Problem 22 - Calculus Single Variable

Question

Find the slope of the tangent line to the graph of the given function at the given point \(P\). $$ f(x)=x^{3}-2 x \quad P=(1,-1) $$

Step-by-Step Solution

Answer

The slope of the tangent line at the point \( P = (1, -1) \) is 1.

Step 1: Understand the Problem

We want to find the slope of the tangent line to the curve defined by the function \( f(x) = x^3 - 2x \) at a specific point \( P = (1, -1) \). The slope of the tangent line at any point on a curve is given by the derivative of the function at that point.

Step 2: Compute the Derivative

To find the slope of the tangent line, we first need to compute the derivative of \( f(x) \). The function given is \( f(x) = x^3 - 2x \). The derivative, \( f'(x) \), is computed as follows: \[ f'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(2x) = 3x^2 - 2 \].

Step 3: Evaluate the Derivative at the Point of Interest

Now that we have the derivative \( f'(x) = 3x^2 - 2 \), we need to evaluate it at the point \( x = 1 \) to find the slope of the tangent line at \( P = (1, -1) \).\[ f'(1) = 3(1)^2 - 2 = 3(1) - 2 = 1 \].

Key Concepts

Tangent Line

To understand a tangent line, imagine gently touching a curve at just a single point. A tangent line is that hypothetical line which "just touches" the curve without slicing through it. It represents the direction in which the curve is heading at that point.

• A tangent line makes contact with the curve at precisely one point, known as the point of tangency.
• In this specific exercise, we are looking for the tangent line to the curve represented by the function \(f(x) = x^3 - 2x\). The point of interest is \(P = (1, -1)\). This means the line should graze the curve exactly at \(x = 1\).

The mathematical expression of a tangent line at a given point provides the best linear approximation or snapshot of the curve at that specific point. Computing this line is essential for comprehending the behavior of the curve locally around that point.

Slope of a Curve

The term "slope of a curve" refers to the steepness or inclination of the curve at a certain point. Rather than having a constant slope like a straight line, a curve's slope might vary depending on where you look at it.

• In calculus, the slope at a specific point is captured by the derivative of the function at that point.
• In our exercise, we calculated the derivative of \( f(x) = x^3 - 2x \) to find the formula for the slope. We found: \( f'(x) = 3x^2 - 2 \).

At the point \(x = 1\), the slope is \( f'(1) = 1 \). Therefore, the slope of the curve at \(P = (1, -1)\) is 1. This means the curve increases at the rate of one unit up for every unit you move to the right at that point, indicating a gentle incline.

Differentiation

Differentiation is the fundamental tool used in calculus to find the slope of a curve. It involves a set of mathematical rules to determine how a function changes as its input changes.

• Through differentiation, we derive another function called the derivative, which represents the rate of change of the original function.
• For our function, \(f(x) = x^3 - 2x\), differentiation helped us calculate \(f'(x) = 3x^2 - 2\).
By differentiating, and then calculating the derivative at a specific point, we not only determine the slope of the tangent line but also get valuable insight into the behavior and trend of the curve at that point. Differentiation is widely used in various fields, including physics, engineering, and economics, to model and analyze dynamic systems and phenomena.