Calculus Basics
Build your calculus foundation with this essential flashcard deck covering the core concepts of differential and integral calculus. Calculus is the mathematics of change and accumulation — it powers physics, engineering, economics, machine learning, and virtually every quantitative field. This deck covers limits and continuity, the definition of the derivative, all major differentiation rules (power, product, quotient, chain), derivatives of standard functions, the Fundamental Theorem of Calculus, indefinite and definite integrals, integration techniques, and applications including maxima/minima and area under a curve. Essential for AP Calculus AB/BC, A-Level Further Maths, and first-year university calculus.
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Question
What is a limit?
Answer
A limit describes the value a function approaches as the input approaches a given value. Written as lim[x→a] f(x) = L. The function does not have to actually reach L — just approach it.
Question
What is the definition of a derivative?
Answer
f'(x) = lim[h→0] (f(x+h) − f(x)) / h — the instantaneous rate of change of f(x). It gives the slope of the tangent line at any point on the curve.
Question
What is the power rule for differentiation?
Answer
If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Multiply by the exponent, then reduce the exponent by 1. Example: d/dx(x⁵) = 5x⁴.
Question
What is the constant rule for differentiation?
Answer
The derivative of any constant is 0. d/dx(c) = 0. Example: d/dx(7) = 0. Constants have no rate of change.
Question
What is the sum/difference rule for differentiation?
Answer
d/dx[f(x) ± g(x)] = f'(x) ± g'(x). Differentiate each term separately and add or subtract.
Question
What is the product rule for differentiation?
Answer
If h(x) = f(x)·g(x), then h'(x) = f'(x)g(x) + f(x)g'(x). Memory: "first times derivative of second plus second times derivative of first."
Question
What is the quotient rule for differentiation?
Answer
If h(x) = f(x)/g(x), then h'(x) = [f'(x)g(x) − f(x)g'(x)] / [g(x)]². Memory: "low d-high minus high d-low over low squared."
Question
What is the chain rule for differentiation?
Answer
If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x). Differentiate the outer function (keeping the inner unchanged) then multiply by the derivative of the inner function.
Question
What is the derivative of sin(x), cos(x), tan(x)?
Answer
d/dx(sin x) = cos x. d/dx(cos x) = −sin x. d/dx(tan x) = sec²x.
Question
What is the derivative of eˣ and ln(x)?
Answer
d/dx(eˣ) = eˣ — the only function that is its own derivative. d/dx(ln x) = 1/x.
Question
What is the derivative of aˣ and logₐ(x)?
Answer
d/dx(aˣ) = aˣ ln a. d/dx(logₐ x) = 1/(x ln a). These are the generalized forms of d/dx(eˣ) and d/dx(ln x).
Question
What does the first derivative tell you about a function?
Answer
f'(x) > 0: function is increasing. f'(x) < 0: function is decreasing. f'(x) = 0: stationary point (possible maximum, minimum, or inflection).
Question
What does the second derivative tell you?
Answer
f''(x) > 0: function is concave up (cup shape) — local minimum. f''(x) < 0: function is concave down (cap shape) — local maximum. f''(x) = 0: possible inflection point.
Question
What is the second derivative test for maxima/minima?
Answer
At a stationary point where f'(x)=0: f''(x) < 0 → local maximum. f''(x) > 0 → local minimum. f''(x) = 0 → inconclusive — use another method.
Question
What is an indefinite integral?
Answer
The reverse of differentiation: ∫f(x)dx = F(x) + C where F'(x) = f(x) and C is the constant of integration. Finding all antiderivatives of f(x).
Question
What is the power rule for integration?
Answer
∫xⁿ dx = xⁿ⁺¹/(n+1) + C (for n ≠ −1). Add 1 to the exponent, then divide by the new exponent. Example: ∫x³ dx = x⁴/4 + C.
Question
What is the Fundamental Theorem of Calculus?
Answer
∫[a to b] f(x) dx = F(b) − F(a) where F is any antiderivative of f. It connects differentiation and integration — they are inverse operations.
Question
What is a definite integral and what does it represent?
Answer
∫[a to b] f(x) dx — the signed area between the curve y=f(x) and the x-axis from x=a to x=b. Area above x-axis is positive, area below is negative.
Question
What is integration by substitution?
Answer
Let u = g(x), then du = g'(x)dx. Rewrite the integral in terms of u, integrate, then substitute back. Used to simplify composite functions. Analogous to the chain rule in reverse.
Question
What is the L'Hôpital's Rule?
Answer
If lim[x→a] f(x)/g(x) gives 0/0 or ∞/∞, then lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x) — differentiate numerator and denominator separately, then take the limit.
Question
What is the area between two curves?
Answer
A = ∫[a to b] [f(x) − g(x)] dx where f(x) ≥ g(x) on [a,b]. The upper curve minus the lower curve, integrated over the interval.
Question
What does continuity of a function require?
Answer
A function is continuous at x=a if: 1. f(a) is defined. 2. lim[x→a] f(x) exists. 3. lim[x→a] f(x) = f(a). All three must be true — no holes, jumps, or asymptotes.
Question
What is implicit differentiation?
Answer
Differentiate both sides of an equation with respect to x, treating y as a function of x. Whenever you differentiate a term with y, multiply by dy/dx. Then solve for dy/dx.
Question
What is the derivative of inverse trig functions?
Answer
d/dx(arcsin x) = 1/√(1−x²). d/dx(arccos x) = −1/√(1−x²). d/dx(arctan x) = 1/(1+x²). These appear frequently in integration problems.