If the tangent to y = e^x at the point x = x₀ intersects the x-axis at x = x₁, show that x₀ - x₁ = 1.

Difficulty rating: 
Special thanks to Calculus with Analytic Geometry by Simmons (c) 1985 for this problem.

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Solution:

This problem deals exclusively with the tangent line to the curve y = e^x . First of all we need to find where this tangent intersects the curve itself. According to the problem, it's at  x = x₀ or the point (x₀, ). (I plugged x₀ into the function to get the ordered pair.) Now, note that the tangent line intersects the x-axis at the point (x₁ , 0), since any point on the x-axis will have zero as a y-value. Because we have two points on the tangent line, we can find the slope of the tangent line by calculating the change in y over the change in x. Doing so gives us:

Don't forget that we know the slope of the tangent line already! The derivative of y evaluated at x = x₀ will give us the slope of the tangent line (because that's what it is by definition). Therefore, the slope must equal m=. If we set these two slopes equal, it's easy to see that the denominator of the first must equal 1 in order that we get the numerators () to be equal. We are done.

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