Single Variable Calculus — Exam Study Notes
Symbols used: f'(x) or dy/dx for the derivative; f''(x) for the second derivative. ∫ f(x) dx is an antiderivative (a family, +C). A definite integral from a to b is a number. lim x→a. ln is natural log. Do not write LaTeX braces; powers are a^n or a^(n).
PART 1 — LIMITS AND CONTINUITY
1.1 Why limits
A derivative is a limit of slopes. An integral is a limit of sums. If you cannot take a limit, you do not yet have calculus — you have algebra with a hole.
1.2 Computing limits
Plug in when the function is continuous there. If you get 0/0, algebra first (factor, conjugate, trig identity), not L'Hôpital yet.
Example 1.1 (verified). lim x→0 sin(x)/x = 1. Numerically: x = 0.1 → 0.9983; x = 0.01 → 0.99998. The squeeze theorem is the proof; the table is the memory.
Example 1.2 (verified). lim x→0 sin(5x)/x = 5. Rewrite as 5 · sin(5x)/(5x). Never cancel the 5 against the x and stop.
Example 1.3 (verified). lim x→0 (1 − cos(x))/x^2 = 1/2. Numerically x = 0.01 → 0.5000. Later L'Hôpital gets the same 1/2; the trig identity 1 − cos(x) = 2 sin^2(x/2) also gets it.
Infinite limits: as x→∞, a rational function is dominated by the highest powers. x/e^x → 0 (exponential beats polynomial).
1.3 Continuity
f is continuous at a if lim x→a f(x) = f(a). Three failures: hole (removable), jump, vertical asymptote.
Differentiable ⇒ continuous. The converse is false: |x| is continuous at 0 and not differentiable there. A corner or a cusp kills the derivative.
1.4 Part 1 checklist
PART 2 — THE DERIVATIVE
2.1 Definition
f'(a) = lim h→0 [f(a+h) − f(a)] / h
Geometry: slope of the tangent. Physics: instantaneous rate. Units: output units per input unit.
Example 2.1. f(x) = x^2. [ (x+h)^2 − x^2 ] / h = 2x + h → 2x. The definition is how you justify the power rule, not how you live.
2.2 What the sign says
- f' > 0: increasing. f' < 0: decreasing.
- f'' > 0: concave up (slope increasing). f'' < 0: concave down.
- f' = 0: critical point — could be max, min, or neither (x^3 at 0).
2.3 Differentiability fails when
A corner (|x|), a cusp (x^(2/3)), a vertical tangent (x^(1/3) at 0), or a jump. On a closed interval the derivative is a one-sided limit at the endpoints.
2.4 Part 2 checklist
PART 3 — DIFFERENTIATION RULES
3.1 The short list (memorize, then never expand by hand)
- (c)' = 0, (x^n)' = n x^(n−1)
- (sin x)' = cos x, (cos x)' = −sin x, (tan x)' = sec^2 x
- (e^x)' = e^x, (ln x)' = 1/x, (a^x)' = a^x ln a
- (arcsin x)' = 1/√(1−x^2), (arctan x)' = 1/(1+x^2)
3.2 Product, quotient, chain
- Product: (uv)' = u'v + uv'
- Quotient: (u/v)' = (u'v − uv') / v^2
- Chain: (f(g(x)))' = f'(g(x)) · g'(x) — “outside derivative, leave inside, times inside derivative”
Example 3.1 (verified). d/dx [x^2 sin x] at x = π: 2x sin x + x^2 cos x = 0 + π^2 (−1) = −π^2 ≈ −9.870.
Example 3.2 (verified). d/dt sin(10t) = 10 cos(10t). At t = 0 this is 10. Forgetting the inner 10 is the most common chain-rule miss.
Example 3.3. d/dx e^(2x) = 2 e^(2x). At x = 0: 2.
You can derive the quotient rule from the product rule and the chain rule on v^(−1). On an exam, either path is legal; the product+chain path is harder to botch.
3.3 Higher derivatives
f'' is the derivative of f'. For position s(t): s' = velocity, s'' = acceleration. The nth derivative of sin x cycles every 4.
3.4 Part 3 checklist
PART 4 — IMPLICIT AND INVERSE FUNCTIONS
4.1 Implicit differentiation
If y is defined by F(x,y) = c, differentiate both sides, treating y as a function of x, then solve for y'.
Example 4.1 (verified). x^2 + y^2 = 25. 2x + 2y y' = 0 ⇒ y' = −x/y. At (3,4): −3/4 = −0.75. Check: the radius to (3,4) has slope 4/3; the tangent is perpendicular, so slope −3/4.
