M4.3b.- Modeling sinusoidal functions

M4.3b.- Modeling sinusoidal functions

TRIGONOMETRY

Profr. Edgar A. Orozco

Pending issues

1.- Check Schedule for Week 10 - Activity 7 (ALEKS) due Thursday, March 18 - Partial Exam 2, on Friday, March 19. 2.- Answers to Activity of Trig. graphs. 3.- Exhibition of infographics (Project 2) in Canvas. 4.- Requirements for the Partial Exam 2. - In Canvas, using Lockdown Browser (must be installed). - Zoom with cellphones, showing the desk and computer. - Exam will be divided in two parts: one with all the questions, and another where students will upload the procedure of the required questions. - Allowed time: 80 min. Estimated time: 45 min.

Index

Objectives of the topic

Simple Harmonic Motion

Periodic phenomena

Sinusoidal functions

Example 3

Example 2

Example 1

Periodic phenomena

Periodic behaviour manifests when the same pattern repeats time and time again, forming cycles. There are many examples of periodic phenomena in nature, where the same cycle appears in said cycles.

Sinusoidal functions

Since sinusoidal functions are also periodical, a lot of the times are very useful to represent periodic behavior.

Basic cosine graph

Basic sine graph

Simple Harmonic Motion

Important elements to consider:

• The independent variable is now the time (t), instead of x. It represents the time that passes during the movement.
• The letter is named "omega" and acts the same as the parameter k.
• The main difference between the sine and cosine functions when describing a phenomenon, lays on the starting point of the movement:
• For sine, is at the equilibrium.
• For cosine, is at the maximum or minimum values.

UNITS

Example 1

INFO

Solve the following exercises:

a) Find the amplitude, period and frequency of the motion, from the equation:

b) Find a function that models the given situation. Give two answers, when t = 0, the displacement is 0 and the displacement is a maximum.

Example 2

INFO

Solve the following problem:

- Each time your heart beats, your blood pressure increases, then decreases as the heart rests between beats. A certain person's blood pressure is modeled by the function: where p(t) is the pressure in mmHg at time t, measured in minutes. a) Find the amplitude, period, and frequency of p. b) Sketch a graph of p. c) If a person is exercising, his or her heart beats faster. How does this affect the period and frequency of p?

Example 3

INFO

Solve the following problem:

- A mass attached to a spring is moving up and down in Simple Harmonic Motion. The graph gives its displacement d(t) from equilibrium at time t. a) Express the function d in the form: b) Express the same function, but as a cosine.

References:

Stewart, J., Redlin, L. & Watson, S. (2010). Precalculus: Mathematics for Calculus. 6th Edition. CENGAGE. EU. Pages 412 - 423.