Math_134_Fall_2023-week 1

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University of Alberta *

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134

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Mathematics

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Feb 20, 2024

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Math 134 - Fall 2023 University of Alberta Welcome to Biocalculus! Week 1
Who am I? My name : Meymanat Farzamirad Email : meymanat@ualberta.ca Office hours : MWF 01:00-02:00 PM Math 134 Unit 1.1 2 / 30
What do I do when I am not teaching? My main area is in Topological Equivariant Methods in Nonlinear Analysis (TEMNA), with special interest in applications to differential equations with symmetries including symmetric models in mathematical biology. Math 134 Unit 1.1 3 / 30
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ePoll: https://epoll.srv.ualberta.ca Math 134 Unit 1.1 4 / 30
Question 1.1 Let f be the function defined by f ( x ) = sin x + cos x and let g be the function defined by g ( u ) = sin u + cos u , for all real numbers x and u . Then (a) f and g are exactly the same functions. (b) if x and u are different numbers, f and g are different functions. (c) there is not enough information given to determine if f and g are the same. Math 134 Unit 1.1 5 / 30
Question 1.2 True or False . If f ( x ) = x 2 4 x 2 and g ( x ) = x + 2, then we can say the functions f and g are equal. (a) True (b) False Math 134 Unit 1.1 6 / 30
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Question 1.3 Find the domain of the function: f ( x ) = 1 + x 1 cos x . Math 134 Unit 1.1 7 / 30
Question 1.3 Find the domain of the function: f ( x ) = 1 + x 1 cos x . (a) ( −∞ , ) (b) . . . ( π, 0) (0 , π ) ( π, 2 π ) . . . (c) . . . ( 2 π, 0) (0 , 2 π ) (2 π, 4 π ) . . . (d) ( −∞ , 0) (0 , ) Math 134 Unit 1.1 7 / 30
Question 1.4 Express as a single logarithm: ln 5 + 5 ln 3 Math 134 Unit 1.1 8 / 30
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Question 1.4 Express as a single logarithm: ln 5 + 5 ln 3 (a) ln(5 · 3 5 ) (b) ln(5 · 3 · 5) (c) ln(5 + 3 5 ) (d) ln(5 + 3 · 5) Math 134 Unit 1.1 8 / 30
Course srtucture Each unit you: Watch all the videos before class. Use class time to participate, learn, ask questions Pre-class videos stack questions (value 5%) Assignments will be due on Friday, 11:00 PM . There are weekly stack assignments on eClass (vlaue 7%). There are 8 written homeworks on eClass (value 8%) There are three in-class quizzes (value 30%). You are encouraged to discuss assignment questions with your friends; collaboration is the key, not competition! But of course you must not just copy answers and must all submit your own solutions. There is more on this in the Course Outline. Math 134 Unit 1.1 9 / 30
Question 1.5 Which of the two following statements are true? (1) ( x + y ) 2 = x 2 + y 2 (2) 1 x y = 1 x 1 y (a) Both (1) and (2) (b) Only (1) (c) Only (2) (d) Neither (1) nor (2) Math 134 Unit 1.1 10 / 30
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Question 1.6 Which of the two following statements are true? (1) x 1 / 7 + x 4 / 7 = x 1 / 7 ( 1 + x 5 / 7 ) (2) ( x 2 2) 1 / 3 ( x 2 2) 2 / 3 = ( x 2 2) 2 / 9 (a) Both (1) and (2) (b) Only (1) (c) Only (2) (d) Neither (1) nor (2) Math 134 Unit 1.1 11 / 30
Question 1.7 True or False: |− x | = x . (a) True (b) False Math 134 Unit 1.1 12 / 30
Question 1.8 True or False: ( x 2 ) 1 / 2 = x . (a) True (b) False Math 134 Unit 1.1 13 / 30
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Question 1.9 Which of the following equations (given in point-slope form) corresponds to the equation of a line that passes through the point ( 4 , 3) and is parallel to the line y = 3 x + 15? (a) y 3 = 3( x 4) (b) y 3 = 3( x + 4) (c) y 3 = 3( x 4) (d) y 4 = 15( x 3) (e) y + 4 = 3( x 3) Math 134 Unit 1.1 14 / 30
Question 1.10 Use the Law of Exponents to rewrite and simplify the following expressions: 1 4 3 2 8 2 1 3 x 4 Select the correct set of answers: Math 134 Unit 1.1 15 / 30
Question 1.10 Use the Law of Exponents to rewrite and simplify the following expressions: 1 4 3 2 8 2 1 3 x 4 Select the correct set of answers: (a) 8 , x 4 / 3 (b) 4 , x 4 / 3 (c) 8 , x 3 / 4 (d) 4 , x 3 / 4 Math 134 Unit 1.1 15 / 30
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Question 1.11 Find the domain of the function f ( x ) = x p 2 x 2 . (a) ( −∞ , ) (b) ( −∞ , 2) ( 2 , ) (c) ( 2 , 2) (d) [ 2 , 2] Math 134 Unit 1.1 16 / 30
Question 1.12 Starting with the graph of y = e x , write the equation of the graph that results from 1 shifting 2 units downward 2 shifting 2 units to the right Select the correct set of answers: Math 134 Unit 1.1 17 / 30
Question 1.12 Starting with the graph of y = e x , write the equation of the graph that results from 1 shifting 2 units downward 2 shifting 2 units to the right Select the correct set of answers: (a) y = e x 2 , y = e x 2 (b) y = e x 2 , y = e x + 2 (c) y = e x 2 , y = e x 2 (d) y = e x + 2 , y = e x 2 Math 134 Unit 1.1 17 / 30