4.2 Inverse functions
If y = f^(−1)(x) and f'(y) ≠ 0, then dy/dx = 1 / f'(y). That is how arcsin and arctan get their formulas.
Example 4.2. d/dx ln x = 1/x, because ln is the inverse of e^x and (e^y)' = e^y = x.
4.3 Exponential, e, and hyperbolic
e is the base whose derivative is itself. lim n→∞ (1 + 1/n)^n = e. Hyperbolic: sinh' = cosh, cosh' = sinh, and cosh^2 − sinh^2 = 1 (a sign flip from trig).
4.4 Part 4 checklist
PART 5 — GRAPHING AND OPTIMIZATION
5.1 Curve sketching
- Domain, intercepts, asymptotes.
- f' = 0 or undefined → critical points; sign chart for increase/decrease.
- f'' = 0 or undefined → concavity; inflection where concavity changes.
- Sketch. First-derivative test or second-derivative test to classify extrema.
Trap: f'' = 0 does not prove an inflection (x^4 at 0: f''=0 but concave up on both sides).
5.2 Closed-interval max/min
On [a,b], extrema occur at critical points or endpoints. Always test the endpoints. An open interval may have no max.
Example 5.1 (verified). 100 m of fence, three sides of a rectangular pen against a wall. If the side parallel to the wall is w and the two ends are x, then w + 2x = 100, A = w x = (100−2x)x = 100x − 2x^2.
A' = 100 − 4x = 0 ⇒ x = 25, w = 50. Max area = 50 · 25 = 1250 m^2. This is not a square: a square would mean all four sides fenced.
Example 5.2 (verified, four-sided). Rectangle of perimeter 100: x + y = 50, A = x(50−x). A' = 50 − 2x = 0 ⇒ x = 25, square, A = 625.
Name the constraint before you differentiate.
5.3 Part 5 checklist
6.1 Related rates
Draw, name, write the equation that is always true, differentiate with respect to time, then plug in the instant.
Example 6.1 (ladder, verified). 10 ft ladder, bottom 6 ft from the wall and sliding out at 2 ft/s. Then height y = 8 (because 6-8-10). 2x x' + 2y y' = 0 ⇒ y' = −x x' / y = −6·2 / 8 = −1.5 ft/s. The top is falling at 1.5 ft/s.
Example 6.2 (balloon, verified). Sphere, dr/dt = 2, r = 10. V = (4/3)π r^3 ⇒ dV/dt = 4π r^2 dr/dt = 4π · 100 · 2 = 800π ≈ 2513.
Trap: plug in the numbers after differentiating. If you plug in x = 6 first, you have differentiated a constant.
6.2 Linear approximation
f(x) ≈ f(a) + f'(a)(x − a). Tangent line, not the function.
Example 6.3 (verified). √4.1 ≈ √4 + (1/(2·2)) · 0.1 = 2.025. True value ≈ 2.02485. The linear estimate is slightly high because √x is concave down.
Quadratic approximation (18.01SC Sessions 25–26): f(x) ≈ f(a) + f'(a)(x − a) + f''(a)(x − a)^2 / 2.
Example 6.3b (verified). For √x at 4, f'' = −1/(4 x^(3/2)) = −1/32. Quadratic: 2.025 + (1/2)(−1/32)(0.01) = 2.025 − 0.00015625 = 2.02484. True √4.1 ≈ 2.02485. The second-order term undoes the linear overshoot.
Newton’s method: x(n+1) = x(n) − f(x(n))/f'(x(n)). For x^2 − 2 = 0, start at 1: 1.5, then 1.41667, then 1.41422, already at √2 ≈ 1.41421.
6.3 Mean value theorem
If f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) has f'(c) = [f(b)−f(a)]/(b−a). Average slope is attained.
Example 6.4 (verified). f(x) = x^3 on [0,2]. Average slope = 4. 3c^2 = 4 ⇒ c = √(4/3) ≈ 1.155.
Rolle is MVT when f(a) = f(b): some c with f'(c) = 0.
6.4 Separable differential equations
18.01SC Exam 2 includes separation of variables. First solve h(y₀) = 0: every root gives a constant equilibrium solution y(x) = y₀. Then, on intervals where h(y) ≠ 0, divide and separate:
∫ dy / h(y) = ∫ g(x) dx
Then pin +C with an initial condition. Dividing by h(y) before checking its zeros can erase valid solutions. The family of antiderivatives is required; a definite integral will not choose C for you.
