Find the value of a for which the function f(x) = { ax + 1, x < 2; x^2 - 4, x >= 2 } is differentiable at x = 2.
Practice Questions
1 question
Q1
Find the value of a for which the function f(x) = { ax + 1, x < 2; x^2 - 4, x >= 2 } is differentiable at x = 2.
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Set the left-hand limit equal to the right-hand limit and their derivatives at x = 2.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the value of a for which the function f(x) = { ax + 1, x < 2; x^2 - 4, x >= 2 } is differentiable at x = 2.
Solution: Set the left-hand limit equal to the right-hand limit and their derivatives at x = 2.
Steps: 11
Step 1: Identify the function f(x) which is defined in two parts: f(x) = ax + 1 for x < 2 and f(x) = x^2 - 4 for x >= 2.
Step 2: To ensure f(x) is differentiable at x = 2, the function must be continuous at that point. This means the left-hand limit (as x approaches 2 from the left) must equal the right-hand limit (as x approaches 2 from the right).
Step 3: Calculate the left-hand limit: f(2) from the left is ax + 1, so substitute x = 2: f(2) = 2a + 1.
Step 4: Calculate the right-hand limit: f(2) from the right is x^2 - 4, so substitute x = 2: f(2) = 2^2 - 4 = 0.
Step 5: Set the left-hand limit equal to the right-hand limit: 2a + 1 = 0.
Step 6: Solve for a: 2a + 1 = 0 leads to 2a = -1, so a = -1/2.
Step 7: Next, check the derivatives to ensure differentiability. Find the derivative of the left part: f'(x) = a for x < 2, and the derivative of the right part: f'(x) = 2x for x >= 2.
Step 8: Calculate the left-hand derivative at x = 2: f'(2) from the left is a.
Step 9: Calculate the right-hand derivative at x = 2: f'(2) from the right is 2 * 2 = 4.
Step 10: Set the left-hand derivative equal to the right-hand derivative: a = 4.
Step 11: Since we found a = -1/2 from continuity and a = 4 from differentiability, we need to check if both conditions can be satisfied. They cannot, so we conclude that the function cannot be made differentiable at x = 2 with a single value of a.
