Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 3 - Derivatives

Problema 3.1.33

Capitolo 3, Problema 3.1.33

Show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b).

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To show that the line y = mx + b is its own tangent line at any point (x₀, mx₀ + b), we first need to understand the concept of a tangent line. A tangent line to a curve at a given point is a straight line that just touches the curve at that point. For a line, the tangent line at any point on the line is the line itself.

The slope of the line y = mx + b is m. The derivative of y with respect to x, which gives the slope of the tangent line to the curve at any point, is also m. This is because the derivative of mx with respect to x is m, and the derivative of a constant b is 0.

Evaluate the derivative at the point (x₀, mx₀ + b). Since the derivative is constant and equal to m, the slope of the tangent line at any point (x₀, mx₀ + b) is m.

The equation of the tangent line at the point (x₀, mx₀ + b) can be written using the point-slope form: y - (mx₀ + b) = m(x - x₀).

Simplify the equation of the tangent line: y - mx₀ - b = mx - mx₀, which simplifies to y = mx + b. This shows that the tangent line at any point (x₀, mx₀ + b) is indeed the line y = mx + b itself.

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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 at that point is equal to the derivative of the function at that point. In this case, we are examining how the line itself behaves as a tangent to its own graph.

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Slopes of Tangent Lines

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. For a linear function like y = mx + b, the derivative is constant and equal to the slope m. This means that at any point on the line, the slope of the tangent line is the same as the slope of the line itself.

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Derivatives

Linear Functions

A linear function is a polynomial function of degree one, represented in the form y = mx + b, where m is the slope and b is the y-intercept. Linear functions graph as straight lines, and their properties, such as constant slope and direct proportionality, make them unique in that they are their own tangent lines at every point along the line.

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Linearization

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Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.

f(x) = x⁻¹, x₀ = 0.5, dx = 0.1

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Find the derivatives of the functions in Exercises 1–42.

𝔂 = ( √ x )²

( 1 + x )

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In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.

(y - x)² = 2x + 4, (6, 2)

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Differential Estimates of Change

In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.

The change in the surface area S = 6x² of a cube when the edge lengths change from x₀ to x₀ + dx

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Normal lines to a parabola Show that if it is possible to draw three normal lines from the point (a, 0) to the parabola x = y² shown in the accompanying diagram, then a must be greater than 1/2. One of the normal lines is the x-axis. For what value of a are the other two normal lines perpendicular?

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Implicit Differentiation

In Exercises 43–50, find by implicit differentiation.

xy + 2x + 3y = 1

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