Question Video: Finding the Linear Approximation of an Exponential Function to Estimate an Exponential Value Mathematics • Higher Education

Video Transcript

By finding the linear approximation of the function 𝑓 of 𝑥 equals 𝑒 to the power of 𝑥 at a suitable value of 𝑥, estimate the value of 𝑒 to the power of 0.1.

We’re told to use the linear approximation of the function 𝑓 of 𝑥 equals 𝑒 to the power of 𝑥. So we recall the formula. If 𝑓 is differentiable at 𝑥 equals 𝑎, then the equation that can be used to find the linear approximation to the function at 𝑥 equals 𝑎 is 𝑙 of 𝑥 equals 𝑓 of 𝑎 plus 𝑓 prime of 𝑎 times 𝑥 minus 𝑎. In this example, we’re trying to approximate the value of 𝑒 to the power of 0.1. This is going to be close to the value of 𝑒 to the power of zero. So we let 𝑎 be equal to zero. This means 𝑓 of 𝑎 is equal to 𝑓 of zero. And substituting zero into our function 𝑓 of 𝑥 equals 𝑒 to the power of 𝑥 gives 𝑒 to the power of zero which is one.

Next, we find 𝑓 prime of 𝑎. First, of course, we need to find an expression for the derivative of our function. So we different 𝑒 to the power of 𝑥 with respect to 𝑥. The first derivative of 𝑒 to the power of 𝑥 is 𝑒 to the power of 𝑥. So 𝑓 prime of 𝑎 becomes 𝑓 prime of zero which is 𝑒 to the power of zero. And once again, that’s one. Substituting what we know into our formula for our tangent line approximation and we see that 𝑙 of 𝑥 is equal to one plus one times 𝑥 minus zero. And that simplifies to 𝑥 plus one.

We’ll use this to approximate the value of 𝑒 to the power of 0.1 by finding 𝑙 of 0.1. That’s 0.1 plus one which is 1.1. And an estimate to the value of 𝑒 to the 0.1 is 1.1. And if we type this into our calculator, 𝑒 to the 0.1 is 1.10517 and so on. That’s very close in value to our estimation. And that’s because 0.1 is fairly close to zero. Had we tried the larger value, our number might not have been so accurate. Let’s check that.

For example, 𝑙 of 0.3 is 0.3 plus one. So according to our approximation, 𝑒 to the 0.3 is approximately 1.3. Typing 𝑒 to the 0.3 into our calculator and we get 1.349858808, still not a bad approximation but not quite as close as 𝑒 to the power of 0.1.