Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problema 28a

Capitolo 3, Problema 28a

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x)= 1/(2x + 1); P (0,1)

Guida verificata passo dopo passo

Step 1: Recall the definition of the derivative as the slope of the tangent line at a point. The derivative of a function f at a point x=a is given by the limit: \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).

Step 2: Identify the function and the point of interest. Here, \( f(x) = \frac{1}{2x + 1} \) and the point \( P \) is \( (0, 1) \). We need to find \( f'(0) \).

Step 3: Substitute \( a = 0 \) into the derivative formula: \( f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} \).

Step 4: Calculate \( f(0+h) \) and \( f(0) \). We have \( f(0) = \frac{1}{2(0) + 1} = 1 \) and \( f(h) = \frac{1}{2h + 1} \).

Step 5: Substitute these into the limit expression: \( f'(0) = \lim_{h \to 0} \frac{\frac{1}{2h + 1} - 1}{h} \). Simplify the expression inside the limit and evaluate the limit as \( h \to 0 \) to find the slope of the tangent line.

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Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is crucial for understanding how the function behaves locally.

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Slopes of Tangent Lines

Derivative

The derivative of a function at a point quantifies how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this context, finding the derivative of f(x) will provide the slope of the tangent line at point P.

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Derivatives

Limit Definition of Derivative

The limit definition of the derivative states that the derivative f'(a) at a point a is the limit of the difference quotient as h approaches zero: f'(a) = lim (h -> 0) [(f(a+h) - f(a))/h]. This definition is fundamental for calculating the slope of the tangent line using the function's values around the point of interest.

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Definition of the Definite Integral

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