Example 6.5 (verified). dy/dx = y has the equilibrium solution y = 0. For the initial condition y(0) = 1, that equilibrium is impossible, so separate: dy/y = dx, ln|y| = x + C. y(0) = 1 ⇒ C = 0, so y = e^x. Check: (e^x)' = e^x and e^0 = 1.
6.5 Part 6 checklist
PART 7 — DEFINITE INTEGRALS AND FUNDAMENTAL THEOREM
7.1 Riemann sums
∫ from a to b of f(x) dx = lim of Σ f(x(i)) Δx. Area if f ≥ 0; net signed area if f dips below.
Example 7.1 (verified). ∫ from 0 to 1 of x^2 dx = x^3/3 from 0 to 1 = 1/3.
7.2 Indefinite vs definite
- Antiderivative: F'(x) = f(x). Write +C. The family matters when you solve DEs.
- Definite integral: a number. +C cancels.
7.3 Fundamental theorem
- FTC 1: if f is continuous, d/dx [ ∫ from a to x of f(t) dt ] = f(x). Differentiating a running integral returns the integrand at points of continuity.
- FTC 2: if f is continuous on [a,b] and F' = f, then ∫ from a to b of f = F(b) − F(a).
If f is continuous at u(x) and u is differentiable, the upper-limit chain rule gives f(u(x)) u'(x).
Example 7.2. d/dx ∫ from 0 to x of t^2 dt = x^2. d/dx ∫ from 0 to x^2 of t^2 dt = (x^2)^2 · 2x = 2x^5.
7.4 Average value
(1/(b−a)) ∫ from a to b of f. For x^2 on [0,1]: 1/3.
7.5 Part 7 checklist
PART 8 — APPLICATIONS OF INTEGRATION
8.1 Area and volume
Area between curves: ∫ (top − bottom) dx, or (right − left) dy.
Volumes of revolution:
- Disk / washer (slicing perpendicular to the axis): V = π ∫ (R^2 − r^2) dx
- Shell (cylinders parallel to the axis): V = 2π ∫ (radius)(height) dx
Example 8.1 (verified). Region under y = x, 0 ≤ x ≤ 1.
- Around the x-axis (disks): V = π ∫ x^2 dx = π/3 ≈ 1.047.
- Around the y-axis (shells): V = 2π ∫ x · x dx = 2π/3 ≈ 2.094.
Same region, different axis, different volume. Draw which way you slice.
Sphere of radius 1 by washers: V = π ∫ from −1 to 1 of (1 − x^2) dx = 4π/3.
8.2 Arc length and surface area
L = ∫ √(1 + (y')^2) dx. For y = x on [0,1]: √(1+1) = √2 ≈ 1.414.
Surface of revolution (18.01SC Session 79): rotate y = f(x) about the x-axis. Each arc ds sweeps a band whose radius is distance to the axis, |y|, so
S = 2π ∫ |y| √(1+(y')^2) dx
If y ≥ 0, this simplifies to the familiar formula without absolute values. About the y-axis, the radius is |x| instead. Do not use the volume disk formula π ∫ y^2 dx for a surface.
Example 8.2 (verified). y = x on [0,1] about the x-axis: S = 2π ∫ from 0 to 1 of x √2 dx = π√2 ≈ 4.443. Same as a cone of radius 1 and slant √2: π r ℓ = π√2.
8.3 Work, springs
W = ∫ F(x) dx. Hooke: F = kx. Stretch a spring with k = 200 N/m by 0.1 m: W = ∫ from 0 to 0.1 of 200x dx = 100 x^2 = 1 J.
8.4 Probability (the 18.01 cameo)
A PDF f ≥ 0 with total integral 1. P(a ≤ X ≤ b) = ∫ from a to b of f. Mean = ∫ x f(x) dx. This is the same integral you already compute; the story is a density, not an area of land.
8.5 Numerical integration
Trapezoid and Simpson when you have data, not a formula. More intervals ⇒ generally less error for smooth f. Simpson needs an even number of intervals.
8.6 Part 8 checklist
PART 9 — TECHNIQUES OF INTEGRATION
9.1 Substitution
u = inside, du = inside' dx. The 18.01 tell: the derivative of the inside is sitting in the integrand, up to a constant.
Example 9.1. ∫ 2x / (x^2 + 1) dx = ln|x^2 + 1| + C.
9.2 Inverse sub and completing the square
√(a^2 − x^2) → x = a sin θ. √(x^2 + a^2) → x = a tan θ. Completing the square turns x^2 + 6x + 13 into (x+3)^2 + 4.
9.3 Partial fractions
Factor the denominator. Linear factors → A/(x−r). Repeated linear → also B/(x−r)^2. Irreducible quadratic → (Cx+D)/(quad).
