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

By finding the linear approximation of the function π(π₯) = π^(π₯) at a suitable value of π₯, estimate the value of π^(0.1).

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### Video Transcript

By finding the linear approximation of the function π of π₯ equals π to the power of π₯ at a suitable value of π₯, estimate the value of π to the power of 0.1.

Weβre told to use the linear approximation of the function π of π₯ equals π to the power of π₯. So we recall the formula. If π is differentiable at π₯ equals π, then the equation that can be used to find the linear approximation to the function at π₯ equals π is π of π₯ equals π of π plus π prime of π times π₯ minus π. In this example, weβre trying to approximate the value of π to the power of 0.1. This is going to be close to the value of π to the power of zero. So we let π be equal to zero. This means π of π is equal to π of zero. And substituting zero into our function π of π₯ equals π to the power of π₯ gives π to the power of zero which is one.

Next, we find π prime of π. First, of course, we need to find an expression for the derivative of our function. So we different π to the power of π₯ with respect to π₯. The first derivative of π to the power of π₯ is π to the power of π₯. So π prime of π becomes π prime of zero which is π to the power of zero. And once again, thatβs one. Substituting what we know into our formula for our tangent line approximation and we see that π of π₯ is equal to one plus one times π₯ minus zero. And that simplifies to π₯ plus one.

Weβll use this to approximate the value of π to the power of 0.1 by finding π of 0.1. Thatβs 0.1 plus one which is 1.1. And an estimate to the value of π to the 0.1 is 1.1. And if we type this into our calculator, π to the 0.1 is 1.10517 and so on. Thatβs very close in value to our estimation. And thatβs because 0.1 is fairly close to zero. Had we tried the larger value, our number might not have been so accurate. Letβs check that.

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