Calculus is essential for majors in biology, chemistry, computer science, mathematics, physics, and environmental science and policy. Part I includes functions, limits, continuity, differentiation of algebraic and trigonometric functions, mean value theorem, and various applications. Part II includes; Riemann sums and integrals, techniques and applications of integration improper integrals, transcendental functions; (logarithms, exponential functions, and inverse trigonometric functions). Part III includes further topics from calculus proper (sequences, series, polar coordinates) and introduces linear algebra (vectors, matrices, and linear systems). Though not all results are derived rigorously, care is taken to distinguish intuitive arguments from rigorous proofs. Math 120, 121, and 122 each fulfill the Formal Analysis requirement. Prerequisite for Math 120: appropriate score on the mathematics placement test, or appropriate grade in Math 119. Prerequisite for Math 121: Math 120, or Math 124, or AP credit in Calculus. Prerequisite for Math 122: Math 121. / Offered every fall (120, 122) and spring (121).

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• Numbers.
  • Integers Z. Mathematical induction for positive integers.
    Natural numbers N.
  • See also appendix A.2 of the text.
  • The rational numbers Q, quotients of two integers.
  • The Real numbers R, a complete ordered field.
    See also appendix A.7 of the text.
    A field has operations of addition, subtraction, multiplication, and division and has the usual properties of those operations.
    A field is ordered if it has a distinguished set of positive elements and satisfies three axioms including the axiom of trichotemy.
    It's complete if every cut is made by some element.
  • Completeness of the real numbers, least upper bounds, greatest lower bounds, completeness axiom
  • Complex numbers C.
    Dave's Short Course on Complex Numbers.
    Dave's Short Trig Course
  • Assignment 2 on complex numbers, due Wed, Sept. 21. Answers
• Infinite sequences and indeterminate forms
  • Infinite sequences of real numbers, definition and examples
  • Limits of sequences, convergence and divergence of sequences.
    Def: A sequence converges to a limit L if for each ε > 0, there is some N such that beyond the N^th term, every term is within ε of L.
    See also § 8.1 of the text.
  • Assignment 3 on lubs and glbs, due Mon, Sept. 26 Answers
  • Convergence theorems
    Uniqueness of limits. Monotone bounded sequences have limits.
    Limits of sums, differences, products, and quotients.
    Continuous versus discrete limits. Limits of continuous functions of convergent sequences converge.
    Pinching theorem.
    Summary of limits of sequences.
  • Assignment 4 on limits of sequences, due Wed, Sept. 28. Answers
  • (1 + 1/n)ⁿ → e. Notes.
  • Indeterminate forms 0/0, ∞/∞, ∞ − ∞, 0⁰, 1^∞, ∞⁰
    See also § 4.6 of the text.
    L'Hôpital's rule.
  • Assignment 5 on L'Hôpital's rule, due Mon. Oct. 3. Answers
  • Improper integrals. Integrals where one or both limits of integration are infinity. Integrals of unbounded functions.
    See also § 7.7 of the text.
  • Assignment 6 on L'Hôpital's rule and improper integrals. Answers
• Infinite series
  • Survey of sequences and series.
    See also § 8.2 of the text.
  • Σ notation
    See also §§ 5.2 and 8.2 of the etxt
  • Infinite series (sums) of real numbers
    S_n, the n^th partial sum
    Definition: A series converges to a sum S if S_n → S; the series diverges if the sequence of partial sums S_n diverges
    Geometric series. a + ar + ar² + ... converges to a/(1-r) when  −1 < r < 1, diverges otherwise.
    The harmonic series  1/2 + 1/3 + 1/4 + ...  diverges
    The term test: only series whose terms approach 0 can converge
  • Various tests for convergence for series with positive terms. Notes
    • Comparison test: if the terms of one positive series are less than the terms of another series, and the second converges, so does the first, but if the first diverges, so does the second
      See also § 8.4 of the text.
    • Integral test: a positive decreasing series converges iff the corresponding integral converges.
      See also § 8.3 of the text.
    • Limit comparison test: If the ratio a_n/b_n of the terms of two positive series converges to a finite number, then they either both converge or both diverge
    • Root test: suppose that a_n^1/n approaches a limit L. If L < 1, the series converges; if L > 1, it diverges; but if L = 1, the test is inconclusive, so use another test
      See also § 8.5 of the text
    • Ratio test: suppose that a_n+1/a_n approaches a limit L. If L < 1, the series converges; if L > 1, it diverges; but if L = 1, the test is inconclusive, so use another test
  • Assignment 7. Answers
  • Alternating series, Absolute and conditional convergence, Leibniz' alternating series test. Notes.
    See also § 8.6 of the text
  • Assignment 8
  • Power series, interval and radius of convergence. Notes.
    See also § 8.7 of the text.
    Differentation and integration of power series
  • Taylor polynomials and Taylor series
    See also § 8.8 of the text.
  • The harmonic series and Euler's constant
• Coordinates, vectors, polar coordinates,and parametric equations
  • Rectangular coordinates. Vectors  a = (a₁,a₂)  in R² and  a = (a₁,a₂,a₃)  in R³.
  • Polar/rectangular conversion of coordinates for points:
     x = r cos θ.  y = r sin θ.  x² + y² = r².  tan θ = y/x.
    See also § 9.1 of the text
  • Polar coordinates and complex numbers.  z = x + yi  = |z| (cos &theta + i sin &theta).
  • Curves in polar coordinates, conversion
    Cardioids and Limaçons, Julia and Mandelbrot Sets at http://aleph0.clarku.edu/~djoyce/julia/,
    Spiral of Archimedes, Equiangular or logarithmic spiral,
    Roulettes also called Rose curves
    See also § 9.2 of the text
  • Area in polar coordinates. The area between the rays  θ = α  and  θ = β  and inside the curve  r = f(θ)  is
    ∫_α^β r²/2 dθ.
    See also § 9.3 of the text
  • Assignment 9. Answers
  • Tangents to curves  r = f(θ)  in polar coordinates.

