Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 3.1.62a

Capitolo 3, Problema 3.1.62a

{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).

Guida verificata passo dopo passo

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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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.

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Percorso guidato

05:43

Definition of the Definite Integral

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.

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The Quotient Rule

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.

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06:30

Derivatives of Other Trig Functions

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