QuestionEvaluate the piecewise function at to check if it's continuous and differentiable. What is true about at ?
Studdy Solution
STEP 1
Assumptions1. The function is defined as a piecewise function with two parts - for - for . We need to determine the continuity and differentiability of at .
STEP 2
First, we need to check if is continuous at . A function is continuous at a point if the limit from the left, the limit from the right, and the function value at that point all equal each other.
So, we need to calculate the following
STEP 3
Let's calculate the limit from the left, , using the first part of the piecewise function, .
STEP 4
Substitute into the equation to find the limit from the left.
STEP 5
Now, let's calculate the limit from the right, , using the second part of the piecewise function, .
STEP 6
Substitute into the equation to find the limit from the right.
STEP 7
Now, let's calculate the function value at , , using the first part of the piecewise function, .
STEP 8
As we can see, the limit from the left, the limit from the right, and the function value at are all equal to . Therefore, the function is continuous at .
STEP 9
Next, we need to check if is differentiable at . A function is differentiable at a point if the derivative from the left and the derivative from the right at that point are equal.
So, we need to calculate the following
STEP 10
Let's calculate the derivative from the left, , using the first part of the piecewise function, .
First, we need to find the derivative of .
STEP 11
Now, substitute into the derivative to find the derivative from the left.
STEP 12
Now, let's calculate the derivative from the right, , using the second part of the piecewise function, .
First, we need to find the derivative of .
STEP 13
Now, substitute into the derivative to find the derivative from the right.
STEP 14
As we can see, the derivative from the left, , and the derivative from the right, , at are not equal. Therefore, the function is not differentiable at .
So, the correct statement is is continuous but not differentiable at .
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