Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
a. log_2(x²)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
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 guidance1
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.
Recommended video:
Guided course
6:39
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.
Recommended video:
05:13Slopes 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.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
3. a. arcsin(-1/2)
Textbook Question
84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.
1
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Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
a. x² + √x
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
In Exercises 41–44:
a. Find f⁻¹(x).
44. f(x) = 2x², x ≥ 0, a = 5