Question: If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f differentiable at x = 1?
Options:
Yes
No
Only continuous
Only left differentiable
Correct Answer: No
Solution:
f\'(1) from left = 2 and from right = 2; hence f is differentiable at x = 1.
If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f differentiable at x
Practice Questions
Q1
If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f differentiable at x = 1?
Yes
No
Only continuous
Only left differentiable
Questions & Step-by-Step Solutions
If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f differentiable at x = 1?
Step 1: Identify the function f(x). It is defined in two parts: f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1.
Step 2: To check if f is differentiable at x = 1, we need to find the derivative from the left side (as x approaches 1 from the left) and from the right side (as x approaches 1 from the right).
Step 3: Calculate the derivative from the left side. For x < 1, f(x) = x^2. The derivative f'(x) = 2x. So, f'(1) from the left is 2 * 1 = 2.
Step 4: Calculate the derivative from the right side. For x ≥ 1, f(x) = 2x - 1. The derivative f'(x) = 2. So, f'(1) from the right is 2.
Step 5: Compare the left and right derivatives. Since both f'(1) from the left and f'(1) from the right are equal to 2, we conclude that f is differentiable at x = 1.
Piecewise Functions – Understanding how to analyze functions defined by different expressions over different intervals.
Differentiability – Determining if a function is differentiable at a point by checking the equality of left-hand and right-hand derivatives.
Continuity – Recognizing that a function must be continuous at a point for it to be differentiable there.
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