The energy (in joules) released by an earthquake of magnitude ...

Question

The energy (in joules) released by an earthquake of magnitude M is given by the equation E=25,000 ⋅ 10^1.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)
Compute dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)

Compute dE/dM and evaluate it for M=3. What does this derivative mean? (M has no units, so the units of the derivative are J per change in magnitude.)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with derivatives. This problem says a certain type of star emits energy according to the formula E is equal to 150 multiplied by 10 rates to the quantity of 0.5 m. Where E is the energy in joules and M is the magnitude of the star's brightness. Find E M and approximate the value of E 2 to a whole number, also providing an interpretation of E2. Now, the first quantity this problem wants us to find is E M. And so that means I'm gonna be taking a derivative of E with respect to M. So that means E M is going to be equal to the derivative with respect to M. Of the quantity of 150, multiplied by 10 raised to the quantity of 0.5 m. Now, in order to take this derivative, we're going to have to recall our one of our rules for derivatives, and that is the derivative with respect to X. Of the quantity of B raised to the power X, and that is equal to B raised to the power X, multiplied by the natural log of B. So, coming back to E M, it is going to be equal to. We're going to use our constant multiple rule for derivatives to pull the 150 outside of our derivative. So we'll have 150 multiplied by the derivative with respect to M of 10 raised to the quantity of 0.5 m. So we're going to use our derivative rule that we just previously written out. And so that means we're going to have 10 raised to the quantity of 0.5 m in quantity. Multiplied by the natural log of 10. And that is gonna get multiplied by the derivative of our exponent with respect to M, since we need to apply our chain rule as well. So, the derivative of 0.5, M with respect to M is 0.5. So we can now simplify our expression, and this is going to be equal to 75, multiplied by 10 raised to the quantity of 0.5 m. In quantity multiplied by the natural log of 10. So, this is the expression that we're looking for for E M. So that's the first part of our question answered. We also need to approximate the value of E 2 to a whole number. So we're going to evaluate our expression of E M when M is equal to 2. So we'll have E of 2, and that's going to be equal to 75, multiplied by 10, raised to the quantity of 0.5 multiplied by 2 in quantity, multiplied by the natural log of 10. So we can use our calculator to calculate this um expression, and rounding it to the nearest whole number, this is going to be approximately equal to 1,727. And this has units of joules. Per unit magnitude. Since our energy was given in joules. And we're taking a derivative with respect to M, which is our magnitude. So that's the reason why we get units of joules per unit magnitude. And finally, we need to provide an interpretation of E2. So what this is saying is that when M is equal to 2, since that is the value that we're looking at. And we call that a derivative is looking at the change of a function. So, in this case, we're looking at the change in our energy function. So, we're gonna have our energy changing. And since um our derivative here is positive, that means that our energy increases. So a positive. Um, quantity for our derivative means it's increasing, and if it was negative, it would be decreasing. But this case is increasing. So we have our energy increases, and it's by the value that we calculated, so by 1000. 727, and that has units of joules for our energy. But recall that our change here is going to be a rate, so that is gonna be per. Unit and here we've taken a derivative with respect to our magnitude. That's that's per unit increase. In our magnitude. Since that is what we're taking a derivative with respect to. And so that is going to be Our explanation. So we have for E of 2, it has a value of 1,727 joules per unit magnitude, and the interpretation is when M is equal to 2, the energy increases by approximately 1,727 joules per unit increase in magnitude. So those are the answers to the questions um that this problem was asking. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Derivative

Exponential Functions

Magnitude Scale

Watch next