Problem 76 - Brief Applied Calculus

Question

True or False: \(\frac{d}{d x} f(x / 2)=\frac{f^{\prime}(x / 2)}{2}\)

Step-by-Step Solution

Answer

True, the statement correctly applies the Chain Rule.

Step 1: Understand the Problem

We need to determine whether the given statement about the derivative is true or false. The statement is about the derivative of a function where the argument of the function is scaled (i.e., divided by 2).

Step 2: Recognize the Chain Rule

To differentiate a composite function like \( f(x/2) \), we use the Chain Rule. The Chain Rule states that if you have a function \( h(x) = f(g(x)) \), then the derivative \( h'(x) \) is given by \( f'(g(x)) \cdot g'(x) \).

Step 3: Apply the Chain Rule

Let \( g(x) = x/2 \), so \( g'(x) = 1/2 \). Therefore, by the Chain Rule:\[ \frac{d}{dx} f(x/2) = f'(g(x)) \cdot g'(x) = f'(x/2) \cdot \frac{1}{2} \].

Step 4: Compare with the Given Statement

The expression obtained by applying the Chain Rule is \( \frac{f'(x/2)}{2} \). This matches exactly with the expression given in the problem statement.

Step 5: Conclude

Since the derived expression using the Chain Rule matches the given expression, the statement is indeed true.

Key Concepts

Derivatives: The Fundamental Building Blocks of Calculus

Derivatives are a core concept in calculus. They represent the instantaneous rate of change of a function. Imagine you have a curve on a graph; the derivative at any point tells you the slope of the tangent line at that point.
For example, if you have a function describing the distance traveled over time, the derivative of that function gives the speed at any moment. Derivatives help us understand how systems change, and they appear everywhere in science and engineering.

• Derivatives measure how much a quantity changes as its input changes.
• They are useful for finding extrema (maxima or minima) of functions.
• You can compute derivatives using rules like the product rule, quotient rule, and the chain rule.

In calculus, understanding derivatives is crucial because they form the basis for more advanced topics like integration and differential equations.

Composite Functions: Combining Simplicity into Complexity

Composite functions are made by plugging one function into another. When you're dealing with composite functions, you're essentially building a chain of functions.
Think of composite functions as a process with multiple steps—you take an input, pass it through the first function, then take the output and pass it through the second function, and so on. For example, if you have a function \( f(x) = x^3 \) and another \( g(x) = x + 2 \), then the composite function \( (f \circ g)(x) \) is \( f(g(x)) = (x+2)^3 \).

• Composite functions are expressed as \( f(g(x)) \).
• They help in breaking down complex problems into simpler steps.
• To differentiate these, we use the chain rule.

Understanding composite functions is crucial because they allow you to approach problems methodically and see how changes in one part of a system can influence another.

Calculus: The Mathematical Study of Change

Calculus is the branch of mathematics that focuses on how things change. It provides tools for modeling and analyzing situations where there is change and movement. Calculus helps in predicting future trends based on current and past data.
There are two main branches of calculus: differential calculus (which involves derivatives) and integral calculus (concerned with areas under curves and accumulations). In differential calculus, we learn about finding slopes of curves, while integral calculus is about finding areas and volumes.

• Calculus allows for solving real-world problems in physics, engineering, and beyond.
• It helps in understanding dynamics of systems, like the motion of planets or the growth of populations.
• Central to calculus are derivatives, which describe change, and integrals, which accumulate quantities.

By learning calculus, one gains the ability to predict and manipulate environments mathematically, offering invaluable insights across both theoretical and practical applications.