Question

True or False? In Exercises \(67-70\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. In exponential growth, the rate of growth is constant.

Short answer

False. In exponential growth, the rate of growth is not constant, it increases over time.

Step-by-step solution

Understanding Exponential Growth and Rate of Growth

Exponential growth refers to an amount of quantity that increases gradually, but at an accelerating rate over a period of time. The rate of growth here denotes the speed at which a certain variable grows over a certain period of time.

Analyzing the Statement

The statement suggests that the rate of growth in exponential growth is constant. This means that the speed at which a variable grows remains the same throughout the period. This is clearly against the definition of exponential growth, where the growth rate is said to be accelerating, not constant.

Concluding the Statement

Therefore, considering the definition of exponential growth, the statement 'In exponential growth, the rate of growth is constant.' is false.

Key concepts

Rate of Growth

Understanding the rate of growth is crucial when dealing with exponential functions. Exponential growth is a concept where the quantity grows by a certain factor in each equal successive period. Here's an easy way to picture it: imagine you have \(100 in a savings account with an interest rate that doubles your money every year; after the first year, you'll have \)200, after the second year, \(400, and so on. The money grows, and the growth itself gets larger - the actual amount you gain each year is increasing, not just the initial sum.

This is different from linear growth, where you might gain a fixed amount, say \)100, each year, regardless of how much money you have in the account. So for exponential growth, the key point is that the percentage growth is constant (say, doubling every year), but the amount by which the quantity grows is increasing over time - this is what we describe as an accelerating growth rate.

False Statements

In mathematics, and especially in real-world scenarios, it's important to recognize and correct false statements to avoid misconceptions. The given false statement, 'In exponential growth, the rate of growth is constant,' allows us to delve into the common confusion between the rate of growth and the amount of growth. A constant rate of growth implies that the amount by which a quantity grows over equal time intervals is the same, which is not the case in exponential growth.

To help students avoid these misunderstandings, it's necessary to emphasize the definition of terms. In exponential growth, while the relative growth rate (like doubling every period) is constant, the actual amount of growth is increasing because it is always a fixed percentage of the larger previous total. Recognizing the falsity in statements helps clarify definitions and reinforces learning through correction.

Calculus Concepts

Calculus concepts offer a formal toolset to describe and analyze the behavior of functions, including exponential growth. The exponential function, often written as a base (e.g., 2 or the natural base 'e') raised to the power of a variable exponent, is a standard topic in calculus. The tools of calculus, such as derivatives and integrals, help us clarify the nature of exponential growth.

Taking the derivative, in particular, shows how the rate of change of a function corresponds to the function itself in exponential cases - a unique property of exponential functions known as continuous compounding. When we take the derivative of an exponential function, we find that it is proportional to the original function, which substantiates the idea of the amount increasing at an accelerating rate, rather than the rate itself being constant. This is a prime example of how calculus concepts help us distinctly characterize mathematical phenomena like exponential growth.