In Exercises 41–44, determine whether the piecewise-defined fu...

Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x − 1, x ≥ 0

x² + 2x + 7, x < 0

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says determine whether the following piece wise function is differentiable at X equal to 2. We're given the function G of X is equal to 3 X plus 2 if X is greater than 2, and X2 minus 4 if X is less than or equal to 2. And 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 for a function to be differentiable at a point, it also needs to be continuous at that point. So we're going to check for continuity first. So we call for a function to be continuous at a point. We need the left-handed limit to be equal to the right-handed limit to be equal to the value of the function at that point. So we're going to look at the limit as X approaches 2 from the left of G of X. And that needs to be equal to the limit, as X approaches to from the right of G of X. Which has to be equal to G of 2, or a function to be continuous at X equal to 2. So we're gonna need to evaluate all three of these quantities. So we're gonna start off by looking at G of 2. So for GF2, Since X is equal to 2, we're going to be using the second piece of our piece-wise function, which is X2 minus 4, since that is the piece that contains X equal to 2. So G of 2 is going to be equal to 22 minus 4, and when we evaluate that expression, this is going to be equal to 0. Next we're gonna look at the left-handed limits, so we'll have the limit. As X approaches 2 from the left of G of X, and here, since X is going to be slightly less than 2, we're going to be using the second piece of our piece-wise function. So this is going to be the limit as X approaches 2 from the left of the quantity of X2 minus 4 in quantity. And we can evaluate this limit by direct substitution, so this is going to be equal to Q2 minus 4. Which is going to be equal to 0, which is the same value as our function GF2. The next Um, quantity that we need to calculate is going to be the right-handed limit. So we'll have the limit as x approaches 2 from the right of G of X, and here X is going to be slightly greater than 2. So we're going to use the first piece of our piecewise function. So we'll have the limit as X approaches 2 from the right of the quantity of 3 X plus 2. Again, we can evaluate this limit by direct substitution, and so this is going to be equal to 3, multiplied by 2. Plus 2, which evaluating the expression is equal to 8. Now, if we look and compare our values for the left-hand limit and the right-handed limit, we see that they're not equal. So the limit as X approaches you from the left of G of X. is not equal to the limit. As X approaches to from the right of G of X. So that means that our function is not continuous at X equal to 2, and since our function is not continuous at X equal to 2, that means it's also not differentiable at that point. So the answer for this problem is going to be no, it is not differentiable. And if we compare our answer to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x − 1, x ≥ 0
x² + 2x + 7, x < 0

Key Concepts

Piecewise Functions

Differentiability

Continuity

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