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 virtua...
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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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.
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.
If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹. Multiply by the exponent, then reduce the exponent by 1. Example: d/dx(x⁵) = 5x⁴.
The derivative of any constant is 0. d/dx(c) = 0. Example: d/dx(7) = 0. Constants have no rate of change.
d/dx[f(x) ± g(x)] = f'(x) ± g'(x). Differentiate each term separately and add or subtract.
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."
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."
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.
d/dx(eˣ) = eˣ — the only function that is its own derivative. d/dx(ln x) = 1/x.
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).
f'(x) > 0: function is increasing. f'(x) < 0: function is decreasing. f'(x) = 0: stationary point (possible maximum, minimum, or inflection).
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.
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.
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).
∫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.
∫[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.
∫[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.
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.
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.
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.
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.
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.
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.