    dy dx

    =

    dy/dθ dx/dθ

    =

    r' sin θ + r cos θ; r' cos θ − r sin θ

  • Curves given parametrically: (x(t),y(t)).
    See also § 3.5 of the text
    Tangents to parametric curves:

    dy dx

    =

    dy/dθ dx/dθ

    Velocity, the vector-valued function (x'(t),y'(t)).
    Speed: the square root of (x' ² + y' ²)
    Acceleration: (x''(t),y''(t)).
  • Areas under parametric curves. ∫_a^b y x' dt.
  • Arclength. The path element is ds, the square root of dx² + dy².
    Thus, the speed is ds/dt is the speed, the square root of (dx/dt)² + (dy/dt)².
    The length of a path is ∫_a^b ds. = ∫_a^b(speed) dt.
    See also § 6.3 of the text
  • Summary of polar coordinates and parametric equations
  • Assignment 10 on velocity, speed, acceleration, and arclength. Answers
  • Area of a surface of revolution.. ∫_a^b 2π y ds. = ∫_a^b2π y (speed) dt.
    See also § 6.4 of the text
  • Cycloids.
    See also § 9.6 of the text
• Vectors
  • Vectors v in the plane R² and space R³
    Vector operations: addition v + w, subtraction v − w, and multiplication by scalars av
    Geometric interpretation of vector operations
    Properties of vector operations.
  • Norm (length) ||v|| of a vector v
    Triangle inequality ||v + w|| ≤ ||v|| + ||w||
  • Dot products (a.k.a inner products or scalar products) v · w
    Angles between vectors, parallel and perpendicular vectors. Notes.
  • Assignment 11. Due Friday Dec 2.
  • Unit vectors, standard bases, vectors in n-space Rⁿ, vector operations, coordinates in physical space. Notes.
  • Cauchy-Schwarz inequality. Notes.
  • Determinants. Notes.
  • Cross products (a.k.a. outer products or vector products) v × w, scalar triple products. Notes.
  • Assignment 12.