Let L(c) be the length of the parabola f(x) = x² from x = 0 to...

Question

Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.

b. Is L concave up or concave down on [0, ∞)?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says let LB be the arc length of the curve G of X, which is equal to 3 X2, from X equal to 0 to X equal to B, where B is greater than or equal to 0. Determine whether L is concave up or concave down on the right open interval from 0 to infinity. And we're given two possible choices as our answers. For choice A, we have concave up, and for choice B, we have concave down. So we need to determine whether a function L of B is going to be concave up or concave down on the right open interval from 0 to infinity. So the first thing we need to do is write down the expression for L B. So LFB is the arc length of a curve. And to recall your formula for finding the arc length of a curve. So that's going to be L of B, which is our arc length, and that's going to be equal to the integral from 0 to B of the square root of the quantity of 1. Plus the quantity of G of X. In quantity squared in quantity D X. So first step is to find G of X and G of X is just the derivative with respect to X of G of X, which is going to be equal to the derivative with respect to X. Of 3 x squared. And when we take this derivative, we see that G X is going to be equal to 6 X. So making this substitution back into our formula, we see that L B is going to be equal to the integral from 0 to B of the square root of the quantity of 1 plus the quantity of 6 X in quantity squared in quantity DX. Which we can simplify as being equal to the integral from 0 to B of the square root of the quantity of 1 plus 36 X squared in quantity DX. So now what we need to do is to take a derivative of LB with respect to B. And so that's going to be L. Of B And this is going to be equal to, and here we can actually apply the fundamental theorem of calculus. So here we're taking a derivative of the integral with respect to the upper bound of the integral. So that means that we're just going to get back the square root of the quantity of 1 plus 36 B squad in quantity. So here we just have our integram evaluated at the upper bound. Since our lower bound here is a constant of 0. So now we can take one more derivative to find the second derivative and then use the second derivative test to see whether our function is concave up or concave down. So, taking another derivative, we'll have L of B. And this is going to be equal to the derivative with respect to B of the square root of the quantity of 1 + 36B squared in quantity, and this is going to be equal to. 1 divided by the quantity of 2 multiplied by the square root of 1 plus 36 B squad in quantity. And here, we've just taken the derivative of the square root function, but we now need to apply our chain rule and take the derivative of what's inside of the square roots function. So we'll multiply this by the derivative with respect to B of 1 + 36B squad, which is just going to be 72B. And simplifying this expression, this is going to be equal to 36B. Divided by the square root of the quantity of 1 plus 36 B squared in quantity. And if we look at this function here, and we're given in the problem that B is greater than or equal to 0, so that means that our numerator is going to be greater than or equal to 0. And if we look at our denominator, we're taking the positive square root of a positive quantity. So that means our denominator is also going to be positive. So that means that L B is going to be greater than or equal to 0 for all values of B that are greater than or equal to 0. And since L of B is positive for B greater than equal to 0, that means that our function LB is going to be concave up on that same interval. So the answer that we're looking for for this problem is going to be concave up. And if we compare that answer to our answer choices, we see that the correct answer choice corresponds to answer choice A. So I hope that this explanation has been useful, and I'll see you in the next video.

Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.
b. Is L concave up or concave down on [0, ∞)?

Key Concepts

Arc Length of a Curve

Concavity

Second Derivative Test

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