{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by
f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>
Let f (x) = √x.
a. Find the exact value of f' (4).

검증된 단계별 안내

Step 1: Recall the definition of the derivative for a function f(x). The derivative f'(x) is defined as the limit: f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{f(x+h) - f(x)}{h}\).

Step 2: Identify the function given in the problem. Here, f(x) = \(\sqrt{x}\). We need to find the derivative at x = 4, so we are looking for f'(4).

Step 3: Substitute f(x) = \(\sqrt{x}\) into the derivative definition. This gives us: f'(4) = \(\lim\)_{h \(\to\) 0} \(\frac{\sqrt{4+h}\) - \(\sqrt{4}\)}{h}.

Step 4: Simplify the expression under the limit. To do this, multiply the numerator and the denominator by the conjugate of the numerator: \(\frac{\sqrt{4+h}\) - 2}{h} \(\cdot\) \(\frac{\sqrt{4+h}\) + 2}{\(\sqrt{4+h}\) + 2}.

Step 5: Simplify the resulting expression and evaluate the limit as h approaches 0. This will give you the exact value of f'(4).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Definition of the Derivative

The derivative of a function at a point quantifies the rate at which the function's value changes as its input changes. Mathematically, it is defined as f′(a) = lim h→0 [f(a + h) - f(a)] / h. This limit represents the slope of the tangent line to the function at the point a, providing insight into the function's behavior near that point.

Forward and Backward Difference Quotients

The forward difference quotient is an approximation of the derivative using values of the function at a point and a small increment h, expressed as f' (a) ≈ [f(a + h) - f(a)] / h for h > 0. Conversely, the backward difference quotient uses a decrement, given by f' (a) ≈ [f(a) - f(a - h)] / h for h < 0. These approximations are particularly useful for estimating derivatives when dealing with complex functions or discrete data.

Calculating Derivatives of Functions

To find the exact value of the derivative for a specific function, such as f(x) = √x, one can either apply the definition of the derivative directly or use known derivative rules. For f(x) = √x, the derivative can be calculated using the power rule, where f'(x) = (1/2)x^(-1/2), allowing for straightforward evaluation at any point, such as x = 4.