Lecture 1 course content PDF

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First lecture about inequality, absolute function, basic function, basic algebra...

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MAT1330 C ALCULUS

FOR THE

L IFE S CIENCES

Nazanin Zaker

1. Introduction & High School Review: Algebraic Methods Overview of Course Syllabus. Course website:

Official Syllabus:

Evaluation: Your final grade is based on the following:

Mobius: ¨

Ask questions and ask for help:

Note on upload formats:

• if it will appear in the body of an email, make sure the image is right side up and of a reasonable size (email it to yourself to test, please!); • to scan with your phone, download and use the very useful CamScanner app (or its equivalent). (Sorry: the default iPhone format is not accepted; convert to jpg or (better) pdf.); • to collate several files into one pdf, use free online tools like ilovepdf.com.

This class will be recorded and made available on Brightspace. It is NOT permitted to share or distribute these recordings beyond students in MAT1330. ∗

These notes are based on ones developed by Dr. Elizabeth Maltais, and are solely for the personal use of students registered in MAT1330.

1

C ALCULUS IN

A NUTSHELL

Calculus was developed in the mid seventeenth century at about the same time by two scientists: Sir Isaac Newton and Gottfried Leibniz. The idea was to develop general solutions to two problems posed by the Greeks: Differential Calculus (about 1/2 of MAT1330)

Integral Calculus (about 1/4 of MAT1330)

Goal: find the slope of a tangent at any point on a given function

Goal: find the area bounded by a function and the x-axis over a given interval

Applications of math and calculus to the Life Sciences:

A lot of the problems you will be considering in this course will require you to be adept with some basic algebraic tricks and rules. You need to know how to  read mathematical notation  simplify exponents  add, subtract, multiply, divide, and simplify fractional expressions  factor or expand algebraic expressions  solve equations involving polynomials  solve equations involving exponents or logarithms  solve equations involving absolute value  solve inequalities involving rational expressions and/or absolute values We’ll focus on just a few of these today. In the first week of DGDs (before your regular TA takes over) we’ll tackle the rest. See the schedule posted on Brightspace, and the blank ”workbook” for each day, to choose which ones might be beneficial for you. Evaluate your skills with the homework assignment! 2

3

Example 1.1. Simplify

p

x1/2 y 5 , where x, y > 0. x5 y −1/4

Example 1.2. Suppose x, y, z > 0. Simplify

Example 1.3. Simplify

p

x3 y 2 + x3 z 2 .

4 + 1k . 5 −2 k

4

A particularly clever trick, that applies only to expressions with square roots, is called rationalization, and it’s based on the identity

Example 1.4. Rationalize the denominator to simplify √

4 . 10 − 3

Answer: • Multiplies 1

• Choosing 1 =

using (a + b)(a − b) = a2 − b2

• Simplify

5

S OLVING EQUATIONS Example 1.5. Solve for x:

4x = 3x. 1+x

162x−1 .

6

S ET NOTATION Sets of numbers show up everywhere in math, including: as solutions to an equation, or an inequality; or as the domain or range of a function. When they are intervals, we can represent them in multiple ways. Set notation

as Intervals

Graphically

4

4

4

4

te the ns to

ntity,

Guidelines to Solve Nonlinear Inequalities: 1. Move all nonzero terms to one side using addition or subtraction so you don’t need to worry about flipping the inequality sign. 2. Factor the nonzero size of the inequality (numerator and denominator if there is one). 3. Chop up the real number line into intervals, broken up at each number that a factor of the numerator and/or denominator equals zero. 4. Make a table to test the sign of each factor within each interval obtained in Step 3. The last row of the table should be the entire nonzero side of your inequality. 5. Check the signs to see on which intervals the inequality is satisfied. These intervals will belong to the

E

8

A BSOLUTE VALUE Absolute Value:

Solving an absolute value equation Example 1.9. Solve for x:

|x2 − 5| = 1

9

S OLVING AN

ABSOLUTE VALUE INEQUALITY

Guidelines to Solve Inequalities involving Absolute Value: Solving an inequality involving absolute value generally falls into one of the following scenarios (where c > 0): Inequality

Set of solutions

Graphical solution

−c

0

c

−c

0

c

−c

0

c

−c

0

c

   x − 3 > 5 2

10...