Calculus I – All These Worlds

Precalculus identities used constantly algebra / trig

(a – b)(a + b) = a² – b²

Difference of squares. Common in limit rationalizations.

a^m · a^n = a^(m+n), a^m / a^n = a^(m-n), (a^m)^n = a^(mn)

Exponent rules used to simplify functions and limits.

ln(ab)=ln a + ln b, ln(a/b)=ln a – ln b, ln(a^r)=r ln a

Log laws. Essential for logarithmic differentiation and integral forms.

sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = csc²x

Pythagorean identities. Appear in trig derivatives/integrals and simplification.

tan x = sin x / cos x, sec x = 1/cos x, csc x = 1/sin x, cot x = cos x / sin x

Reciprocal/quotient identities. Useful for rewriting derivatives and integrals.

sin(a±b)=sin a cos b ± cos a sin b; cos(a±b)=cos a cos b ∓ sin a sin b

Angle-sum/difference identities. Sometimes used for limits or simplification.

You won’t usually “learn” these in Calc I, but you use them constantly to make limits and derivatives tractable.

Limits & continuity limits

lim_x→a f(x) = L

Meaning: f(x) can be made arbitrarily close to L by taking x sufficiently close to a (not necessarily equal to a).

lim (f ± g)=lim f ± lim g; lim (c f)=c lim f; lim (f g)=(lim f)(lim g); lim (f/g)=(lim f)/(lim g) if lim g ≠ 0; lim (f)^n=(lim f)^n

Limit laws. Reduce complex limits to simpler ones when constituent limits exist.

lim_x→a⁺ f(x), lim_x→a⁻ f(x)

Right-hand and left-hand limits. Two-sided limit exists only if both agree.

f is continuous at a iff (1) f(a) defined, (2) lim_x→a f(x) exists, (3) lim_x→a f(x)=f(a)

Operational definition of continuity at a point.

lim_x→0 (sin x)/x = 1, lim_x→0 (1 – cos x)/x = 0, lim_x→0 (1 – cos x)/x² = 1/2

Core trigonometric limits (radians). Foundations for trig derivatives.

e = lim_n→∞ (1 + 1/n)^n and e^x = lim_n→∞ (1 + x/n)^n

Limit definitions tied to exponential growth and the derivative of e^x.

lim_x→a f(x) = ±∞

Infinite limit: f(x) grows without bound near x=a (often indicates a vertical asymptote).

lim_x→∞ f(x) = L, lim_x→-∞ f(x) = L

Limits at infinity. Used for end behavior and horizontal asymptotes.

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

Squeeze (Sandwich) Theorem. Classic for trig-based limits.

In many Calc I courses, you also see “indeterminate forms” (0/0, ∞/∞, etc.) as a motivation for derivative-based tools.

Derivatives: definition & rules derivatives

f′(a) = lim_h→0 (f(a+h) − f(a))/h

Derivative at a point. Slope of tangent line; instantaneous rate of change.

y − f(a) = f′(a)(x − a)

Point-slope form of the tangent line to y=f(x) at x=a.

L(x) = f(a) + f′(a)(x − a), dy = f′(x) dx

Linearization near x=a; differentials approximate small changes: Δy ≈ dy.

(c)′ = 0

Derivative of a constant is zero.

(x^n)′ = n x^(n−1)

Power rule (for real n where defined). Workhorse of polynomial derivatives.

(c f(x))′ = c f′(x)

Pull constants out of derivatives.

(f(x) ± g(x))′ = f′(x) ± g′(x)

Differentiate term-by-term for sums/differences.

(f g)′ = f′g + f g′

Derivative of a product. Don’t distribute derivatives incorrectly.

(f/g)′ = (f′g − f g′)/g²

Derivative of a quotient (g ≠ 0). Often simplified afterward.

(f∘g)′(x) = f′(g(x)) · g′(x)

Derivative of a composition. Core rule behind most “complex” derivatives.

Many Calc I exams test whether you can choose the right rule (especially chain + product/quotient together) and simplify cleanly.

Derivatives: common functions derivative table

(e^x)′ = e^x, (a^x)′ = a^x ln(a) (a>0, a≠1)

Exponential derivatives. e is special because its derivative equals itself.

