MATH1050 Topic 1 Lecture 2

Page 1: Announcements and Important Dates

• Welcome to Lecture 2!

• Important reminders:

  • No quizzes posted for Topic O.

  • Complete Topic I quiz by Sunday, March 9 (not this weekend, next).

  • If you need help downloading materials for MATH1050, reach out on the Ed Discussion Board.

  • Bring pen and paper (or device with stylus) to Topic I Workshop.

  • Lecture recording available for Lecture I.

  • Use www.slido.com, code 20868921 for questions.

  • Deadline to pick up a new unit is Friday, March 7.

  • Deadline to drop a unit without penalty is March 31.

Page 2: Overview of Last and Current Lectures

• Last Lecture Topics:

  • Introduction to MATH1050

  • Scientific notation and significant figures

  • Linear relationships

  • Quadratic relationships

• Current Lecture Topics:

  • Periodic functions

  • Sigmoid functions

  • Application of quadratic equations

  • Buffer solutions

Page 3: Periodic Functions Using Trigonometric Functions

• Focus on the sine and cosine functions.

  • These are periodic functions measured in degrees or radians.

• Important measurements:

  • 180° = π radians

  • To convert degrees to radians, use the formula ( ext{radian} = \frac{\text{degree} \times \pi}{180} )

Page 4: Properties of Periodic Functions

• Definition: A periodic function has a period ( p ) if ( f(x + p) = f(x) ).

• Examples:

  • ( \sin(90° + 360°) = \sin(450°) = \sin(90°) = 1 )

  • ( \cos(1/3 + 2\pi) = \cos(1) )

Page 5: Amplitude and Period/Wave Length

• Amplitude: Represented by ( A ).

  • Sign changes based on the sine and cosine functions.

• Period/Wave Length:

  • Calculated as ( 2\pi \times \text{frequency} ) to find the wave length.

Page 6: Introduction to Sigmoid Curves

• Sigmoid functions:

  • Commonly used in pharmacology (e.g., dose-response curves).

• Important characteristics:

  • Maximal asymptote, steepness parameter, threshold.

  • Expressed mathematically as: ( y = \frac{M}{1 + e^{-k(x-x0)}} ) (Location and steepness parameters).

Page 7: Changing Parameters in Sigmoid Curves

• Effects of changing ( M ):

  • Alters the value the curve approaches.

• Example graph: ( \text{Plot}[4x^3/(2^3+x^3), (x, 0, 8)] )

Page 8: Effects of Changing k and n

• Increasing ( k ):

  • Shifts the curve right.

• Increasing ( n ):

  • Steepens the slope near x = K.

  • Different behavior observed for non-zero slopes at various n values.

Page 9: Understanding Buffer Solutions

• Key characteristic: Maintains a stable pH (acidity level) under dilution or additional solutions.

  • Example buffer solutions with specific pH values.

• Buffer solutions contain weak acids that partially dissociate in water.

• The pH is determined by the concentration of the acid and its dissociation constant ( K_a ).

Page 10: Example of a Buffer System with Chloroacetic Acid

• Reaction representation:

  • ( ext{ClCH}_2 ext{COOH} + ext{H}_2 ext{O} \rightarrow ext{H}_3 ext{O}^+ + ext{ClCH}_2 ext{COO}^- )

• Calculation of equilibrium concentration using ( K_a ) and initial values.

Page 11: Finding Equilibrium Concentration

• Setting up the equation based on the dissociation reaction:

  • ( K_a = \frac{[H_3O^+][ClCH_2COO^-]}{[ClCH_2COOH]} )

• Solving a quadratic equation for equilibrium and concentration results.

• Result: Concentration of ( H_3O^+ ) is approximately ( 4.64 imes 10^{-3} ) indicating a pH of 2.3.