Textbook Question
Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.
Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at
c. x=3
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.8.3c
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. √(x^4 + x^3)
Verified step by step guidance1
Rewrite the function to better understand its growth rate. The function is \(\sqrt{x^4 + x^3}\). Since the square root is the same as raising to the power \(\frac{1}{2}\), rewrite it as \(\left(x^4 + x^3\right)^{\frac{1}{2}}\).
Factor out the highest power of \(x\) inside the parentheses to simplify the expression. The highest power inside is \(x^4\), so factor it out: \(\left(x^4\left(1 + \frac{1}{x}\right)\right)^{\frac{1}{2}}\).
Use the property of exponents to separate the factors: \(\left(x^4\right)^{\frac{1}{2}} \cdot \left(1 + \frac{1}{x}\right)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}} \cdot \left(1 + \frac{1}{x}\right)^{\frac{1}{2}} = x^2 \cdot \left(1 + \frac{1}{x}\right)^{\frac{1}{2}}\).
Analyze the behavior of \(\left(1 + \frac{1}{x}\right)^{\frac{1}{2}}\) as \(x \to \infty\). Since \(\frac{1}{x} \to 0\), this term approaches \(1\).
Conclude that the function behaves like \(x^2\) times a term approaching \(1\), so its growth rate is asymptotically similar to \(x^2\). Therefore, it grows at the same rate as \(x^2\) as \(x \to \infty\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Asymptotic Growth Rates
Asymptotic growth rates describe how functions behave as the input variable approaches infinity. Comparing growth rates helps determine which functions increase faster, slower, or at the same rate by analyzing dominant terms and their exponents.
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Intro To Related Rates
Dominant Term in Polynomials
In polynomials, the term with the highest power of x dominates the function's behavior for large x. For example, in x^4 + x^3, the x^4 term grows faster and dictates the overall growth rate as x approaches infinity.
Recommended video:
07:00Taylor Polynomials
Simplifying Expressions with Radicals
Simplifying expressions involving square roots, such as √(x^4 + x^3), often involves factoring out the highest power inside the root to identify dominant behavior. This helps compare the function's growth rate to simpler functions like x².
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6:36Simplifying Trig Expressions
Related Practice
Textbook Question
b. Find the center of mass if, instead of being constant, the density function is δ(x)=4/√x.
1
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Textbook Question
75. b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
g(x) = x(ln x)²
Textbook Question
c. Find the slopes of the tangent lines to the graphs of h and k at (2, 2) and (−2, −2).
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:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
Textbook Question
23. Human evolution continues The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years.
c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?