Problem 79 - Calculus an Applied Approach

Question

True or False? determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } y=(1-x)^{1 / 2}, \text { then } y^{\prime}=\frac{1}{2}(1-x)^{-1 / 2} $$

Step-by-Step Solution

Answer

The statement is False because the correct derivative of the function \(y = (1-x)^{1/2}\) is \(-\frac{1}{2}(1-x)^{-1/2}\), not \(\frac{1}{2}(1-x)^{-1/2}\).

Step 1: Identify the function and stated derivative

The function given is \(y = (1-x)^{1/2}\) and the stated derivative is \(y' = \frac{1}{2}(1-x)^{-1/2}\).

Step 2: Determine the derivative of the function

Use the power rule of derivatives: \(d/dx[a*f(x)]^{n}=n*a*f(x)^{n-1}*f'(x)\) where \('a'\) is a constant, \('f(x)'\) is the function and \('f'(x)'\) is the derivative of \('f(x)'\). For the function \(y=(1-x)^{1/2}\), \(n=1/2\), \(a=-1\), and \(f(x)=x\). Hence, the derivative is calculated as follows: \(y' = \frac{1}{2} * -1 * (1-x)^{-1/2} * -1 = -\frac{1}{2}(1-x)^{-1/2}\).

Step 3: Compare the calculated derivative and the stated derivative

The calculated derivative, \(-\frac{1}{2}(1-x)^{-1/2}\), does not match the stated derivative, \(\frac{1}{2}(1-x)^{-1/2}\). Therefore, the statement is false.

Key Concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This field forms the basis for much of modern mathematics and is instrumental in many scientific and engineering problems. One of the most fundamental concepts in calculus is the derivative. A derivative represents the rate at which a function is changing at any given point and is a critical tool in finding the slope of a curve, optimizing functions, and solving differential equations. By understanding derivatives, we can predict and model the behavior of dynamically changing systems in real-world applications.

Derivative Calculation

Derivative calculation is an essential part of calculus that involves finding the derivative of a function. Techniques for computing derivatives vary depending on the complexity of the function. The power rule, product rule, quotient rule, and chain rule are some of the basic strategies used to tackle different types of functions. Computing derivatives accurately requires a clear understanding of these rules and their appropriate applications. Incorrectly applying these rules can lead to errors, hence it's crucial to practice derivative calculation systematically to gain proficiency.

False Statements in Calculus

In the study of calculus, it's crucial to recognize false statements as they can lead to misunderstandings and incorrect applications of mathematical principles. A common source of confusion comes from misapplying rules and forgetting to consider negative signs, constants, or more complex parts of a function when using derivative formulas. For instance, overlooking the chain rule or applying the power rule incorrectly could lead to a false statement about the derivative of a function. It is important to scrutinize each step in the calculation process to ensure accuracy and to understand why a statement might be false to prevent similar mistakes.

Power Rule Examples

The power rule is a fundamental derivative rule used to find the derivative of a function with the form \(f(x)=x^n\), where \(n\) is a real number. According to the power rule, the derivative of \(f\) with respect to \(x\) is \(f'(x)=nx^{n-1}\). For example, if you have \(f(x)=x^3\), then the derivative is \(f'(x)=3x^{3-1}=3x^2\). However, when using the power rule, it's crucial to apply the chain rule if the function is more complex, as in the case of \(f(x)=(1-x)^{1/2}\). Here, the outer function is \(x^{1/2}\) and the inner function is \(1-x\), so the derivative involves the derivative of the outer function multiplied by the derivative of the inner function, leading to a negative sign in the correct derivative \(f'(x)=-\frac{1}{2}(1-x)^{-1/2}\). Understanding these nuances is key to mastering the power rule in calculus.