Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
THE CHAIN RULE
The following problems require the use of the chain rule. The chain rule is a rule for
differentiating compositions of functions. In the following discussion and solutions
the derivative of a function h(x) will be denoted by
or h'(x) . Most
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
THE METHOD OF INTEGRATION BY POWER SUBSTITUTION
The following problems involve the method of power substitution. It is a method for
finding antiderivatives of functions which contain th roots of or other expressions.
Examples of such expressions are
and
.
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
DETERMINING LIMITS USING L'HOPITAL'S RULES
The following problems involve the use of l'Hopital's Rule. It is used to circumvent
the common indeterminate forms "0"0"0"0 and " when computing limits. There
are numerous forms of l"Hopital's Rule, whose verifi
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
IMPLICIT DIFFERENTIATION PROBLEMS
The following problems require the use of implicit differentiation. Implicit
differentiation is nothing more than a special case of the wellknown chain rule for
derivatives. The majority of differentiation problems in fi
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
CALCULUS II
Practice Problems
Sequences and Series
Paul Dawkins
Calculus II
Table of Contents
Preface . 1
Sequences and Series . 2
Introduction . 2
Sequences . 3
More on Sequences . 3
Series The Basics .
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
Chapter 1
Centroid
Prepared by Kahenya NP
Definition
Centre of gravity (C.O.G) centre of mass ? it is a point in a body where all the
mass can be thought to be concentrated without altering the effect that the earth?s
gravity has upon it. For regular bodi
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
Binomial expansion, power series,
limits,
approximations, Fourier series
Notice: this material must not be used as a substitute for attending
the lectures
1
1
Binomial expansion
We know that
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2 b
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
1
Eigenvalue and Eigenvector; Prepared by Kahenya NP
Definition
Given a square matrix A an eigenvalue and its associated nonzero eigenvector x are by
definition such that ;
A =
Alternatively it can be written a;
where I is the identity matrix.
A I
=
D
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
CALCULUS II
Solutions to Practice Problems
Sequences and Series
Paul Dawkins
Calculus II
Table of Contents
Preface . 1
Sequences and Series . 1
Introduction . 2
Sequences . 2
More on Sequences . 6
Series The Basics .
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
CALCULUS II
Assignment Problems
Sequences and Series
Paul Dawkins
Calculus II
Table of Contents
Preface . 1
Sequences and Series . 1
Introduction . 1
Sequences . 2
More on Sequences . 3
Series The Basics .
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
10.3 pseries and the ratio test
Let p be a positive number. An infinite series of the form
X
1
1
1
1
= p + p + p + .
p
n
1
2
3
n=1
is called a pseries. If p = 1, the series is called the harmonic series
X
1
1 1 1
= 1 + + + + .
n
2 3 4
n=1
Classifying pser
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
Tests for Convergence of Series
1) Use the comparison test to confirm the statements in the following exercises.
P 1
P
1
1.
n=4 n diverges, so
n=4 n3 diverges.
2.
P
converges, so
P
3.
P
converges, so
P
1
n=1 n2
1
n=1 n2
1
n=1 n2 +2
en
n=1 n2
converges.
co
Jomo Kenyatta University of Agriculture and Technology
Calculus
ECE 2173

Fall 2016
SMA 2270
W126016
JOMO KENYATTA UNIVERSITY
OF
AGRICULTURE AND TECHNOLOGY
University Examinations 2011/2012
FIRST SEMESTER EXAMINATION FOR THE DEGREE OF BACHELOR
OF SCIENCE IN MINING AND MINERAL PROCESSING
ENGINEERING
SMA 2270 : CALCULUS III
DATE: AUGUS