A function is differentiable if it has a derivative at every point in its domain. This means that you can find the slope of the tangent line at any point on its graph.
Imagine walking along a path with varying terrain. If you can smoothly walk along the path without stumbling or falling, then it's differentiable. However, if there are sudden cliffs or steep drops, it's not differentiable.
Derivative: The rate at which a function is changing at a particular point. It represents the slope of the tangent line to the graph of the function at that point.
Continuity: A property of functions where there are no breaks, jumps, or holes in its graph. Differentiability implies continuity, but continuity does not necessarily imply differentiability.
Critical Point: A point on a function where its derivative is either zero or undefined. These points can be local maxima, local minima, or points of inflection.
If a function is differentiable at a point, it must also be:
Which of the following functions is both continuous and differentiable at x = 0?
If a function is differentiable on the interval (a, b), which of the following must be true?
If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), which of the following statements is true?
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