Mathematical Foundations

Juan Carlos Ponce Campuzano, Mar 23, 2015

Supplementary material for Mathematical Foundations (MATH1050) University of Queensland

Table of Contents

  1. Functions
    1. Real functions
    2. Domain and range of real functions
    3. Representation of real functions
    4. Solving inequalities
    5. Exponential and logarithmic functions
    6. Solving inequalities
    7. A mathematical model (gravity and water)
  2. Differentiation
    1. Relationship between a function and its derivative
    2. Mathematical optimization: Building a rectangular pen
    3. Box problem (right square prism)
    4. Cylinder container
  3. Integration
    1. Riemann sums for continuous functions
    2. Riemann sums
  4. Vectors
    1. Relative velocity: Boat problem
    2. Inclined plane forces
    3. Addition of vectors
    4. Scalar multiplication
    5. Dynamic Frenet-Serret frame
  5. Complex numbers
    1. Roots of a complex number

1.

Functions

  • 1. Real functions
  • 2. Domain and range of real functions
  • 3. Representation of real functions
  • 4. Solving inequalities
  • 5. Exponential and logarithmic functions
  • 6. Solving inequalities
  • 7. A mathematical model (gravity and water)

1.1.

Real functions

Move the Slider: [b]Apply function[/b] or Click on [b]Animate[/b] button to see how it changes. This worksheet shows an animated representation of real functions [math]f: D\subseteq \mathbb R\rightarrow R\subseteq \mathbb R[/math]. When the function is applied to any value [math]x\in D[/math], it will determine a number [math]f(x)\in R[/math]. The Domain and Range of [math]f[/math] will be determine by the algebraic expression in the box.
– Juan Carlos Ponce Campuzano

1.2.

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1.4.

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1.5.

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1.6.

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1.7.

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2.

Differentiation

  • 1. Relationship between a function and its derivative
  • 2. Mathematical optimization: Building a rectangular pen
  • 3. Box problem (right square prism)
  • 4. Cylinder container

2.1.

Relationship between a function and its derivative

Visual relationship between a given function [math]f[/math], [math]f'[/math] and [math]f''[/math]
– Juan Carlos Ponce Campuzano

2.2.

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2.3.

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2.4.

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3.

Integration

  • 1. Riemann sums for continuous functions
  • 2. Riemann sums

3.1.

Riemann sums for continuous functions

Riemann Sums: The sum is calculated by dividing the region up into shapes (rectangles or trapezoids) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together. Because the region filled by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing up the region more finely, using smaller and smaller shapes. As the number of shapes grows towards infinity, the sum approaches the Riemann integral.

You can change the function. Also, it is possible to change the interval of integration, the number of sub-intervals for getting a better calculation of the area. The slider 'd' determines the position of the point [math]\xi_d[/math] on the interval [math][x_{i-1},x_i][/math].
– Juan Carlos Ponce Campuzano

3.2.

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4.

Vectors

  • 1. Relative velocity: Boat problem
  • 2. Inclined plane forces
  • 3. Addition of vectors
  • 4. Scalar multiplication
  • 5. Dynamic Frenet-Serret frame

4.1.

Relative velocity: Boat problem

A river flows due West at a speed of 2.5 metres per second and has a constant width of 1 km. You want to cross the river from point [b]A[/b] (South) to a point [b]B[/b] (North) directly opposite with a motor boat that can manage a speed of 5 metres per second.[br][br]Things to try: [br][list][*]Drag the sliders to change the magnitude and direction of the vectors velocity.[br][/*][*]Click the [b]Start[/b] button to activate the motion of the boat.[br][/*][*]Click the [b]Reset[/b] button to put the boat to its original position.[br][/*][/list]
a) If you head out pointing your boat at an angle of 90 degrees to the bank, how long does it take to cross the river? After reaching the bank, how far will the boat be from point [b]B[/b]?[br][br]b) In what direction should you point your motor boat in to travel directly from point [b]A[/b] to point [b]B[/b]? How long does it take in such case?
– Juan Carlos Ponce Campuzano

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4.5.

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5.

Complex numbers

  • 1. Roots of a complex number

5.1.

Roots of a complex number

To solve the equation [math]z^n=a+bi[/math] for [math]a,b\in \mathbb R[/math] and [math]n\in \mathbb N[/math]
Change the values of a, b, and n.
– Juan Carlos Ponce Campuzano

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