Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.3.155a

Capitolo 7, Problema 7.3.155a

155. Which is bigger, πᵉ or e^π?
Calculators have taken some of the mystery out of this once-challenging question.
(Go ahead and check; you will see that it is a very close call.)
You can answer the question without a calculator, though.
a. Find an equation for the line through the origin tangent to the graph of
y = ln(x).

Guida verificata passo dopo passo

Identify the function given: \(y = \ln(x)\).

Find the derivative of \(y = \ln(x)\) to get the slope of the tangent line at any point \(x\). The derivative is \(y' = \frac{1}{x}\).

Since the tangent line passes through the origin (0,0), find the point of tangency \(x = a\) where the tangent line touches the curve \(y = \ln(x)\).

The slope of the tangent line at \(x = a\) is \(\frac{1}{a}\), and the tangent line passes through \((a, \ln(a))\) and the origin \((0,0)\).

Use the point-slope form of the line through the origin with slope \(\frac{1}{a}\): \(y = \frac{1}{a} x\). This line is tangent to \(y = \ln(x)\) at \(x = a\).

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Comparing Exponential Expressions

To compare expressions like π^e and e^π without a calculator, use logarithms to transform the comparison into a difference of products involving natural logs. This approach leverages properties of exponents and logarithms to analyze which value is larger.

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Simplifying Exponential Expressions

Tangent Line to a Curve

A tangent line to a curve at a point touches the curve without crossing it locally and has the same slope as the curve at that point. For y = ln(x), the slope at any x is given by the derivative 1/x, which helps find the equation of the tangent line.

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Percorso guidato

05:13

Slopes of Tangent Lines

Derivative of the Natural Logarithm Function

The derivative of y = ln(x) is y' = 1/x, which gives the slope of the tangent line at any point x > 0. This derivative is essential for finding tangent lines and understanding the behavior of the logarithmic function.

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Derivative of the Natural Logarithmic Function

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