Determine the value of a for which the function f(x) = { x^2 + a, x < 1; 2x +

Determine the value of a for which the function f(x) = { x^2 + a, x < 1; 2x + 3, x >= 1 } is differentiable at x = 1.

Determine the value of a for which the function f(x) = { x^2 + a, x < 1; 2x + 3, x >= 1 } is differentiable at x = 1.

Step 1: Understand that the function f(x) is defined in two parts: one for x < 1 and another for x >= 1.
Step 2: Identify the two parts of the function: f(x) = x^2 + a for x < 1 and f(x) = 2x + 3 for x >= 1.
Step 3: To find the value of a for which the function is differentiable at x = 1, we need to ensure that the function is continuous at x = 1.
Step 4: Calculate f(1-) which is the value of the function as x approaches 1 from the left: f(1-) = 1^2 + a = 1 + a.
Step 5: Calculate f(1+) which is the value of the function as x approaches 1 from the right: f(1+) = 2(1) + 3 = 5.
Step 6: Set f(1-) equal to f(1+) to ensure continuity: 1 + a = 5.
Step 7: Solve for a: a = 5 - 1 = 4.
Step 8: Next, we need to check the derivatives to ensure differentiability at x = 1.
Step 9: Calculate the derivative from the left: f'(x) = 2x, so f'(1-) = 2(1) = 2.
Step 10: Calculate the derivative from the right: f'(x) = 2, so f'(1+) = 2.
Step 11: Set f'(1-) equal to f'(1+) to ensure differentiability: 2 = 2, which is true.
Step 12: Since both conditions (continuity and equal derivatives) are satisfied, we conclude that a = 4 for the function to be differentiable at x = 1.

Piecewise Functions – Understanding how to evaluate and analyze functions defined in pieces based on the input value.
Continuity and Differentiability – Knowing that for a function to be differentiable at a point, it must first be continuous at that point.
Limits and Derivatives – Applying the concept of limits to find the left-hand and right-hand derivatives at a specific point.