Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.R.84
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
ln x and log₁₀ x
Verified step by step guidance1
Step 1: Understand the problem by identifying the two functions involved: ln(x) and log₁₀(x). These are logarithmic functions with different bases. ln(x) is the natural logarithm with base e, and log₁₀(x) is the common logarithm with base 10.
Step 2: Recall the change of base formula for logarithms, which allows us to express log₁₀(x) in terms of ln(x). The formula is: log₁₀(x) = ln(x) / ln(10). This will help us compare the growth rates of the two functions.
Step 3: Analyze the growth rates by considering the derivative of each function. The derivative of ln(x) with respect to x is 1/x, and the derivative of log₁₀(x) with respect to x is 1/(x ln(10)).
Step 4: Compare the derivatives: Since 1/x is greater than 1/(x ln(10)) for all x > 0, ln(x) grows faster than log₁₀(x) as x increases.
Step 5: Conclude that ln(x) grows faster than log₁₀(x) because the derivative of ln(x) is larger than the derivative of log₁₀(x) for all x > 0, indicating a faster rate of increase.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Growth Rates of Functions
The growth rate of a function describes how quickly its value increases as the input approaches infinity. In calculus, we often compare functions by analyzing their limits or using L'Hôpital's rule to determine which function grows faster. Understanding growth rates is essential for comparing functions like ln(x) and log₁₀(x).
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Natural Logarithm vs. Common Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base e (approximately 2.718), while log₁₀(x) is the logarithm to the base 10. These two logarithmic functions have different properties and growth rates, which can be analyzed using their derivatives or limits. Recognizing the differences between these logarithmic functions is crucial for the comparison.
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05:18Derivative of the Natural Logarithmic Function
Limits and L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. By taking the derivative of the numerator and denominator, we can simplify the limit and determine the growth rates of functions. This technique is particularly useful when comparing ln(x) and log₁₀(x) as x approaches infinity.
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Related Practice
Textbook Question
45–46. Linear approximation
a. Find the linear approximation to f at the given point a.
b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?
ƒ(x) = x²⸍³ ; a =27; ƒ(29)
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Textbook Question
Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m³ needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.
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Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
a . Give the approximate coordinates of the local maxima and minima of ƒ
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Textbook Question
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.
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Textbook Question
104–107. Functions from derivatives Find the function f with the following properties.
ƒ'(t) = sin t + 2t; ƒ(0) = 5
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