SF1625 Calculus in One Variable 7.5 credits

Envariabelanalys

  • Educational level

    First cycle
  • Academic level (A-D)

    A
  • Subject area

    Mathematics
    Technology
  • Grade scale

    A, B, C, D, E, FX, F

Course offerings

Autumn 17 CELTE1 m.fl. for programme students

Autumn 17 CMETE1 m.fl. for programme students

Autumn 17 CMATD1 m.fl. for programme students

Autumn 17 CDEPR1 m.fl. for programme students

Autumn 17 CBIOT1 m.fl. for programme students

Autumn 17 CMEDT1 for programme students

Autumn 17 CMAST ITSY1 for programme students

Spring 18 CDATE1 m.fl. for programme students

Spring 18 CINTE1 for programme students

Intended learning outcomes

After completing the course the student a passing grade to:

  • Use, explain and apply fundamental concepts and problem solving methods of one variable calculus, especially:
    - Describe the fundamental characteristics of elementary functions, such as power laws, logarithm lawss and trigonometric formulas and use them in problem solving and calculations
    - Calculating derivatives using, among other things, the product, quotient and the chain rule
    - Using derivatives to investigate the properties of a function, for example, decide questions of increasing and decreasing behavior, sketch function graphs, determine the tangent, prove inequalities and find extreme values
    - Using Taylor's formula to approximate functions with polynomials to desired degree of accuracy
    - Account for the definition of the Riemann Integral and its applications, as well as approximate integrals with Riemann sums
    - Calculate integrals using primitive functions, integration by parts, variable substitution and partial fractions
    - Account for the fundamental theorem of calculus about the relationship between the derivative and integral, and use it in problem solving and calculations
    - Solve some linear ordinary differential equations with constant coefficients and explain how they arise in applications
    - Calculate limits and use these to study the behavior of functions locally or asymptotically
    - Determine whether a given function is invertible and if possible calculate the inverse function
    - Determine if certain series converges or diverges and if possible calculate them
  • Set up simple mathematical models for applications that can be described with the help of functions of one variable, and discuss such models relevance, plausibility and accuracy.
  • Read and understand mathematical text about functions of one variable and their applications, communicate mathematical reasoning and computations within this filed orally as well as in writing in such a way that they are easy to follow.

For higher grades, the student should be able to:

  • Account for the theory of one variable calculus, with definitions, theorems and proofs
  • Generalize and adapt methods to fit in new situations
  • Solve problems requiring a combination of methods, or more extensive calculations in several steps
  • Solve more advanced problems in, for example, limits, series, integrals and applications

Course main content

Function, function graph, domain, range. Increasing and decreasing functions, odd and even functions. Inverse functions. The class of elementary functions. Unit circle, trigonometric formulas and equations, exponential and logarithmic functions, power laws, logarithms. Limits, rules for calculating limits, standard limits. Continuity, theorems on continuous functions. Derivative, rules of differentiation and applications: rate of change, linear approximation, tangent, extreme value problems, sketching the graph of a function, inequalities etc. Taylor's formula with error estimates. Linear differential equations with constant coefficients and their applications. Riemann integral, primitive functions, variable substitution, integration by parts, partial fractions. Riemann sums, geometric and other applications of integrals, improper integrals. Something about sequences and series. Something about numerical methods (eg Newton's method).

Eligibility

Basic and specific requirements for engineering program.

Mandatory for first year, can not be read by other students

Literature

Robert A. Adams, Christopher Essex, Calculus - A Complete Course, 8th edition. ISBN 978-0-321-78107-9.

Examination

  • TEN1 - Examination, 7.5, grade scale: A, B, C, D, E, FX, F

Requirements for final grade

Written exam, possibly with the possibility of continuous examination.

Offered by

SCI/Mathematics

Contact

Roy M Skjelnes (skjelnes@kth.se)

Examiner

Lars Filipsson <lfn@kth.se>

Roy M Skjelnes <skjelnes@kth.se>

Version

Course syllabus valid from: Autumn 2015.
Examination information valid from: Autumn 2007.