For the following exercises, determine whether the equation represents continuous growth, contin

For the following exercises, determine whether the equation...

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. They are used to model situations where changes occur at rates proportional to the current level of a quantity, which is common in natural growth and decay processes.

Continuous Growth and Decay

In continuous growth or decay models, changes occur smoothly over time, and the rate of change is directly proportional to the current value of the quantity. A positive exponent indicates growth where the quantity increases over time, while a negative exponent indicates decay where the quantity decreases.

The Role of the Base e

The constant e is used as the base in natural exponential functions because it provides a natural model for continuous change. Its properties make calculations in growth and decay processes particularly simple and elegant, with the sign of the exponent's coefficient clearly indicating whether the function models growth or decay.

Transcript

00:02 All right, so determine whether the equation represents a continuous growth, continuous decay, or in neither.

00:08 So we have 3742 times e to the 0 .5t.

00:16 So let's see, 3742 times e to the .75t.

00:31 Let me make sure that's right.

00:33 The exact numbers don't really matter for this particular question, but this is good to have it.

00:39 Okay, yep.

00:43 So now let's see if this is increasing, decreasing what's going on.

00:47 So remember what an exponential equation holds.

00:51 We have the principal amount right over here, or the initial amount, and that's being multiplied by the rate essentially, right? so this is the value, the base value right here, and that's being raised to some power, 0 .75t in this case.

01:04 So if t was 1, it would be 0 .75, it was 2, then it would be 1 .5, 3, so on.

01:12 And then so basically it's being multiplied by itself a bunch of times and this right here is being multiplied by 3 -742.

01:19 And then finally we have our final answer, which is going to be something like this or like this, something like this over time.

01:27 Now we have to figure out if this is increasing or decreasing.

01:29 And the way to do that is you look at the base value.

01:32 Look at this value right over here.

01:34 And you look at it and see is it less than one? or sorry, yeah, is it less than one equal to one or greater than one? than one.

01:45 Well, if it's less than one, right, it would be a smaller number, like a fraction being multiplied every single time, which becomes even a smaller fraction.

01:53 So the end result is going to be a smaller number.

01:56 So it's going to go down.

01:58 One multiplied by itself a bunch of times is just one.

02:01 And a number bigger than one is actually going to be increasing, like two times two times two i think it was 2 .71828, so on.

02:19 So like, you know that it's about, let's say, 2 .72, for simplicity, 2 .72.

02:25 This is greater than one, meaning the rate right over here is going to be increasing.

02:32 It's going to be a number greater than 2 .72, because 2 .72, let's say t was like 5 or something, right? it would be 5 times 0 .75, which is 3 .75...

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