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 graph of 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: Analyze the graph of the function f(x) at x=1. Observe whether there is a break, jump, or hole in the graph at x=1 to determine continuity. A function is continuous at a point if there is no interruption in the graph at that point.
Step 2: Check if the left-hand limit and right-hand limit of f(x) as x approaches 1 are equal. If they are equal and the function value f(1) exists, the function is continuous at x=1.
Step 3: Examine the slope of the graph at x=1. If the graph has a sharp corner or cusp at x=1, the function is not differentiable at that point. Differentiability requires the graph to be smooth without abrupt changes in direction.
Step 4: Verify if the derivative exists at x=1. A function is differentiable at a point if the derivative can be calculated and is finite. Sharp corners or vertical tangents indicate non-differentiability.
Step 5: Based on the observations, conclude whether the function is continuous and/or differentiable at x=1. Use the definitions of continuity and differentiability to justify your conclusion.
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