Example 9.2 (verified). 1/(x(x+1)) = 1/x − 1/(x+1). ∫ from 1 to 2 = [ln x − ln(x+1)] from 1 to 2 = ln(2/3) − ln(1/2) = ln(4/3) ≈ 0.2877.
9.4 Integration by parts
∫ u dv = uv − ∫ v du. LIATE for choosing u: Log, Inverse trig, Algebraic, Trig, Exponential.
Example 9.3 (verified). ∫ from 0 to π/2 of x cos x dx. u = x, dv = cos x dx ⇒ uv − ∫ v du = x sin x − ∫ sin x = x sin x + cos x. Evaluate: (π/2 · 1 + 0) − (0 + 1) = π/2 − 1 ≈ 0.5708.
9.5 Trig integrals
Odd power of sine or cosine: peel one factor, use sin^2 + cos^2 = 1. Integrals of tan and sec have their own short list.
9.6 Part 9 checklist
10.1 Parametric
x(t), y(t). dy/dx = (dy/dt) / (dx/dt) when dx/dt ≠ 0. Arc length: ∫ √( (dx/dt)^2 + (dy/dt)^2 ) dt.
10.2 Polar
x = r cos θ, y = r sin θ. Area: (1/2) ∫ r^2 dθ. Do not use (1/2)∫ r dθ.
10.3 L'Hôpital
For 0/0 or ∞/∞: lim f/g = lim f'/g' when the latter exists. Not for 0·∞ until you rewrite as a quotient. Not a substitute for algebra when algebra is one line.
Example 10.1 (verified). lim x→0 (1 − cos x)/x^2. L'Hôpital twice: sin x / (2x) then cos x / 2 → 1/2. Same as Example 1.3.
Trap: applying L'Hôpital to 0/1 or to a product you did not rewrite. Differentiating top and bottom is not the quotient rule — it is a special limit tool.
10.4 Part 10 checklist
PART 11 — IMPROPER INTEGRALS AND SERIES
11.1 Improper integrals
Replace ∞ or a singularity by a limit.
Example 11.1 (verified). ∫ from 1 to ∞ of 1/x^2 dx = lim b→∞ [−1/x] from 1 to b = 1. Converges.
∫ from 1 to ∞ of 1/x dx = lim ln b = ∞. Diverges (harmonic).
p-integrals at infinity: ∫ 1/x^p from 1 to ∞ converges iff p > 1.
11.2 Infinite series
Σ a(n) converges if the partial sums approach a number. Necessary condition: a(n) → 0 (not sufficient: harmonic).
Geometric: Σ from n = 0 to ∞ of a r^n = a/(1−r) when |r| < 1. The sum starts at n = 0. If it starts at n = 1, the sum is a r/(1−r). a = 1, r = 1/2 from n = 0: sum = 2.
Tests 18.01 expects you to name: geometric, p-series, comparison / limit comparison, integral test, ratio test (for later Taylor radius). Alternating series: decrease to 0.
11.3 Taylor polynomials and Taylor series
The degree-N Taylor polynomial is P_N(x) = Σ from n=0 to N of f^(n)(a)/n! · (x−a)^n. It approximates f near a, with remainder R_N(x) = f(x) − P_N(x).
The infinite Taylor series is the limit of these polynomials. You may write f(x) = Σ from n=0 to ∞ of f^(n)(a)/n! · (x−a)^n only at x-values where the series converges and R_N(x) → 0. Infinite differentiability alone does not guarantee equality.
Standard expansions at 0:
- e^x = 1 + x + x^2/2! + x^3/3! + ⋯
- sin x = x − x^3/3! + x^5/5! − ⋯
- cos x = 1 − x^2/2! + x^4/4! − ⋯
- 1/(1−x) = 1 + x + x^2 + ⋯ for |x| < 1
Example 11.2 (verified). e ≈ 1 + 1 + 1/2 + 1/6 + 1/24 = 2.7083 (true e ≈ 2.7183). Five terms already leave an error of about 0.01.
Remainder: if f has the required derivatives on the interval, Lagrange form is R_N(x) = f^(N+1)(c)(x−a)^(N+1)/(N+1)! for some c between a and x. Use it to bound error and prove series equality when R_N(x) → 0. Always state the interval where a power series converges; equality to f still requires the remainder test.
