MIT 18.01SC Single Variable Calculus Notes

Single Variable Calculus — Exam Study Notes

Course: MIT 18.01SC Single Variable Calculus (David Jerison, MIT OpenCourseWare)

Style: one complete set of study notes organized the way you actually compute — limit, derivative, integral, then the infinite. Every numbered example is machine-verified. This is a study guide, not a lecture transcript.

Notation: 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).

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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

• 0/0 is not an answer — it is a signal to rewrite.
• sin(x)/x → 1 is the trig-limit engine.
• Continuous does not mean differentiable.

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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

• Can write the limit definition and compute it for a polynomial.
• Can read increasing / decreasing / concavity from f' and f''.
• |x| is the standard “continuous but not differentiable” example.

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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

• Chain rule on every composition, including e^(u), sin(u), ln(u).
• Product rule has two terms; do not differentiate only one factor.
• Quotient: minus in the numerator, square in the denominator.

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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

• After differentiating implicitly, collect every y' term before dividing.
• Inverse derivative: 1 over the original derivative, evaluated at the inverse.
• Circle example: tangent ⊥ radius.

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PART 5 — GRAPHING AND OPTIMIZATION

5.1 Curve sketching

1. Domain, intercepts, asymptotes.
2. f' = 0 or undefined → critical points; sign chart for increase/decrease.
3. f'' = 0 or undefined → concavity; inflection where concavity changes.
4. 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

• Endpoints on a closed interval.
• Constraint first, then A'(x) = 0, then check it is a max.
• f'' = 0 is not an inflection by itself.

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PART 6 — RELATED RATES AND APPROXIMATIONS

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. If dy/dx = g(x) h(y) and h(y) ≠ 0,

∫ dy / h(y) = ∫ g(x) dx

then pin +C with an initial condition. The family of antiderivatives is required; a definite integral will not choose C for you.

Example 6.5 (verified). dy/dx = y, y(0) = 1. 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

• Related rates: equation, d/dt, then numbers.
• Linear approx uses f and f' at the nearby easy point; quadratic adds the f'' term.
• MVT needs continuity on the closed interval and differentiability inside.
• Separable DE: split variables, integrate, then pin C.

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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: d/dx [ ∫ from a to x of f(t) dt ] = f(x). Differentiating a definite integral in the upper limit returns the integrand.
• FTC 2: ∫ from a to b of f = F(b) − F(a) if F' = f.

If the upper limit is u(x), chain rule: 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

• +C on indefinite; never on a definite evaluation.
• Upper-limit chain rule when the bound is not x.
• Average value is an integral, not f at the midpoint (unless f is linear).

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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 of radius y, so

S = 2π ∫ y √(1+(y')^2) dx

About the y-axis, the radius is x instead of y. 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

• Name the axis before choosing disk vs shell.
• Arc length has √(1 + (y')^2), not √(1 + y^2).
• Surface of revolution is 2π ∫ (radius) ds, not the volume disk formula.
• Work is force integrated along the path, not F times the final x blindly.

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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

• Sub before parts when the inside derivative is visible.
• Parts: differentiate u (simpler), integrate dv (still integrable).
• Partial fractions: include every repeated-factor term.

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PART 10 — PARAMETRIC POLAR AND INDETERMINATE FORMS

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

• Parametric slope is a ratio of derivatives, not y/x.
• Polar area uses r^2.
• L'Hôpital only after you have 0/0 or ∞/∞.

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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 series

f(x) ≈ Σ f^(n)(a) / n! · (x−a)^n. Near a, the first terms are the linear and quadratic approximations you already used.

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: Lagrange form uses the next derivative at some c between a and x. For exams, knowing the first three terms plus the interval of convergence is the usual ask.

11.4 Part 11 checklist

• Improper = a limit. Write the limit.
• a(n) → 0 does not prove convergence.
• Taylor of e, sin, cos at 0 from memory.
• Geometric |r| < 1 is the cleanest series.

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APPENDIX A — FORMULA SHEET

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: 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: ∫ dy/h(y) = ∫ g(x) dx if y' = g(x) h(y)

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
  e^x, sin x, cos x Taylor at 0

APPENDIX B — WORKED MINI-EXAM

1. lim x→0 sin(5x)/x ? → 5
2. lim x→0 (1−cos x)/x^2 ? → 1/2
3. f(x)=x^2 from the definition. f'(x)? → 2x
4. d/dx (x^2 sin x) at x=π ? → −π^2
5. d/dt sin(10t) at t=0 ? → 10
6. x^2+y^2=25 at (3,4). y'? → −3/4
7. Perimeter 100 rectangle, max area? → 625
8. 10 ft ladder, x=6, x'=2. y'? → −1.5
9. √4.1 by linearization at 4? → 2.025
10. Balloon r=10, r'=2. dV/dt? → 800π
11. Integral from 0 to 1 of x^2 dx ? → 1/3
12. d/dx of integral from 0 to x^2 of t^2 dt ? → 2x^5
13. y=x, 0..1, around x-axis. Volume? → π/3
14. Same region around y-axis. Volume? → 2π/3
15. Arc length of y=x on [0,1]? → √2
16. Integral from 0 to π/2 of x cos x dx ? → π/2 − 1
17. Integral from 1 to 2 of dx/(x(x+1)) ? → ln(4/3) ≈ 0.2877
18. Integral from 1 to ∞ of x^(−2) dx ? → 1
19. Σ (1/2)^n from n=0 to ∞ ? → 2
20. Four-term Taylor of e at 0, plus the x^4/24 term, vs e? → 2.708 vs 2.718
21. Newton for x^2=2, start 1, two steps? → 1.5 then 1.41667
22. MVT for x^3 on [0,2]. c? → √(4/3) ≈ 1.155
23. Spring k=200, stretch 0.1. Work? → 1
24. |x| at 0: continuous? differentiable? → yes, no
25. L'Hôpital on a product 0·∞ without rewriting? → illegal

APPENDIX C — TRAP CHECKLIST

1. 0/0 is unfinished.
2. Forgetting the inner derivative (chain).
3. Product rule with only one term differentiated.
4. Implicit: plugging numbers before d/dt or before solving for y'.
5. Skipping endpoints on a closed interval.
6. Calling f''=0 an inflection automatically.
7. Linearization at the wrong point.
8. +C dropped on an antiderivative; +C kept on a definite integral.
9. FTC with a non-x upper limit and no chain rule.
10. Disk vs shell: wrong axis.
11. Arc length missing the 1 inside the square root.
12. Polar area with r instead of r^2.
13. L'Hôpital on a form that is not 0/0 or ∞/∞.
14. Harmonic series “converges because terms go to 0.”
15. Taylor without saying the expansion point.
16. Geometric series without saying whether n starts at 0 or 1.
17. Surface of revolution with π ∫ y^2 (that is volume, not surface).