Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.3.155a

Chapter 7, Problem 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).

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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

Related Practice

Textbook Question

Suppose that the function g and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function g^(-1) is differentiable, find the slope of g^(-1)(x) at

a. x=1

Textbook Question

Verify the integration formulas in Exercises 37–40.

37. a. ∫sech(x)dx = tan⁻¹(sinh x) + C

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Textbook Question

143.

a. Show that ∫ ln(x) dx = x ln(x) − x + C.

Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

a. Plot the function y=f(x) together with its derivative over the given interval. Explain why you know that f is one-to-one over the interval.

72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2

Textbook Question

Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

69. ∫(from 5/4 to 2)dx/(1-x²)

Textbook Question

In Exercises 41–44:

a. Find f⁻¹(x).

44. f(x) = 2x², x ≥ 0, a = 5