Calculus I: Limits & Derivatives
42 cardsEnglish → EnglishIntermediate~22 minUpdated Jul 21, 2026
An early-university level review of introductory Calculus I, covering 41 key definitions, rules, and formulas. Topics include limits, continuity, indeterminate forms and L'Hôpital's Rule, the formal definition of the derivative, differentiation rules (power, product, quotient, chain), derivatives of common functions, critical points, concavity, inflection points, tangent lines, and the Mean Value Theorem. Ideal for reinforcing the foundations before moving on to integral calculus.
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Cards in this deck
- Q:What is a limit?
- A:The value a function approaches as its input approaches a given point.
- Q:How do you write "the limit of f(x) as x approaches a"?
- A:lim(x→a) f(x)
- Q:What is a one-sided limit?
- A:The value a function approaches from only one direction, either the left or the right.
- Q:What does it mean for a limit to not exist at a point?
- A:The left-hand and right-hand limits are not equal, or the function grows without bound.
- Q:What is the definition of continuity at x = a?
- A:f(a) is defined, lim(x→a) f(x) exists, and lim(x→a) f(x) = f(a).
- Q:What is a removable discontinuity?
- A:A point where the limit exists but does not equal the function's value there (a "hole" in the graph).
- Q:What is an indeterminate form?
- A:An expression such as 0/0 or ∞/∞ whose limit cannot be determined directly.
- Q:What is L'Hôpital's Rule used for?
- A:Evaluating limits of indeterminate forms (0/0 or ∞/∞) by differentiating the numerator and denominator.
- Q:What is the formula for L'Hôpital's Rule?
- A:lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x), when the limit is of the form 0/0 or ∞/∞.
- Q:What is the formal definition of a derivative?
- A:f'(x) = lim(h→0) [f(x+h) - f(x)] / h
- Q:What does the derivative of a function represent geometrically?
- A:The slope of the tangent line to the function's graph at a given point.
- Q:What does the derivative of a function represent physically?
- A:The instantaneous rate of change of a quantity.
- Q:What is another common notation for the derivative of y with respect to x?
- A:dy/dx
- Q:What is the Power Rule for derivatives?
- A:d/dx[xⁿ] = n·xⁿ⁻¹
- Q:What is the derivative of a constant?
- A:0
- Q:What is the Constant Multiple Rule for derivatives?
- A:d/dx[c·f(x)] = c·f'(x)
- Q:What is the Sum Rule for derivatives?
- A:d/dx[f(x) + g(x)] = f'(x) + g'(x)
- Q:What is the Difference Rule for derivatives?
- A:d/dx[f(x) - g(x)] = f'(x) - g'(x)
- Q:What is the Product Rule for derivatives?
- A:d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
- Q:What is the Quotient Rule for derivatives?
- A:d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]²
- Q:What is the Chain Rule for derivatives?
- A:d/dx[f(g(x))] = f'(g(x)) · g'(x)
- Q:What is the derivative of sin(x)?
- A:cos(x)
- Q:What is the derivative of cos(x)?
- A:-sin(x)
- Q:What is the derivative of tan(x)?
- A:sec²(x)
- Q:What is the derivative of eˣ?
- A:eˣ
- Q:What is the derivative of ln(x)?
- A:1/x
- Q:What is the derivative of aˣ for a constant a > 0?
- A:aˣ · ln(a)
- Q:What is a critical point of a function?
- A:A point where f'(x) = 0 or f'(x) is undefined.
- Q:How does the First Derivative Test identify a local maximum?
- A:f'(x) changes from positive to negative at that critical point.
- Q:How does the First Derivative Test identify a local minimum?
- A:f'(x) changes from negative to positive at that critical point.
- Q:What does the Second Derivative Test tell you about a critical point?
- A:If f''(x) > 0 it's a local minimum; if f''(x) < 0 it's a local maximum.
- Q:What is concavity?
- A:The direction a curve bends: concave up (like a cup) or concave down (like a frown).
- Q:How do you determine concavity using the second derivative?
- A:f''(x) > 0 means the graph is concave up; f''(x) < 0 means it is concave down.
- Q:What is an inflection point?
- A:A point on a curve where the concavity changes.
- Q:What is the equation of the tangent line to f at the point (a, f(a))?
- A:y - f(a) = f'(a)(x - a)
- Q:What is the formula for the average rate of change of f(x) over [a, b]?
- A:[f(b) - f(a)] / (b - a)
- Q:What does the Mean Value Theorem state?
- A:If f is continuous on [a,b] and differentiable on (a,b), there exists some c in (a,b) where f'(c) equals the average rate of change over [a,b].
- Q:What does it mean for a function to be differentiable at a point?
- A:The derivative exists at that point, meaning the function has a defined, finite slope there.
- Q:What is the second derivative of a function?
- A:The derivative of the first derivative, denoted f''(x) or d²y/dx².
- Q:What is implicit differentiation used for?
- A:Finding dy/dx when y cannot easily be isolated and solved explicitly in terms of x.
- Q:What is a vertical asymptote?
- A:A vertical line x = a that a function's graph approaches, with the function's value going to ±∞.
- Q:What is a horizontal asymptote?
- A:A horizontal line that a function's graph approaches as x → ∞ or x → -∞.
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