11.4 Part 11 checklist
LIMITS
sin(x)/x → 1, (1−cos x)/x^2 → 1/2, (1+1/n)^n → e
DERIVATIVES
(x^n)' = n x^(n−1) (sin x)' = cos x (cos x)' = −sin x
(e^x)' = e^x (ln x)' = 1/x (arctan x)' = 1/(1+x^2)
(uv)' = u'v+uv' (u/v)' = (u'v−uv')/v^2
chain: f(g(x))' = f'(g(x)) g'(x)
implicit: differentiate both sides, solve for y'
APPLICATIONS OF f'
increasing where f'>0 concave up where f''>0
closed interval: critical points AND endpoints
related rates: equation, d/dt, then plug in
linear: f(a)+f'(a)(x−a)
MVT: f'(c) = [f(b)−f(a)]/(b−a)
Newton: x − f(x)/f'(x)
INTEGRALS
FTC (f continuous): d/dx ∫_a^x f = f(x); ∫_a^b f = F(b)−F(a)
average = (1/(b−a)) ∫ f
disk: π ∫ R^2 dx shell: 2π ∫ (radius)(height) dx
arc length: ∫ √(1+(y')^2) dx
surface (about x-axis): 2π ∫ |y| √(1+(y')^2) dx
work: ∫ F(x) dx
separable: first check h(y₀)=0 equilibria; where h(y)≠0, ∫ dy/h(y) = ∫ g(x) dx
TECHNIQUES
sub: u = inside
parts: ∫ u dv = uv − ∫ v du
partial fractions after factoring
L'Hôpital: 0/0 or ∞/∞ only
INFINITE
∫_1^∞ x^(−p) converges for p>1
geometric Σ_{n=0}^∞ a r^n = a/(1−r) for |r|<1
Taylor: P_N + R_N; series equals f only where it converges and R_N→0
APPENDIX B — WORKED MINI-EXAM
- lim x→0 sin(5x)/x ? → 5
- lim x→0 (1−cos x)/x^2 ? → 1/2
- f(x)=x^2 from the definition. f'(x)? → 2x
- d/dx (x^2 sin x) at x=π ? → −π^2
- d/dt sin(10t) at t=0 ? → 10
- x^2+y^2=25 at (3,4). y'? → −3/4
- Perimeter 100 rectangle, max area? → 625
- 10 ft ladder, x=6, x'=2. y'? → −1.5
- √4.1 by linearization at 4? → 2.025
- Balloon r=10, r'=2. dV/dt? → 800π
- Integral from 0 to 1 of x^2 dx ? → 1/3
- d/dx of integral from 0 to x^2 of t^2 dt ? → 2x^5
- y=x, 0..1, around x-axis. Volume? → π/3
- Same region around y-axis. Volume? → 2π/3
- Arc length of y=x on [0,1]? → √2
- Integral from 0 to π/2 of x cos x dx ? → π/2 − 1
- Integral from 1 to 2 of dx/(x(x+1)) ? → ln(4/3) ≈ 0.2877
- Integral from 1 to ∞ of x^(−2) dx ? → 1
- Σ (1/2)^n from n=0 to ∞ ? → 2
- Four-term Taylor of e at 0, plus the x^4/24 term, vs e? → 2.708 vs 2.718
- Newton for x^2=2, start 1, two steps? → 1.5 then 1.41667
- MVT for x^3 on [0,2]. c? → √(4/3) ≈ 1.155
- Spring k=200, stretch 0.1. Work? → 1
- |x| at 0: continuous? differentiable? → yes, no
- L'Hôpital on a product 0·∞ without rewriting? → illegal
APPENDIX C — TRAP CHECKLIST
- 0/0 is unfinished.
- Forgetting the inner derivative (chain).
- Product rule with only one term differentiated.
- Implicit: plugging numbers before d/dt or before solving for y'.
- Skipping endpoints on a closed interval.
- Calling f''=0 an inflection automatically.
- Linearization at the wrong point.
- +C dropped on an antiderivative; +C kept on a definite integral.
- FTC with a non-x upper limit and no chain rule.
- Disk vs shell: wrong axis.
- Arc length missing the 1 inside the square root.
- Polar area with r instead of r^2.
- L'Hôpital on a form that is not 0/0 or ∞/∞.
- Harmonic series “converges because terms go to 0.”
- Taylor without saying the expansion point, convergence domain, or why the remainder tends to 0.
- Geometric series without saying whether n starts at 0 or 1.
- Surface of revolution with π ∫ y^2 (that is volume, not surface), or using signed y instead of radius |y|.
- Dividing by h(y) in a separable DE before checking h(y)=0 equilibrium solutions.
These are original HyNote study notes for people taking MIT OpenCourseWare’s 18.01SC Single Variable Calculus. They follow the public syllabus. They are not official MIT materials, and they are not lecture transcripts.