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Question 1.13 Starting with the graph of y = e x , write the equation of the graph that results from 1 reflecting about the x -axis 2 reflecting about the y -axis Select the correct set of answers: Math 134 Unit 1.1 18 / 30
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Question 1.13 Starting with the graph of y = e x , write the equation of the graph that results from 1 reflecting about the x -axis 2 reflecting about the y -axis Select the correct set of answers: (a) y = e x , y = e x (b) y = e x , y = e x (c) y = e x , y = e x (d) y = e x , y = e x Math 134 Unit 1.1 18 / 30
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Functions are everywhere in Biology 1. Average C02 level in the atmosphere versus year: (Data from Mauna Loa Observatory were fit using linear regression to give C = 1 . 71262 t 3054 . 14.) Math 134 Unit 1.1 19 / 30
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Functions are everywhere in Biology 2. Vertical displacement of zebra finches as they start to fly (Tobalske et al, 1999) (Digitized points representing the position of the bird’s eye were used in fitting the curve.) Math 134 Unit 1.1 20 / 30
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Functions are everywhere in Biology 3. The number of different bat species in a cave is related to the size of the cave (Brunet et al., 2001): (Model is for bats living in caves in central Mexico.) Math 134 Unit 1.1 21 / 30
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Functions are everywhere in Biology Exponential model for world population growth P = (1436 . 53) × (1 . 0135) t (The period of relatively slow population growth is explained by the two world wars and the Great Depression of the 1930s.) Math 134 Unit 1.1 22 / 30
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Different functions In the following m , a , b , c are constants. CO2 level: a linear function: C ( t ) = mt + b . Zebra finch flight: quadratic function: y ( x ) = ax 2 + bx + c . Bat species in caves: Power law (power function): S ( A ) = aA c . World population: Exponential growth: P ( t ) = ce at . Math 134 Unit 1.1 23 / 30
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Question 1.14 Cyclic neutropenia is a blood disorder in humans characterized by periodic fluctuations in the density of neutrophils . The density of neutrophils reaches highs of around 2000 cells/ µ L of blood and lows near zero. The period of fluctuations is approximately three weeks. Find a function N ( t ) that describes the amount of neutrophils in the blood as function of time (in days), assuming that the density is at its highest on day zero. Math 134 Unit 1.1 24 / 30
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max = 2000, min = 0, period = 21 days, density = max at day 0. Math 134 Unit 1.1 25 / 30
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Question 1.15 Antibiotic dosage: Antibiotics are used to treat bacterial sinus infections. 1 If a dosage of x mg is taken orally, suppose that the amount absorbed into the bloodstream through the stomach is h ( x ) = 8 x x +8 mg. 2 If x mg enters the bloodstream, the amount surviving filtration by the liver is g ( x ) = 1 4 x . 3 Finally, if x mg survives filtration by the liver, then f ( x ) = x 1 mg is absorbed into the sinus cavity, provided that x > 1 (otherwise f ( x ) = 0). Use functional composition to derive the function that relates oral dosage to the amount of drug that reaches the sinus cavity. Math 134 Unit 1.1 26 / 30
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1 Stomach h ( x ) = 8 x x +8 mg. 2 Liver g ( x ) = 1 4 x mg 3 Sinus f ( x ) = ( x 1) mg Math 134 Unit 1.1 27 / 30
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Question 1.16 find the domain of the composite function f g h for f ( x ) = lnx + 1 1 x , g ( x ) = 1 x , h ( x ) = x 2 . (a) ( −∞ , ) (b) ( 1 , 0) (0 , 1) (c) [ 1 , 1] (d) [ 1 , 0) (0 , 1] Math 134 Unit 1.1 28 / 30
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Question 1.17 Express as a single logarithm: 1 3 ln( x + 2) 3 + 1 2 [ln x ln( x 2 + 3 x + 2) 2 ] Math 134 Unit 1.1 29 / 30
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Question 1.17 Express as a single logarithm: 1 3 ln( x + 2) 3 + 1 2 [ln x ln( x 2 + 3 x + 2) 2 ] (a) ln x 3( x + 2) (b) ln x x + 1 (c) ln x + 1 x (d) ln 3( x + 2) x Math 134 Unit 1.1 29 / 30
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Question 1.18 Find the domain of the function: f ( x ) = 1 e x 2 1 e 1 x 2 . Math 134 Unit 1.1 30 / 30
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Question 1.18 Find the domain of the function: f ( x ) = 1 e x 2 1 e 1 x 2 . (a) ( −∞ , ) (b) ( −∞ , 1) ( 1 , 1) (1 , ) (c) ( −∞ , 1) (1 , ) (d) ( −∞ , 1) (1 , ) Math 134 Unit 1.1 30 / 30
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