Ch. 7 - Transcendental Functions

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

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.8.17

Chapter 7, Problem 7.8.17

17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

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Identify the functions given: \( f(x) = \sqrt{10x + 1} \) and \( g(x) = \sqrt{x + 1} \). We want to compare their growth rates as \( x \to \infty \).

Recall that to show two functions grow at the same rate as \( x \to \infty \), we can compare their ratios to a common function that represents the growth rate. Here, the common function is \( h(x) = \sqrt{x} \).

Compute the limit of the ratio \( \frac{f(x)}{h(x)} = \frac{\sqrt{10x + 1}}{\sqrt{x}} \) as \( x \to \infty \). Simplify the expression inside the limit by factoring out \( x \) inside the square root in the numerator.

Similarly, compute the limit of the ratio \( \frac{g(x)}{h(x)} = \frac{\sqrt{x + 1}}{\sqrt{x}} \) as \( x \to \infty \). Simplify this expression by factoring out \( x \) inside the square root in the numerator.

Evaluate both limits to find finite, nonzero constants. Since both limits exist and are finite and nonzero, conclude that \( f(x) \) and \( g(x) \) grow at the same rate as \( h(x) = \sqrt{x} \) as \( x \to \infty \), and therefore \( f(x) \) and \( g(x) \) grow at the same rate as each other.

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Key Concepts

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

Asymptotic Growth and Limits

Asymptotic growth describes how functions behave as the input grows very large. Using limits, we compare the ratio of two functions as x approaches infinity to determine if they grow at the same rate. If the limit of their ratio is a finite nonzero constant, the functions have the same growth rate.

Square Root Function Behavior

The square root function, √x, grows slower than linear functions but faster than logarithmic ones. Understanding how expressions like √(10x+1) and √(x+1) relate to √x helps in comparing their growth rates by factoring out √x and analyzing the remaining terms as x becomes large.

Limit of a Ratio to Compare Growth Rates

To show two functions grow at the same rate, we examine the limit of their ratio as x approaches infinity. For example, evaluating lim(x→∞) √(10x+1)/√x simplifies to √10, a finite constant, indicating both functions grow proportionally to √x and thus at the same rate.

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Related Practice

Textbook Question

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

17. lim(x→∞)arcsec(x)

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Textbook Question

Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.

4. cosh x = 13/5, x>0

Textbook Question

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

63. tanh⁻¹(-1/2)

Textbook Question

In Exercises 139–142, find the length of each curve.

139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.

Textbook Question