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.