객관식
Determine if the graph of the function is continuous and/or differentiable at .
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1
rank
2. Intro to Derivatives
Differentiability
Multiple Choice
Determine if the function is continuous and/or differentiable at .
A
Continuous and non-differentiable
B
Continuous and differentiable
C
Discontinuous and non-differentiable
D
Discontinuous and differentiable
검증된 단계별 안내1
Step 1: To determine if the function is continuous at x=2, check if the left-hand limit (as x approaches 2 from the left) equals the right-hand limit (as x approaches 2 from the right) and if both equal f(2). For x < 2, f(x) = x^3, and for x ≥ 2, f(x) = (x - 2)^2.
Step 2: Calculate the left-hand limit of f(x) as x approaches 2 from the left. Substitute x=2 into the expression x^3 to find the value of the limit.
Step 3: Calculate the right-hand limit of f(x) as x approaches 2 from the right. Substitute x=2 into the expression (x - 2)^2 to find the value of the limit.
Step 4: Compare the left-hand limit, right-hand limit, and f(2). If all three values are equal, the function is continuous at x=2. If not, the function is discontinuous at x=2.
Step 5: To determine differentiability at x=2, check if the derivative of f(x) from the left-hand side equals the derivative from the right-hand side at x=2. Compute the derivative of x^3 for x < 2 and the derivative of (x - 2)^2 for x ≥ 2, then evaluate both derivatives at x=2 and compare their values.
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