Problem 68 - Calculus Volume 1

Question

The given limit represents the derivative of a function \(y=f(x)\) at \(x=a\). Find \(f(x)\) and \(a\). $$ \lim _{h \rightarrow 0} \frac{(1+h)^{2 / 3}-1}{h} $$

Step-by-Step Solution

Answer

\( f(x) = x^{2/3} \), \( a = 1 \).

Step 1: Recognize the Limit Definition of Derivative

The given limit is in the form \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \), which is the definition of the derivative of a function \( f(x) \) at \( x = a \). Here, \( f(a+h) = (1+h)^{2/3} \) and \( f(a) = 1 \).

Step 2: Identify the Function and the Value of 'a'

From Step 1, it's evident that \( f(x) = x^{2/3} \) and \( f(1) = 1 \). Therefore, the function \( f(x) \) is \( x^{2/3} \), and \( a = 1 \).

Step 3: Verify by Computing the Derivative

To verify, let's compute the derivative of \( f(x) = x^{2/3} \). The derivative is \( f'(x) = \frac{2}{3} x^{-1/3} \). Evaluating this at \( x = 1 \) gives \( f'(1) = \frac{2}{3} \cdot 1^{-1/3} = \frac{2}{3} \).

Step 4: Confirm the Original Limit Equals the Derivative

The limit \( \lim_{h \to 0} \frac{(1+h)^{2/3} - 1}{h} \) should equal \( f'(1) = \frac{2}{3} \). Since we've shown that this is correct, our values for \( f(x) \) and \( a \) are verified.

Key Concepts

Understanding the Derivative

The derivative is a fundamental concept in calculus, representing the rate at which a function is changing at any point. It is the primary tool for mathematical modeling of change. To find the derivative of a function at a particular point, we use a limit process. The derivative of a function \( f(x) \) at point \( x = a \) is determined using the formula:\[\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\]This equation allows us to calculate the instantaneous rate of change, or the slope of the tangent line, to the curve at that specific point. By understanding how to apply this limit, we can evaluate the behavior of functions in various contexts. It is a gateway to much deeper and nuanced analyses in calculus.

Evaluating the Function

Function evaluation involves finding the value of a function for a particular input. This is usually done by substituting the input value into the function. In the given example, the function is \( f(x) = x^{2/3} \). Evaluating at a specific point like \( x = 1 \) involves calculating \( f(1) = 1^{2/3} = 1 \).

• First, identify the function from the given limit problem. Here, \((1 + h)^{2/3}\) matches the function form \( f(a + h) \).
• Notice that when \( h = 0 \), the value is 1, which correlates to finding \( f(a) \).
• Therefore, it becomes apparent that \( a = 1 \) and \( f(x) = x^{2/3} \).

Understanding how to evaluate functions is crucial as it serves as the foundation for finding derivatives and integrals.

Defining the Limit

The concept of a limit is central to calculus and analysis. Limits describe the behavior of a function as its inputs approach a particular value. In the context of derivatives, limits help us find instantaneous rates of change. The definition used in the derivative calculation is:\[\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}\]It captures how the function approaches a particular value as the input variable gets infinitely close to a point. In our example, we had to find the limit of \( \frac{(1+h)^{2/3}-1}{h} \) as \( h \to 0 \). This limit reveals the function's behavior at \( a = 1 \), which gives us the derivative and, consequently, the rate of change at that point.

Verifying the Derivative

Verification of the derivative involves checking if the computed derivative matches the initial formulation given by the limit. In our example, we first determined the function and point \( a \) using the provided limit.

• First, we verify by computing the derivative using differentiation rules. For the function \( f(x) = x^{2/3} \), the derivative \( f'(x) \) is calculated as \( \frac{2}{3} x^{-1/3} \).
• Next, evaluate this derivative at \( x = 1 \) to see if it matches the limit's result. Here, \( f'(1) = \frac{2}{3} \), which confirms the limit given in the problem statement.
• This matching step is critical in confirming that our function and point \( a \) are correctly identified from the limit definition.

Confirming that these values are consistent ensures that our calculus process and results are accurate.