Problem 39 - Calculus

Question

Estimate the change in \(y\) for the given change in \(x\).\(y=f(x), f^{\prime}(12)=30, x\) increases from 12 to 12.2.

Step-by-Step Solution

Answer

The change in y is approximately 6.

Step 1: Identify Given Information

We are given that the function is represented as \( y = f(x) \), and the derivative at \( x = 12 \) is \( f'(12) = 30 \). We also know \( x \) changes from 12 to 12.2.

Step 2: Understand the Meaning of the Derivative

The derivative \( f'(x) \) gives us the rate of change of \( y \) with respect to \( x \). In this problem, \( f'(12) = 30 \) means that for very small changes in \( x \), \( y \) changes approximately by \( 30 \) times that change in \( x \).

Step 3: Calculate the Change in \( x \)

The change in \( x \) is from 12 to 12.2. Therefore, the change \( \Delta x = 12.2 - 12 = 0.2 \).

Step 4: Estimate the Change in \( y \)

The estimated change in \( y \), \( \Delta y \), can be calculated using the derivative: \( \Delta y \approx f'(12) \cdot \Delta x = 30 \cdot 0.2 \).

Step 5: Perform the Calculation

Multiply the values from the previous step: \( 30 \cdot 0.2 = 6 \). Therefore, the estimated change in \( y \) is 6.

Key Concepts

Understanding Derivatives

In differential calculus, derivatives play a crucial role. A derivative, denoted as \( f'(x) \), represents the rate at which a function's value changes as its input changes. Essentially, the derivative is a tool that helps us understand how a function behaves at different points.
For example, if we have a function \( y = f(x) \), the derivative \( f'(x) \) gives us the slope of the tangent line to the function at any given point. This slope tells us how steeply the function is rising or falling at that particular point.

• Derivatives can be thought of as a measure of sensitivity. This is how responsive the function is to changes in \( x \).
• A large derivative value indicates a steep slope, while a small derivative value indicates a gentle slope.
• Understanding the concept of derivatives is essential for analyzing and predicting the behavior of functions.

Rate of Change

The phrase "rate of change" is commonly used in mathematics to describe how one quantity changes in relation to another.
In the context of our topic, the derivative is used as a tool to measure this rate of change. It essentially helps to quantify how the dependent variable \( y \) changes for a small change in the independent variable \( x \).
In our exercise, \( f'(12) = 30 \) indicates that for every small unit change in \( x \) near 12, \( y \) changes by about 30 times that change.

• This means that the function is quite sensitive to changes at \( x = 12 \).
• By understanding the rate of change, we can make predictions about the behavior of functions in different contexts, such as physics, economics, or biology.

Grasping the concept of rate of change is vital for describing real-world phenomena through mathematical models.

Exploring Functions

Functions are a cornerstone of mathematics and are often depicted as \( y = f(x) \).
A function defines a specific relationship between inputs (\( x \)) and outputs (\( y \)). For each input \( x \), there is a unique output \( y \) determined by the function's rule.

• Functions can model a wide range of natural and man-made processes. They can be linear, quadratic, exponential, and so forth.
• The goal of studying functions is to understand these relationships and predict the outcomes when inputs change.
• The function & its derivative together help us decide how one change affects another. They give important insights into the nature of the changes.

In the exercise given, the function \( y = f(x) \) helps to establish how \( y \) varies as \( x \) adjusts from 12 to 12.2.
Learning to distinguish these relationships is vital for solving real-world problems using mathematical tools.