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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 31b
Capitolo 3, Problema 31b

Equations of tangent lines by definition (2)
b. Determine an equation of the tangent line at P.
f(x) = √x+3; P (1,2)

Guida verificata passo dopo passo
1
Step 1: Understand that the equation of a tangent line to a curve at a given point is given by the formula: y - y_1 = m(x - x_1), where m is the slope of the tangent line, and (x_1, y_1) is the point of tangency.
Step 2: Identify the function f(x) = \(\sqrt{x}\) + 3 and the point P(1, 2). Here, x_1 = 1 and y_1 = 2.
Step 3: To find the slope m of the tangent line, calculate the derivative of the function f(x). The derivative f'(x) represents the slope of the tangent line at any point x.
Step 4: Differentiate f(x) = \(\sqrt{x}\) + 3. The derivative f'(x) = \(\frac{1}{2\sqrt{x}\)}. This is because the derivative of \(\sqrt{x}\) is \(\frac{1}{2\sqrt{x}\)}, and the derivative of a constant is 0.
Step 5: Evaluate the derivative at x = 1 to find the slope of the tangent line at P. Substitute x = 1 into f'(x) to get m = f'(1).

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Tangent Line Definition

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. Mathematically, it represents the instantaneous rate of change of the function at that point, which is equivalent to the derivative of the function evaluated at that point.
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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 calculated as the limit of the average rate of change of the function as the interval approaches zero. For the function f(x) = √(x + 3), the derivative will provide the slope of the tangent line at point P.
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Derivatives

Point-Slope Form

The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for writing the equation of a tangent line once the slope (derivative) and the point of tangency are known.
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Slope-Intercept Form
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