(ln x)′ = 1/x (x>0), (log_a x)′ = 1/(x ln a)

Log derivatives. For ln|x| you can extend to x≠0 (common in practice).

(sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = sec²x

Basic trig derivatives (x in radians).

(sec x)′ = sec x tan x, (csc x)′ = −csc x cot x, (cot x)′ = −csc²x

Other trig derivatives, frequently needed for chain rule practice.

(arcsin x)′ = 1/√(1−x²), (arccos x)′ = −1/√(1−x²), (arctan x)′ = 1/(1+x²)

Inverse trig derivatives (principal branches). Domains matter.

(arcsec x)′ = 1/(|x|√(x²−1)), (arccsc x)′ = −1/(|x|√(x²−1)), (arccot x)′ = −1/(1+x²)

Often optional depending on syllabus. Absolute value appears in arcsec/arccsc formulas.

(sinh x)′=cosh x, (cosh x)′=sinh x, (tanh x)′=sech²x

Sometimes included late in Calc I or early Calc II; included here for completeness.

Implicit & related derivatives implicit

If F(x,y)=0, then d/dx[F(x,y)] = 0 and solve for dy/dx

Differentiate both sides w.r.t. x, treating y as y(x), then isolate dy/dx.

(f^{-1})′(a) = 1 / f′(f^{-1}(a))

Derivative of an inverse function, assuming f′ ≠ 0 and f is invertible near the point.

If y = f(x) > 0, then ln y = ln f(x) and y′ = y · (d/dx[ln f(x)])

Technique: take logs to simplify products/powers, then differentiate implicitly.

Differentiate an equation relating variables w.r.t. time: d/dt[F(x(t), y(t), …)] = 0

Related rates problems: variables change over time; apply chain rule with t.

Implicit differentiation is basically the chain rule under a “solve for dy/dx” workflow.

Key theorems about derivatives theorems

If f is continuous on [a,b], then f attains an absolute max and min on [a,b]

Existence of global extrema on a closed interval. Used in optimization and analysis.

If f continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃c∈(a,b): f′(c)=0

Rolle’s Theorem: equal endpoints imply a horizontal tangent somewhere in between.

If f continuous on [a,b] and differentiable on (a,b), then ∃c: f′(c) = (f(b)−f(a))/(b−a)

Mean Value Theorem: some instantaneous slope equals the average slope.

f′(x)>0 ⇒ f increasing; f′(x)<0 ⇒ f decreasing

Monotonicity test. Used in curve sketching and optimization.

f″(x)>0 ⇒ concave up; f″(x)<0 ⇒ concave down; inflection where concavity changes

Concavity/inflection analysis via the second derivative.

If f′(c)=0 and f″(c)>0 ⇒ local min; if f″(c)<0 ⇒ local max

Second derivative test at a critical point c. Inconclusive if f″(c)=0.

x_n+1 = x_n − f(x_n)/f′(x_n)

Newton’s Method for approximating roots. Often included in Calc I applications.

Some courses add: “If f′(x)=0 for all x in an interval, then f is constant there” (a corollary of MVT).

Indeterminate forms & L’Hôpital’s Rule limits

Indeterminate forms: 0/0, ∞/∞, 0·∞, ∞−∞, 0^0, 1^∞, ∞^0

Forms where naive substitution doesn’t determine the limit. Often require algebra, logs, or L’Hôpital.

If lim f/g gives 0/0 or ∞/∞ and conditions hold, then lim_x→a f(x)/g(x) = lim_x→a f′(x)/g′(x) (if the latter exists).

L’Hôpital’s Rule. Requires differentiability near a and g′ not zero near a (standard course conditions).

Many Calc I syllabi include L’Hôpital only briefly, or move it to Calc II. It’s included because students often want it on a “Calc I formula sheet.”

Antiderivatives & indefinite integrals integrals

F′(x) = f(x) ⇔ F is an antiderivative of f

Antiderivative reverses differentiation (up to a constant).

∫ f(x) dx = F(x) + C where F′(x)=f(x)

Indefinite integral notation for the family of antiderivatives.

∫ x^n dx = x^(n+1)/(n+1) + C (n≠−1)

Power rule for integrals. Special case n = −1 leads to ln|x|.

