Theory and Examples

In Exerc...

Question

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

b. Does f'(3) exist?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says, given the function F of X, which is equal to the absolute value of the quantity of X 4th minus a X in quantity, does F2 exist? We're given two possible choices as our answers. For choice A, we have yes, and for choice B, we have no. Now, in order to determine if F2 exists, we're going to be using the difference quotient. So we call for the difference quotient that we have F of 2. And this could be equal to the limit. As each approaches 0. Of the quantity of F of 2 + H. Minus F of 2 in quantity divided by H. So, in order for the derivative to exist, that means that the limit must also exist. So, one of the conditions for a limit to exist, if you recall, is that the left-hand limit must approach the same value as the right-hand limit. So we're going to be calculating the left-hand limit and the right-hand limit of our quotient here. But before we do that, we're going to calculate F of 2 and F of 2 + H, since it will appear in both our left-hand limit and the right-hand limit. So, we'll start by looking at FF 2, and this is going to be equal to the absolute value of the quantity of 2 raised to the 4th power, minus 8 multiplied by 2 in quantity. And when we evaluate this expression, this is going to be equal to 0. We also need to look at F of 2 + H. And this is going to be equal to the absolute value of the quantity, and for the first term, we'll have the quantity of 2 + H in quantity to the 4th. Minus for the second term, we'll have 8 multiplied by the quantity of 2 + H in quantity. We'll want to multiply everything out in that expression, so this is going to be equal to the absolute value of the quantity of 16. Plus 32 H. Plus 24 squared plus 8 H cubed. Plus 4th minus 16 minus 8 H in quantity. Collecting like terms, this is going to be equal to the absolute value of the quantity of 24 H plus 24 squared plus 8 H cubed plus H fourth in quantity. So now we're gonna need to calculate both the left-hand limit and the right-hand limit of our quotient. So we'll start off with the left hand limit. So we'll have the limit. As H approaches 0 from the left of the quantity, and for the first term we'll have F 2 + H. But if we look at F 2 + H, the quantity inside the absolute value sign is going to be slightly negative when H is slightly negative, which is what H is going to be if H is approaching 0 from the left. So in order to remove the absolute value sign, we'll want to multiply the quantity inside the absolute value sign by -1. So F of 2 + H is going to be equal to minus the quantity of 24 H plus 24 H2 + 8 H cubed plus H 4th in quantity. Then we'll have minus F of 2, which is 0, and all of that is divided by H. So we can simplify this expression by dividing each of the terms in the numerator by H. So this is going to be equal to the limit, as H approaches 0 from the left of the quantity of minus 24. Minus 24 H. Minus 8 H squad. Minus H cubed in quantity. And we can evaluate this limit by direct substitution, so this is going to be equal to -24. Minus 24 multiplied by 0, -8 multiplied by 0 squared, minus 0 cut. And evaluating this expression, this is going to be equal to -24. So now we need to calculate the right-hand limits. So we'll have the limit as H approaches 0 from the right of the quantity, and for F of 2 + H, if H is slightly positive, which is the case here when H is approaching 0 from the right, the quantity inside the absolute value sign is also going to be positive, so we can just remove the absolute value sign. So we'll have 24 H plus 24 H squared. Plus 8 H cubed, plus H 4th as F of 2 + H. Then we'll have minus F of 2, which again is 0, and all of that is divided by H. Again, we can simplify this this expression by dividing each of the terms in the numerator by H. And so this will give us the limit as H approaches 0 from the right. Of the quantity of 24 plus 24 H. Plus 8 squad plus HQ. In quantity. Again, we can evaluate this limit by direct substitution, so this is going to be equal to 24 plus 24 multiplied by 0. Plus 8 multiplied by 0 squared. Plus 0 cubed and evaluating the expression that it's going to be equal to 24. And here we see that the left hand limit does not equal the right hand limit. Since -24 is not equal to 24, and so therefore, the limit in our portion rule does not exist and therefore our derivative does not exist. So the answer for this problem is going to be no, and that actually corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

b. Does f'(3) exist?

Key Concepts

Derivative

Absolute Value Function

Differentiability and Continuity

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