Calculus Reference Sheet

For every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε

1

A fundamental limit used to derive trig derivatives

e (Euler's number, approximately 2.71828)

If a limit gives 0/0 or ∞/∞, you can take the derivative of numerator and denominator separately

f(a) is defined, the limit as x→a exists, and the limit equals f(a)

If g(x) ≤ f(x) ≤ h(x) near a, and lim g(x) = lim h(x) = L, then lim f(x) = L

The value a function approaches as x approaches a from only the left (x→a⁻) or only the right (x→a⁺)

0

The instantaneous rate of change of the function, or the slope of the tangent line at a point

f'(x) = lim[h→0] (f(x+h) - f(x))/h

The tangent line is horizontal at x = a — it may be a local maximum, minimum, or inflection point

If a function is differentiable at a point, it must be continuous there (but continuity does not guarantee differentiability)

f''(x) is the derivative of f'(x); it measures concavity — positive means concave up, negative means concave down

d/dx[xⁿ] = nxⁿ⁻¹

cos(x)

-sin(x)

eˣ

The only function that is its own derivative

1/x

d/dx[f·g] = f'·g + f·g'

d/dx[f/g] = (f'g - fg')/g²

Low d-high minus high d-low, over the square of what's below

d/dx[f(g(x))] = f'(g(x)) · g'(x) — derivative of the outside times derivative of the inside

sec²(x)

aˣ · ln(a)

Differentiating both sides of an equation with respect to x, treating y as a function of x and using the chain rule

Find where f'(x) = 0 or is undefined (critical points), then use the first or second derivative test

If f' changes from positive to negative at c, f has a local max; negative to positive gives a local min

If f'(c) = 0 and f''(c) > 0, local min; if f''(c) < 0, local max; if f''(c) = 0, test is inconclusive

Finding the rate of change of one quantity using the known rate of change of a related quantity, via implicit differentiation with respect to time

If f is continuous on [a,b] and differentiable on (a,b), there exists c where f'(c) = (f(b)-f(a))/(b-a)

A point where the concavity of the function changes (f'' changes sign)

Endpoints of the domain (if it's a closed interval) — the absolute max or min could occur there

A function F(x) whose derivative is f(x): F'(x) = f(x)

xⁿ⁺¹/(n+1) + C

The reverse of the power rule

If F is an antiderivative of f, then ∫[a to b] f(x)dx = F(b) - F(a)

d/dx[∫[a to x] f(t)dt] = f(x) — differentiation and integration are inverse operations

ln|x| + C

eˣ + C

A technique that reverses the chain rule: substitute u = g(x), du = g'(x)dx to simplify the integral

∫u dv = uv - ∫v du — reverses the product rule

Remember LIATE for choosing u

sin(x) + C

-cos(x) + C

Evaluate the definite integral ∫[a to b] f(x)dx when f(x) ≥ 0

∫[a to b] (f(x) - g(x))dx, where f(x) ≥ g(x) on [a,b]

(1/(b-a)) · ∫[a to b] f(x)dx

An approximation of a definite integral using the sum of areas of rectangles under the curve

V = π∫[a to b] [f(x)]² dx — rotating a region around an axis creates disc-shaped cross sections

V = π∫[a to b] ([R(x)]² - [r(x)]²)dx — for volumes of revolution with a hollow center

An equation that involves a function and one or more of its derivatives

y = Ceᵏˣ (exponential growth or decay)

This models population growth and radioactive decay

A method of solving DEs by moving all y terms to one side and all x terms to the other, then integrating both sides

A visual representation showing small line segments at grid points indicating the slope (dy/dx) at each point

A differential equation together with a specific initial condition y(x₀) = y₀ that determines the particular solution

Exponential decay — such as radioactive decay, cooling, or depreciation