Textbook Question
Evaluate the integrals in Exercises 31–78.
49. ∫x3^(x²)dx
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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.PE.110c
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
c. f(x) = 10x^3 + 2x^2, g(x) = e^x
Verified step by step guidance1
Identify the given functions: \(f(x) = 10x^{3} + 2x^{2}\) and \(g(x) = e^{x}\).
Recall that as \(x \to \infty\), polynomial functions like \(x^{3}\) grow at a rate proportional to a power of \(x\), while exponential functions like \(e^{x}\) grow much faster than any polynomial.
To compare growth rates, consider the limit of the ratio \(\frac{f(x)}{g(x)} = \frac{10x^{3} + 2x^{2}}{e^{x}}\) as \(x \to \infty\).
Apply L'Hôpital's Rule repeatedly if necessary, differentiating numerator and denominator until the limit can be evaluated or the behavior is clear.
Since the denominator \(e^{x}\) grows faster than any polynomial, the limit of \(\frac{f(x)}{g(x)}\) as \(x \to \infty\) will approach zero, indicating that \(f\) grows slower than \(g\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Growth Rates of Functions
Understanding how functions grow as x approaches infinity involves comparing their dominant terms. Polynomial functions like 10x³ grow at a rate determined by the highest power of x, while exponential functions like eˣ grow much faster, eventually outpacing any polynomial.
Recommended video:
04:16
Intro To Related Rates
Limits at Infinity
Evaluating the limit of the ratio f(x)/g(x) as x approaches infinity helps determine relative growth rates. If the limit is zero, f grows slower; if infinite, f grows faster; if finite and nonzero, they grow at comparable rates.
Recommended video:
03:07Cases Where Limits Do Not Exist
Exponential vs Polynomial Growth
Exponential functions increase faster than any polynomial function as x becomes very large. This is because exponentials multiply by a constant factor repeatedly, while polynomials increase by powers of x, making exponentials dominant in the long run.
Recommended video:
09:29Exponential Growth & Decay
Related Practice
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
f. f(x) = sech(x), g(x) = e^(-x)
Textbook Question
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
11. y = 5x^(3.6)
Textbook Question
In Exercises 79–84, solve for y.
79. 3^y = 2^(y+1)
Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
102. lim(x→0) (x sin(x²))/(tan³x)