Class Notes (1,100,000)

CA (620,000)

UOttawa (30,000)

MAT (1,000)

MAT 1332 (90)

Robert Smith (6)

Lecture 2

This

**preview**shows pages 1-3. to view the full**103 pages of the document.**MAT 1332: Calculus for Life Sciences

A course based on the book

Modeling the dynamics of life

by F.R. Adler

Supplementary material

University of Ottawa

Frithjof Lutscher, Jing Li and Robert Smith?

April 13, 2010

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

MAT 1332: Additional Course Notes 1

The inverse tangent function

The tangent function is deﬁned as

tan(x) = sin(x)

cos(x),

and its derivative can be compute by the quotient rule as

d

dx tan(x) =

d

dx sin(x) cos(x)−sin(x)d

dx cos(x)

cos2(x)=cos2(x) + sin2(x)

cos2(x)=1

cos2(x).

In particular, the function is deﬁned for all xthat are not odd multiples of π, and the function

is monotone increasing, see Figure 1.

The inverse of the tangent is denoted as arctan or tan−1and it is deﬁned in the usual way

as

arctan(tan(x)) = x, tan(arctan(x)) = x.

See Figure 1. What is its derivative? We diﬀerentiate the ﬁrst equality above by the chain

rule and ﬁnd d

dx[arctan(tan(x))] = d

dy arctan(y)d

dx tan(x)=1,

with y= tan(x), since d

dx x= 1.Hence, we can divide

d

dy arctan(y) = 1

d

dx tan(x)=cos2(x)

cos2(x) + sin2(x)=1

1 + sin2(x)

cos2(x)

=1

1 + tan2(x)=1

1 + y2.

Application to integration

We can now integrate the derivative of the arctan function to get

Z1

1 + x2dx = arctan(x) + C.

1

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

MAT 1332: Additional Course Notes 2

MAT 1332: Frithjof Lutscher 2

!5!4!3!2!1012345

!15

!10

!5

0

5

10

15

x

y

y=tan(x)

x=!!/2

x=!/2

!10 !5 0 5 10

!4

!3

!2

!1

0

1

2

3

4

x

y

y=arctan(x)

y=!/2

y=!!/2

Figure 1: Graphs of the tangent function and its inverse, the arctangent function

2

Figure 1: Graphs of the tangent function and its inverse, the arctangent function

2

###### You're Reading a Preview

Unlock to view full version