155. Which is bigger, πᵉ or e^π?
Calculators have taken ...

Question

155. Which is bigger, πᵉ or e^π?

Calculators have taken some of the mystery out of this once-challenging question.

(Go ahead and check; you will see that it is a very close call.)

You can answer the question without a calculator, though.

a. Find an equation for the line through the origin tangent to the graph of

y = ln(x).

Accepted Answer

Welcome back everyone. I'm the equation of the line passing through the origin that is tangent to the graph of y equals a line of 2 x. A says y equals 2 divided by e multiplied by x. B 1 divided by e multiplied by x. Ce divided by 2 multiplied by x, and D 1/2 x. So for this problem, let's remember the equation of the tangent line. If we have a function in this case y equals ln of 2 x, we can simply say that it is equal to f of x. The equation of the tangent line is y equals f at x0 multiplied by x minus x0 plus f of x0. So first of all, what we're going to do is simply identify the slope that would be F at x0, where x0 is the point of tangency. We don't know what it is, right? So we can just leave it as xt. But what we can do is identify the derivative f of x in a general form, and as the derivative of ln of 2 x we can differentiate using the chain rule. We're going to get 1 divided by 2 x multiplied by 2 x derivative, right? And this is 1 divided by 2 x multiplied by 2 which is 1 divided by x. So now What we can do is use the point of tangency as x0 and show that f at x0 is equal to 1 divided by x0 using the previous equation and now the value of the function f of x0 is going to be ln of 2 x 0. From here, we are going to identify the point of tangency. Let's substitute everything into the equation Y equals. We have F of x0, which is 1 divided by x0, right? It is multiplied by x minus x. And we are going to add f of x0, which is ln of 2 x 0. In order to solve for x 0, we can use the fact that our line passes through the origin, and that's a 0.00. So we're going to set x equals 0 and y equals 0 to get 0 equals 1 divided by x0, which is multiplied by 0 minus x0 or basically. g x0 plus ln of 2 x 0. And then we can rearrange, first of all, g x0 multiplied by 1 and divided by x0 gives us -1, we can just write -1. And we can move -1 to the left-hand side. So a -1. Is now going to become positive. 1 equals ln of 2 x 0 and therefore we can exponentiate to find x e to the power of 1, which is e is going to be equal to 2 x 0. And therefore x0 is e divided by 2. Well done. So we have identified the x knot coordinate that would be. E divided by 2. And we can substitute. Into the form that we had, right? We're simply going to replace X not with E divided by 2. Which gives us y equals 1 divided by x knot, so we take the reciprocal value, and that's 2 divided by e. Multiplied by. X minus. X, which is x minus e divided by 2. And plus ln of 2 x 0. So we're adding the natural logarithm of 2 multiplied by e divided by 2, that's L of E, right? And Eon of E is basically equal to 1. So now, let's go ahead and simplify. Y is equal to distributing. We're going to get 2 divided by e multiplied by x minus 2 divided by e multiplied by e divided by 2. D is basically cancel out to give us minus 1 plus L of e is 1. So now notice that we're left with. 2 divided by e multiplied by x which corresponds to the answer choice a. Thank you for watching.

Watch next