Differential CalculusLaajuus (3 ECTS)

Objective

After completing this course, the student will be familiar with the notion of limit of a function, the student will be able to calculate some simple limits, and he or she will understand what is meant by the continuity of a function. The student will understand the derivative as the limit of the difference quotient and he or she will be able to calculate the derivative for the most common real functions, as well as expressions composed of them. The student will be able to study the monotonicity and extrema of a function with the aid of a derivative. The student will be able to distinguish between the limit of a function and the limit of a sequence of real numbers. The student will understand the sum of an infinite series as the limit of the sequence of partial sums. The student will be able to determine the sum of a geometric series and some other series. The student will understand what is meant by the series expansion of a function. After completing this course, the student will be able to construct and solve mathematical models associated with variable quantities and the rate of change. The student will also be able to interpret the solutions given by a model.

Content

1) Limit and continuity
2) Derivative and applications
3) Infinite sequences and series

Qualifications

Functions and complex numbers. Complex numbers are not necessary for understanding the theory.

Assessment criteria, satisfactory (1)

1) Limit and continuity
The student is familiar with the idea of limit and continuity. The student is able to calculate the limit of an elementary function e.g. by dividing both the numerator and denominator by a common factor that tends to zero.
2) Derivative and applications
The student has got an idea of the connection between the derivative and the rate of change in the value of function and whether the change is upwards or downwards. The student is able to calculate the derivatives of common real functions. The student is able to determine the maxima and minima of an elementary differentiable function.
3) Infinite sequences and series
The student is familiar with the notions of a number sequence and an infinite series. The student has got an idea of the convergence of an infinite series. The student is able to determine the sum of a convergent geometric series.

Assessment criteria, good (3)

1) Limit and continuity
The student understands the notions limit and continuity with generalizations (one-sided limit, unbounded growth and so on). The student is able to solve ordinary limit problems and study phenomena with the aid of limits. The student is familiar with the Bolzano's theorem and he/she is able to find an approximate value to a zero of a function by using the bisection method.
2) Derivative and applications
The student is able to examine the differentiability of a function with the aid of the difference quotient. The student has a good command of usual derivative calculations and extrema problems. The student understands the notion of differential and he/she is able to estimate the change in a function value by computing the differential. The student is familiar with the Newton's method and is able to solve equations numerically by means of the Newton's method.
3) Infinite sequences and series
The student is able to distinguish between the limit of a function and the limit of a number sequence. He/she understands the notions of a series and its convergence. The student is able to determine the sum of a geometric series and some other elementary series or alternatively show that the series diverges. The student understands what is ment by the series expansion of a function. The student is able to model real-world phenomena by means of sequences and series (e.g. when comparing the complexity of two algorithms).

Assessment criteria, excellent (5)

1) Limit and continuity
The student has a clear intuitive idea of limit and continuity (mathematica epsilon-delta-definitions are not required). The student is able to solve challenging limit problems.
2) Derivative and applications
The student has a deep understanding of the derivative and its applications. For instance, the student is able to derive formulas for the derivatives of ordinary functions via calculating the difference quotient.
3) Infinite sequences and series
The student has a clear idea of convergence and he/she is able to solve challenging problems. On demand the student is able to derive the formula for the sum of a geometric series.

Further information

Computer exercises will be included.