MAT 1234Calculus I

Section 2.1 Part II

Derivatives and Rates of Change

HW

WebAssign HW 2.1 II Be sure to read the instructions carefully. Turn in the written HW at the end of your

handout.

What do we care?

How fast “things” are going• The velocity of a particle• The “speed” of formation of chemicals• The rate of change of charges in a capacitor

Recall

Limit of the following form is important

Geometrically, for the graph , the limit represents the slope of the tangent line at

h

afhafh

)()(lim

0

Recall

Limit of the following form is important

displacement function of a particle moving in a line at time

The limit represents the velocity of the particle at time

h

afhafh

)()(lim

0

So…in a chemical reaction

Limit of the following form is important

amount of a chemical formed at time The limit represents how fast the

chemical is formed - the rate of change of the amount of chemical at time

h

afhafh

)()(lim

0

Definition (rephrase)

Let represents certain physical quantity, the (instantaneous) rate of change of that physical quantities at is

if it exists.

(This represents how fast the quantity is changing.)

h

afhafh

)()(lim

0

Definition (New Notation)

The derivative* of a function at is defined as

(*Introduced in Lab 3)

h

afhafaf

h

)()(lim)(

0

Example 1

Compare the values of )3(),2(),1(),0(),1( fffff

𝑥

𝑦

1

𝑦= 𝑓 (𝑥 )

-1 2 30

Example 1

Compare the slopes of the tangent lines at 3 ,2 ,1 ,0 ,1x

𝑥

𝑦

1

𝑦= 𝑓 (𝑥 )

-1 2 30

Example 2

Suppose we model the amount of certain drug inside a patient’s body by mg after hours of injection.

)(tQ

Example 2(a)

)(aQWhat is the meaning of ?

Example 2(b)

mg/hour? 4)3( QWhat is the meaning of

Example 2(c)

mg/hour? 4)3( Q

How to determine the units (mg/hour)?

Example 3

(a) Find if

(b) Find

)(af 1)( xxf

(1), (2), and (3)f f f

Example 3

(a) Find if)(af 1)( xxf

0

( ) ( )( ) lim

h

f a h f af a

h

Example 3

(a) Find if

(b) Find

)(af 1)( xxf

(1), (2), and (3)f f f

(1)

(2)

(3)

f

f

f

Remarks

1)( xxf 12

1)(

aaf

If we want to find , ,and , we do not need to compute 3 limits. We only need to substitute 1, 2, and 3 into the formula of above.

This practice treats as a function:

given , the formula gives

Classwork Hint

#2 Do NOT expand the denominator

0 3

1li

3m

h h a h a

Quiz

Please take a look at the grader’s comments

Some of you did really well Some of you have a lot of room to

improve !!!! Explaining your work clearly and

carefully is VERY important It is not too late to get help