math160 lecture 1.1: calculus introduction

MATH160Lecture 1.1:

Calculus Introduction

Sarah Wakes

University of Otago

24 August 2021

Basic Information

Lecturer

Sarah Wakes - Head of Department for Mathematics & Statistics

Office: Room 2.10 (Science III building)

Email: [email protected]

Office hours: open door (email to check I am available)

Research: Numerical modelling of fluid flow - Computational FluidDynamics (complex topographies such as coastal dunes, engineeringapplications)

MATH160 - Lecture 1.1 1

Quizzes

Quizzes continue after mid-semester break for 6 weeks oncalculus material.

You will have from Monday 9am to Friday 12pm (noon) tocomplete the quizzes (4) each week.


Resources

Essential

Available for download on the course’s Resources page.

Study NotesLecture slides with problem solutions included (when teachingonline)Lecture slides with scans of work in lectures (when teaching ina classroom)

Recommended

Textbook

Calculus by J. Stewart (consider buying if you are planning totake MATH170).Some copies are available in the libraries.

Online resources

tread carefully!


Reminders

No Tutorials this week

No Quizzes this week

Zoom drop in session 9-10am Wednesday - Fridayhttps://otago.zoom.us/j/91754783305?pwd=

ZWFFREh0N2d5Z1kxOTQwNmJBQ3Z6QT09

Password: calculus

Assignment 3:

Monday 16 August 9am - Friday 10 September 12pm (noon)Half algebra and half calculus


https://otago.zoom.us/j/91754783305?pwd=ZWFFREh0N2d5Z1kxOTQwNmJBQ3Z6QT09

https://otago.zoom.us/j/91754783305?pwd=ZWFFREh0N2d5Z1kxOTQwNmJBQ3Z6QT09

What is Calculus?

Branch of Mathematics studying the changes/variations experiencedby functions.

The Scientific Method:

Calculus is involved at each stage of the process!

Our goal is to introduce Calculus in an intuitive manner with a focus on

applications. For a more rigorous description, you can take MATH201

(Real Analysis).


Dunedin weather

Temperature and rain accumulation rate in Dunedin

Data obtained from MetService website on 20/02/2018

t (hours past 12am) 2 4 6 8 10 12T (◦C) 13 12 10 11 11 13r (mm per hour) 0 0.2 2.5 1.4 1.5 0.8

time2 4 6 8 10 12

Tem

pera

ture

10 °C

11 °C

12 °C

13 °C

14 °C

time2 4 6 8 10 12

Rai

nac

cum

ulat

ion

rate

0 mm/h

1 mm/h

2 mm/h

3 mm/h


First calculations

Questions

1 How fast was the temperature changing between 6am and 8am?

Answer Calculation

Question is asking for a rate of change with time (t) of temperature (T).

change in temperature

change in time=

change in T

change in t

=T8 − T6

t8 − t6=

11 − 10

8 − 6=

1

2= 0.5◦C/hr

The temperature changed at an averaged rate of 0.5◦C per hour between6am and 8am.


First calculations

Questions


Answer Calculation



change in time=

change in T

change in t

=T8 − T6

t8 − t6

=11 − 10

8 − 6=

1

2= 0.5◦C/hr



First calculations

Questions


Answer Calculation



change in time=

change in T

change in t

=T8 − T6

t8 − t6=

11 − 10

8 − 6

=1

2= 0.5◦C/hr



First calculations

Questions


Answer Calculation



change in time=

change in T

change in t

=T8 − T6

t8 − t6=

11 − 10

8 − 6=

1

2= 0.5◦C/hr



First calculations

Question

1 How much has it rained between 10am and 12pm?

Answer calculation

Question is asking for the net accumulation in rain between 10am and12pm - call this ∆R.Have been given the rate of change of rain with time (r) so r = ∆R

∆twhich when rearranged gives ∆R = r × ∆t.Therefore

∆R ≈ r10−12 × ∆t = (1.5 + 0.8

2) × 2 = 2.3mm

The rain accumulation between 10am and 12pm was approximately 2.3mm.


First calculations

Question


Answer calculation

Question is asking for the net accumulation in rain between 10am and12pm - call this ∆R.

Have been given the rate of change of rain with time (r) so r = ∆R∆t

which when rearranged gives ∆R = r × ∆t.Therefore

∆R ≈ r10−12 × ∆t = (1.5 + 0.8

2) × 2 = 2.3mm



First calculations

Question


Answer calculation


∆t

which when rearranged gives ∆R = r × ∆t.Therefore

∆R ≈ r10−12 × ∆t = (1.5 + 0.8

2) × 2 = 2.3mm



First calculations

Question


Answer calculation


∆twhich when rearranged gives ∆R = r × ∆t.

Therefore

∆R ≈ r10−12 × ∆t = (1.5 + 0.8

2) × 2 = 2.3mm



First calculations

Question


Answer calculation



∆R ≈

r10−12 × ∆t = (1.5 + 0.8

2) × 2 = 2.3mm



First calculations

Question


Answer calculation



∆R ≈ r10−12 × ∆t

= (1.5 + 0.8

2) × 2 = 2.3mm



First calculations

Question


Answer calculation



∆R ≈ r10−12 × ∆t = (1.5 + 0.8

2) × 2 = 2.3mm



Is it the whole story?

Temperature and rain accumulation rates change continuously over time!

time2 4 6 8 10 12

Tem

pera

ture

10 °C

11 °C

12 °C

13 °C

14 °C

time2 4 6 8 10 12

Rai

nac

cum

ulat

ion

rate

0 mm/h

1 mm/h

2 mm/h

3 mm/h

At its core, Calculus consists of a set of tools to calculate (i) the rate of

change and (ii) the accumulation of quantities varying continuously.


Calculus in a nutshell

Calculus consists of 2 themes:

Differentiation: rates of change/slopes of curves

Integration: accumulation (net change)/area under curves

It turns out these two themes are connected through the FundamentalTheorem of Calculus!

Applications

Language to describe the physical world, i.e. modelling.

Methods to solve complex problems, e.g. optimisation, curvefitting, computing lengths, areas and volumes, ...


Before Starting

Little background knowledge is assumed, but you are expected to becomfortable with the basic operations and algebraic manipulations:

+,−,×,÷

fractions

expanding and factoring expressions

square root, cubic root, ...

exponents

inequalities

trigonometry

Some resources available to you are listed in the study notes.


math160 lecture 1.1: calculus introduction

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