AP Calculus AB conceptual understanding of limits, derivative rules, and the Fundamental Theorem of Calculus, explained in plain language rather than as bare formulas.
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- What does it mean for a limit to exist at a point?
- The left and right limits must both exist and be equal to the same value.
- How does a one-sided limit differ from a two-sided limit?
- A one-sided limit approaches from only one direction (left or right), while a two-sided limit requires approach from both directions to the same value.
- What is the definition of continuity at a point?
- A function is continuous at a point if the limit exists there, the function is defined there, and the limit equals the function value.
- Name three types of discontinuities.
- Removable (hole), jump (step), and infinite (vertical asymptote).
- What does it mean when a limit goes to infinity?
- The function values grow without bound as the input approaches a value, or the function has a vertical asymptote.
- What is the primary algebraic strategy for evaluating limits?
- Try direct substitution first; if that gives 0/0 (indeterminate), factor, rationalize, or simplify to resolve it.
- What does the derivative represent geometrically?
- The slope of the tangent line to the curve at a point.
- What does the derivative represent in a real-world context?
- The instantaneous rate of change, such as velocity (change in position), acceleration (change in velocity), or any other rate.
- How is the derivative formally defined as a limit?
- f'(x) = lim(h->0) [f(x+h) - f(x)] / h, the limit of the average rate of change as the interval shrinks to zero.
- What does a negative f'(x) tell you about the function?
- The function is decreasing at that point; the output values are going down as x increases.
- What does f'(x) = 0 typically indicate?
- A critical point where the tangent line is horizontal; the function may have a local maximum, local minimum, or a horizontal inflection point.
- What does the second derivative f''(x) tell you about the graph?
- It describes the concavity: positive f''(x) means concave up (curving upward), negative means concave down (curving downward).
- When is a function not differentiable at a point?
- When the function has a corner, cusp, vertical tangent, or discontinuity at that point.
- What is the relationship between continuity and differentiability?
- If a function is differentiable at a point, it must be continuous there; but a function can be continuous without being differentiable.
- State the power rule for derivatives in plain language.
- The derivative of x^n is n times x to the power (n-1); bring down the exponent and reduce the exponent by one.