Skip to main content
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.2.13
Chapter 7, Problem 7.2.13

13–14. Absolute and relative growth rates Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant.


f(t) = 100 + 10.5t, g(t) = 100e^(t/10)

Verified step by step guidance
1
Identify the given functions: the linear function \(f(t) = 100 + 10.5t\) and the exponential function \(g(t) = 100e^{\frac{t}{10}}\).
Calculate the growth rate of the linear function \(f(t)\) by finding its derivative with respect to \(t\): \(f'(t) = \frac{d}{dt}(100 + 10.5t)\).
Observe that the derivative \(f'(t)\) is a constant value, which shows that the absolute growth rate of the linear function is constant.
Calculate the relative growth rate of the exponential function \(g(t)\) by first finding its derivative: \(g'(t) = \frac{d}{dt}\left(100e^{\frac{t}{10}}\right)\).
Express the relative growth rate of \(g(t)\) as \(\frac{g'(t)}{g(t)}\) and simplify to show that this ratio is a constant, confirming the relative growth rate of the exponential function is constant.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Absolute Growth Rate

The absolute growth rate of a function measures how much the function's value changes per unit increase in the input variable. For a linear function like f(t) = 100 + 10.5t, this rate is constant and equals the slope, which is 10.5. It represents a steady, fixed increase over time.
Recommended video:
04:16
Intro To Related Rates

Relative Growth Rate

The relative growth rate is the rate of change of a function relative to its current value, often expressed as (f'(t)/f(t)). For exponential functions like g(t) = 100e^(t/10), this rate is constant, reflecting proportional growth. It indicates the percentage increase per unit time.
Recommended video:
04:16
Intro To Related Rates

Differentiation of Functions

Differentiation is the process of finding the derivative, which gives the instantaneous rate of change of a function. Calculating derivatives of f(t) and g(t) allows us to determine their growth rates. Understanding how to differentiate linear and exponential functions is essential to analyze their growth behavior.
Recommended video:
05:53
Finding Differentials
Related Practice
Textbook Question

Population of Texas Texas was the third fastest growing state in the United States in 2016. Texas grew from 25.1 million in 2010 to 26.47 million in 2016. Use an exponential growth model to predict the population of Texas in 2025.

Textbook Question

Express sinh⁻¹ x in terms of logarithms.

Textbook Question

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.


lim x → 0⁺ (tanh x)ˣ

Textbook Question

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.


∫₀ˡⁿ ² (e^{3x} − e^{−3x}) / (e^{3x} + e^{−3x}) dx

Textbook Question

What is the inverse function of ln x, and what are its domain and range?

Textbook Question

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.


∫ 3^{-2x} dx