 5 = 0 \) using this formula yields \( x = 1 \pm \sqrt{6} \), revealing precisely where the slope of 1 can occur.
The quadratic formula ensures that you never miss finding the roots, which are key to comprehending the intersections and behavior of quadratic functions on a graph.

Slope

The slope of a line in mathematics represents its steepness and direction. In terms of functions, the slope at a point on the curve is given by the derivative, illustrating how much \( y \) changes for a unit change in \( x \). Consider these points:

• When the slope is positive, the line rises from left to right; when negative, it falls.
• A zero slope indicates the line is horizontal, showing no change in \( y \) for any change in \( x \).

Setting the derivative equal to a specific value, like 1, identifies where on the graph that slope occurs. This pinpoints exactly where the tangent line is parallel to a line with that slope value.
Understanding the concept of slope is crucial when dealing with linear relationships within calculus. It helps unravel the rate and direction at which changes occur, giving deeper insights into dynamic systems.