Using differentials, find the approximate change in the surface area of a soap bubble when its radius increases from eight to 8.1 inches.
Now, we are actually told exactly how we’re going to solve this problem. We’re told to use differentials. Given a function 𝑦 is equal to 𝑓 of 𝑥, we call d𝑦 and d𝑥 differentials. And the relationship between them is given by d𝑦 equals 𝑓 prime of 𝑥 d𝑥. So in this question, we’re going to need to begin by defining a function. And we’re going to define our function for surface area. Now, the surface area, let’s call that capital 𝐴, of a sphere with a radius 𝑟 is given by four 𝜋𝑟 squared. And so, we’re going to need to redefine a little.
We have 𝐴 as being some function of 𝑟. So in this case, our differentials are d𝐴 and d𝑟. And then, the relationship between them is d𝐴 equals 𝑓 prime of 𝑟 d𝑟. But what is d𝑟? Well, essentially, the differential of 𝑟 is just a very small change in 𝑟. So in here, 𝑟 changes from eight to 8.1 inches. So the change in 𝑟 is the difference; it’s 0.1. Since the radius is increasing from eight, we can say that 𝑟 is equal to eight. And d𝐴 is the small change in 𝐴 that we’re trying to find. And so, since we know d𝑟 and we know 𝑟, it’s quite clear we need to find 𝑓 prime of 𝑟. We defined 𝐴 to be equal to 𝑓 of 𝑟. So that’s four 𝜋𝑟 squared. 𝑓 prime of 𝑟 is the first derivative of this function with respect to 𝑟.
Since four 𝜋 is simply a constant, to differentiate four 𝜋𝑟 squared, we multiply the entire term by the exponent and reduce that exponent by one. So we get two times four 𝜋𝑟 to the power of one, which is simply eight 𝜋𝑟. In our formula, we said that 𝑟 is equal to eight. So we need to find 𝑓 prime of eight. That’s found by replacing 𝑟 with eight. So we get eight 𝜋 times eight, which is 64𝜋. Then d𝐴 — remember, that’s what we’re trying to find, it’s the change in the surface area — is the product of 64𝜋 and d𝑟 which is 0.1.
We can now type this into our calculator. And we find that d𝐴 is equal to 20.10619 and so on, which, correct to three decimal places, is 20.106. And the approximate change in the surface area of the soap bubble is 20.106 square inches.