∫ (1/x) dx = ln|x| + C

Key special case. Absolute value handles x<0 as well.

∫ e^x dx = e^x + C, ∫ a^x dx = a^x/ln(a) + C

Exponential antiderivatives.

∫ cos x dx = sin x + C, ∫ sin x dx = −cos x + C

Basic trig antiderivatives.

∫ sec²x dx = tan x + C, ∫ csc²x dx = −cot x + C,
∫ sec x tan x dx = sec x + C, ∫ csc x cot x dx = −csc x + C

Common trig integral patterns.

∫ f(g(x)) g′(x) dx = ∫ f(u) du where u=g(x)

Substitution (reverse chain rule). The fundamental Calc I integration technique.

Integration techniques beyond substitution (integration by parts, trig substitution, partial fractions) are usually Calc II, not Calc I.

Definite integrals, Riemann sums & FTC definite integral

∫_a^b f(x) dx = lim_n→∞ Σ f(x_i^*) Δx, Δx = (b−a)/n

Definite integral as the limit of Riemann sums (area/accumulation).

∫(f±g)=∫f ± ∫g; ∫ c f = c ∫ f; ∫_a^b f = −∫_b^a f; ∫_a^c f + ∫_c^b f = ∫_a^b f

Core definite-integral properties: linearity, reversing bounds, splitting intervals.

f_avg = (1/(b−a)) ∫_a^b f(x) dx

Average (mean) value of f on [a,b].

If F(x)=∫_a^x f(t) dt, then F′(x)=f(x)

FTC Part 1: differentiation cancels integration for continuous f.

∫_a^b f(x) dx = F(b) − F(a), where F′=f

FTC Part 2: evaluate definite integrals using an antiderivative.

∫_a^b f′(x) dx = f(b) − f(a)

Net change: accumulation of a rate of change equals total change.

d/dx [ ∫_g(x)^h(x) f(t) dt ] = f(h(x)) h′(x) − f(g(x)) g′(x)

Leibniz rule (often taught as an FTC application). Handles variable bounds.

In Calc I, area is usually “signed area” for ∫ f, and “geometric area” is ∫ |f| when requested.

Classic Calc I application formulas applications

v(t)=s′(t), a(t)=v′(t)=s″(t)

Position–velocity–acceleration relationships in 1D motion.

Displacement: ∫_a^b v(t) dt; Distance traveled: ∫_a^b |v(t)| dt

Displacement counts direction; total distance uses absolute value.

Work: W = ∫_a^b F(x) dx

Work by a variable force along a line (basic model).

Area = ∫_a^b (top − bottom) dx

Area between curves using vertical slices (assuming top≥bottom on [a,b]).

Area = ∫_c^d (right − left) dy

Area between curves using horizontal slices, when that setup is simpler.

V = π ∫_a^b [R(x)]² dx, Washer: V = π ∫_a^b (R(x)² − r(x)²) dx

Volumes of revolution about an axis (disk/washer methods).

Shells: V = 2π ∫ (radius)(height) dx or dy

Cylindrical shells method. Sometimes taught in Calc I, sometimes Calc II.

Arc length: L = ∫_a^b √(1 + (f′(x))²) dx

Arc length of y=f(x). Frequently Calc II, but some Calc I syllabi include it.

Surface area: S = 2π ∫_a^b f(x) √(1 + (f′(x))²) dx

Surface area generated by revolving y=f(x) about the x-axis (common form).

Application coverage varies by institution. The items marked “optional” are commonly Calc II topics but appear in some Calc I tracks.

Taylor / linear approximations (optional) approximation

f(x) ≈ f(a) + f′(a)(x−a)

First-order Taylor polynomial (same as linearization).

f(x) ≈ f(a) + f′(a)(x−a) + (f″(a)/2)(x−a)²

Second-order Taylor polynomial. Adds curvature via f″(a).

P_n(x) = Σ_k=0ⁿ f^(k)(a)/k! · (x−a)^k

Nth-degree Taylor polynomial about x=a. Usually Calc II, but sometimes previewed.

If your Calc I course doesn’t cover Taylor polynomials, treat this section as a bridge into Calc II.