Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

4. Applications of Derivatives

Differentials

Calculus

4. Applications of Derivatives

Differentials

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1:51 minutes

Problem 121b

Textbook Question

Exponential growth rates
b. Compare the growth rates of eˣ and eᵃˣ as x→∞ , for a > 0.

Verified step by step guidance

To compare the growth rates of \( e^x \) and \( e^{ax} \) as \( x \to \infty \), we first need to understand the behavior of exponential functions. Both functions are exponential, but they have different exponents.

Consider the function \( e^x \). As \( x \to \infty \), \( e^x \) grows exponentially without bound. The base \( e \) is a constant greater than 1, which means the function increases rapidly.

Now, consider the function \( e^{ax} \). Here, \( a \) is a positive constant greater than 0. The exponent \( ax \) means that the rate of growth is scaled by \( a \). If \( a > 1 \), \( e^{ax} \) grows faster than \( e^x \). If \( 0 < a < 1 \), \( e^{ax} \) grows slower than \( e^x \).

To compare the growth rates more formally, we can take the limit of the ratio of the two functions as \( x \to \infty \): \( \lim_{x \to \infty} \frac{e^{ax}}{e^x} = \lim_{x \to \infty} e^{(a-1)x} \).

Evaluate the limit: If \( a > 1 \), \( e^{(a-1)x} \to \infty \), indicating \( e^{ax} \) grows faster. If \( 0 < a < 1 \), \( e^{(a-1)x} \to 0 \), indicating \( e^x \) grows faster. If \( a = 1 \), the limit is 1, indicating both grow at the same rate.

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Key Concepts

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

Exponential Functions

Exponential functions are mathematical expressions of the form f(x) = a * e^(bx), where 'e' is the base of the natural logarithm, approximately equal to 2.71828. These functions exhibit rapid growth or decay, depending on the sign of 'b'. Understanding their behavior as x approaches infinity is crucial for analyzing growth rates.

Growth Rate Comparison

When comparing growth rates of functions, we often look at their limits as x approaches infinity. For the functions e^x and e^(ax) where a > 0, we can determine which function grows faster by evaluating the limit of their ratio. This comparison reveals that e^(ax) grows faster than e^x as x becomes very large.

Limit Analysis

Limit analysis is a fundamental concept in calculus that helps us understand the behavior of functions as they approach a certain point, often infinity. By applying limit techniques, we can rigorously determine the growth rates of functions like e^x and e^(ax) as x approaches infinity, providing insights into their relative rates of increase.

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Textbook Question

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x²⁰ ; 1.0001ˣ

Textbook Question

Consider the lim_x→∞ (√ ax + b) / √cx + d where a, b, c, and d are positive real numbers. Show that l’Hôpital’s Rule fails for this limit. Find the limit using another method.

Textbook Question

Prove that lim_x→∞ (1 + a/x)ˣ = eᵃ , for a ≠ 0 .

Textbook Question

Exponential growth rates

a. For what values of b > 0 does bˣ grow faster than eˣ as x→∞?

Textbook Question

Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.

lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)

Textbook Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_x→∞ ln ((x +1) / (x-1))

Textbook Question