Lectures on Autumn 2007: Tue-Wed-Thu 8am-10am, main building lecture hall M. Lectured by Teijo Arponen and Tuomo Kuusi.

The topics of K1 on 2007

1. week (ti 11.9 - to 13.9)
   • High School revision, mathematical notation, proving theorems (Ad: P1)
   • Planar vectors, inner product. (Ad: P2, 10.2)
   • Trigonometric functions, unit circle (Ad: P7)
   • Complex numbers (Ad: App. I, Lay: App. B) Summary
2. week
   • Complex numbers (roots!)
   • Euclidian spaces, vectors, projection (Ad: 10.1, 10.2) Formulae
   • Planes in 3-dim space (Ad. 10.4)
3. week
   • Cross product in 3-dim space (Ad: 10.3)
   • Planes and lines in 3-dim space (Ad. 10.4)
4. week
   • Linear systems and matrices (Lay: 1.1-1.2) Gaussian algorithm
   • Linear algebra: Lay 1.2-1.6.
   • Linear in/-dependency (Lay: 1.7) Summary
   • 1st mid-term exam (Adams P1, P2, P7, 10.1-10.4, App.I, Lay 1.1-1.7, App.B, laskuharjoituskierrokset 1-4.)
5. week
   • Linear mappings (Lay: 1.8-1.9). Matrix algebra (Lay 2.1-2.3). Summaries: matrix algebra and inverse matrix.
6. week
   • Determinant (Lay 3.1-3.2) det Summary
   • Eigen values and -vectors. (Lay: 5.1-5.2) Eigentheory summary. Also eigentheory notes by Juha Koljonen.
   • Eigentheory visualisations: one and another.
7. week (Tue-Wed 23-24.10, Thu 1.11)
   • Diagonalisation (Lay: 5.3)
   • Diagonalisation in analysing linear mappings (Lay: 5.4-5.5)
   • Complex eigenvalues (Lay: 5.5) Esimerkki.
   • Limits (Adams: Chapter 1)
   • Summary on linear algebra (weeks 5-7).
8. week
   • Continuity (Adams Ch. 1)
   • Derivation and applications (Ad: Ch. 2)
   • Intermediate Value Theorem (Ad: App.III)
   • Chain rule (Ad: 2.4)
   • Mean Value Theorem (Ad: 2.6)
   • Linear approximation (Ad: 2.7)
   • 2nd midterm exam (Lay 1.8-1.9, 2.1-2.3, 3.1-3.2, 5.1-5.5. Adams P4, App.III, 1.1-1.5, 2.1-2.7, 2.11, exercises 5-8.)
9. week
   • Implicit derivation (Ad: 2.9)
   • Transcendental functions (sin, arcsin, cos, arccos, tan, arctan, log, exp). (Ad.: Ch. 3)
   arctan etc. Summary.
   • Extreme values and turning points (Ad: 4.2-4.5).
10. week
    • Extreme values and turning points (Ad: 4.2-4.5).
    Extreme values, convexity.
    • Numerical solution of scalar equations (Ad: 4.6) Newton's method
    • The accuracy of a linear approximation (Ad: 4.7)
11. week
    • Taylor polynomials and limits (indefinite forms) (Ad: 4.8-4.9) Summaries: 1., and 2.
    • Big-O-notation.
    • Integration (Ad: 5.1-5.5 and App.IV). Summary
12. week
    • Integration techniques (Ad: 5.6, 6.2, 6.1 ja 6.5). Summaries:
      • Substitution.
      • Integration by parts.
      • Unbounded functions and/or regions.
    • Numerical integration, trapetzoid method (Ad: 6.6). Summary
    • 3rd mid-term exam (Adams 2.11, 3.1-3.3, 3.5-3.6, 4.1-4.3, 4.5-4.9, 5.1-5.6, 6.1-6.2, 6.5-6.6, exercises 9-12)