Problem 15 - Single Variable Calculus

Question

Find the instantaneous rate of change of the given function when \(x=a .\) \(f(x)=2 x^{2}+1 ; \quad a=1\)

Step-by-Step Solution

Answer

The instantaneous rate of change of the function \(f(x) = 2x^2 + 1\) when \(x = a = 1\) is \(4\).

Step 1: Find the first derivative

To find the first derivative of \(f(x) = 2x^2 + 1\) with respect to \(x\), we apply the power rule: \(f'(x) = \frac{d}{dx} (2x^2 + 1) = \frac{d}{dx}(2x^2) + \frac{d}{dx}(1)\)

Step 2: Apply the power rule

The power rule states that if \(f(x) = x^n\), where \(n\) is a constant, then \(f'(x) = nx^{n-1}\). Applying this rule to our expression: \(f'(x) = 2(2x^{2-1}) + 0 = 4x\) Now we have the first derivative of the function, \(f'(x) = 4x\).

Step 3: Evaluate the first derivative at \(x = a\)

We are given that \(a = 1\). To find the instantaneous rate of change at \(x = 1\), we simply plug in \(a\) into the first derivative: \(f'(a) = f'(1) = 4(1) = 4\) Thus, the instantaneous rate of change of the function \(f(x) = 2x^2 + 1\) when \(x = a = 1\) is \(4\).

Key Concepts

Understanding the First Derivative

Finding the first derivative is a crucial concept in calculus. It represents the rate at which a function is changing at any given point. Think of it like speed in driving; while the function tells you where you are, the derivative tells you how fast you're getting there.

In this exercise, the original function is \( f(x) = 2x^2 + 1 \). To find the first derivative, we need to differentiate the function with respect to \( x \). Essentially, this means we're looking to determine the function that describes the slope at any point on the curve. We use differentiation techniques such as the power rule to achieve this.

Applying the Power Rule to Differentiate

The power rule is a simple and efficient way to find the derivative of polynomial functions. It states that for any function in the form \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \). This rule helps break down complex expressions into manageable parts.

In this problem, we apply the power rule to the term \( 2x^2 \) in the function \( f(x) = 2x^2 + 1 \). Using the power rule:

• The derivative of \( 2x^2 \) is \( 2 \times 2x^{2-1} = 4x \).
• The derivative of a constant, like \( 1 \), is \( 0 \). This is because constants do not change, so their rate of change is zero.

Thus, the first derivative of the function is \( f'(x) = 4x \), describing how \( f(x) \) changes with \( x \).

Evaluating Derivatives at a Given Point

Once we have the first derivative formula, \( f'(x) = 4x \), the next step is to determine the instantaneous rate of change at a specific point. For this, we substitute the given value of \( x \), which is \( a = 1 \), into the derivative.

By computing \( f'(1) \), we find:

• \( f'(1) = 4 \times 1 = 4 \)

This result tells us that at \( x = 1 \), the function's rate of change is 4. This means if you were to look at a small segment of the curve at this point, it would appear to climb steadily at a slope of 4.

An Introduction to Single Variable Calculus

Single variable calculus deals with functions that depend on a single variable. This area of math helps you understand changes and rates of change, concepts that are widely applicable in science and engineering.

The concept of a derivative, specifically the first derivative, is a cornerstone in this field. It allows us to quantify how a function behaves, detailing aspects like whether it's increasing or decreasing at a particular point. Using tools like the power rule, you can break down and analyze relationships within the function.

• Single variable calculus establishes fundamental techniques that pave the way for more advanced topics.
• It provides the mathematical framework to tackle real-world problems involving rates of change and motion.

Mastering these basics allows for greater exploration into richer mathematical concepts and applications.