Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of ...

Question

Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. During a deep sea dive, a diver tracks their depth every 5 minutes. The data recorded is shown in the table below. Table 1 depth D versus time T. Depth D in meters, 2025, 30. Time T minutes 750, 800, 850. Simultaneously, a nearby oceographic station measures the water pressure P in Pascal's PA. At various depths D and meters, measured by the diver and creates the following table shown below. Table 2 pressure P versus depth D. Depth D in meters 750, 800, 850.res P in pascals, 100,000, 120,000, 145,000. Calculate the rate of change of the water pressure with respect to time experienced by the diver when descending for 25 minutes during the dive. Estimate the required derivatives using the forward difference quotient. Awesome. So it appears for this particular problem, we're ultimately trying to figure out what the rate of change of the water pressure is with respect to time. So that's what we're ultimately trying to solve for is this rate of change of the water pressure with respect to time during this diver's descent for 25 minutes. Awesome. So with that said, let's read off our multiple choice answers to see what our final answer might be. Noting that they're all getting the same units of pascals per minute. So A is 55,000, B is 15,000, C is 10,000, and D is 5000. Awesome. So first off, we need to note that we're trying to find the rate of change of pressure with respect to time, so we're trying to solve for DP by DT. At the time T equals 25 minutes. And in order to do that, we will need to recall and use the forward difference quotient. So with that said, in order to calculate DP by DT as we said, once again, we need to recall and use the forward difference. Formula. So as we should recall, the four words. Difference, so once again, forward difference formula, DP by DT is going to be equal to, as we should recall, P of D2 minus P of D1, all divided by D2 minus D1. And using our table, we should be able to note that P of D2 is gonna be equal to 145,000 P of D1 is gonna be equal to 120,000. And we should also recall that D2 is gonna be equal to 850, and finally, D1 is going to be equal to 800. Fantastic. So, with that said, We now need to substitute in our known variables into our equation for the forward difference formula, and when we do that, we will find that DP by DT it's going to be equal to 1,145,000. So 145,000 minus 120,000. Fantastic, and let's put it in parentheses so we don't. It can, so we'll make sure we get the precise answer when we plug it into our calculator. So we're gonna take 145,000 minus 120,000, and then you need to divide it by 850 minus 800. Fantastic. So when we plug that into our calculator, we should get 500 pascals per minute. Fantastic. So now our next step is we need to calculate DD by DT. So we need to recall and use the forwards difference formula once again to solve for DD by DT, which this equation is going to be equal to D of T2 minus D of T1, all divided by T2 minus T1. Which recalling. Or like we're calling the formula, sorry, not the formulas, recalling what we've seen in our tables and plugging in what we know into our formula, we will find that we need to take like D of T2, so we're told that D of T2 is going to be equal to 850, and so we don't get confused. Let's make a note of it off to the side here. So D of T2 is gonna be equal to 850. And D of T1 is going to be equal to 800, and it's also note that T2 is going to be equal to 30, and T1 is going to be equal to 25. So if you plug in these known variables into our equation, we will get 850 minus 800 all divided by 30 minus 25. Fantastic. And when we plug that into our calculator, we will find that it's gonna be equal to 10 and its units of measurement are gonna be in meters per minute. fantastic. So 10 m per minute. So now, our next step is we need to recall and apply the chain rule. So when we do that, we will find that DP by DT is equal to DP by DD multiplied by DD by DT. So now we need to substitute in our known variables, and when we do, we will take 500 multiplied by 10, and this will be equal to 5000 pascals per meter, and that is our final answer. Hooray, we did it. So therefore, without a doubt, the rate of change in water pressure with respect to time at T equals 25 minutes is 5000 pascals per minute. And this corresponds with multiple choice answer letter D, which once again is 5000 pascals per minute. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.

Key Concepts

Chain Rule

Forward Difference Quotient

Rate of Change

Watch next