n
Practice Final
1. Use the limit denition of a derivative to find f(:1:) if f(:n) = -:t: + 2.
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FINAL REVIEW: Chapter 3
Critical Number: For a function f , the critical number, c is where f 0 (c) = 0 or f 0 (c) does not exist.
Critical numbers are where local maximum and local minimum occur.
NOTE: A local max or local min doesnt occur at every criti
FINAL REVIEW: Chapter 2
Definition Let f (x) be a function. The tangent line to the curve y = f (x) at the point (a, f (a) at is the
line through (a, f (a) with slope
f (a + h) f (a)
m = lim
h0
h
provided the limit exists.
Definition Let f be a function.
Hill 1
Sean Fentress Hill
Professor Hallam
MTH 111
18 November 2016
Part 1: Researching Newtons Method
Newtons method allows me to approximate the root of the function [f (x) = 0]. Every
time I reckon the tangent line of my approximate value, I get arbitr
Practice Test 2
1. Find the derivatives of the following functions:
(i) p(x) = 8x5 + 4x4 x3/2
(ii) f (x) = sin x tan x
(iii) f (x) =
x
x2 + 4
(iv) h(t) = sec2 t3
1
(v) g(x) =
q
x+ x
(vi) g(t) = sin(cos x)
2. For the equation below, find
dy
:
dx
xy tan(xy
List Of Theorems and Some Definitions
Theorem: Suppose that c is a constant and that the limits
lim f (x)
and
xa
lim g(x)
xa
exist. Then
1. lim (f (x) + g(x) = lim f (x) + lim g(x)
xa
xa
xa
2. lim (f (x) g(x) = lim f (x) lim g(x)
xa
xa
xa
3. lim cf (x) =
Practice Test 2 Answers
1. (i) p0 (x) = 40x4 + 16x3
3
x
2
(ii) f 0 (x) = sec2 x sin x cos x tan x
(iii) f 0 (x) = p
4
(x2
+ 4)3
(iv) h0 (t) = 6t2 sec2 (t3 ) tan(t3 )
2 x+1
(v) g 0 (x) = p
4 x x+ x
(vi) g 0 (x) = sin x cos(cos x)
2.
y
dy
=
dx
x
3. (i)
(ii
FINAL REVIEW: Chapter 1
The final exam is cumulative and counts for 25% of your grade. It is scheduled for December
16th. Please remember to bring your IDs.
8 am class: Meet in Kirby Hall 101. Final starts at 9 am (until noon)
9 am class: Meet in Kirby Ha
Chapter 8
Decimals
8.1 Introduction
Savita and Shama were going to market to buy some stationary items.
Savita said, I have 5 rupees and 75 paise. Shama said, I have 7 rupees
and 50 paise.
They knew how to write rupees and paise using decimals.
So Savita
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CHAPTER 3
is
COORDINATE GEOMETRY
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Whats the good of Mercators North Poles and Equators, Tropics, Zones and
Meridian Lines? So the Bellman would cry; and crew would reply They are
merely conventional signs!
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LEWIS CARROLL,
Chapter 11
Algebra
11.1 Introduction
Our study so far has been with numbers and shapes. We have learnt numbers,
operations on numbers and properties of numbers. We applied our knowledge
of numbers to various problems in our life. The branch of mathematics
Chapter 4
Basic Geometrical
Ideas
4.1 Introduction
Geometry has a long and rich history. The term Geometry is the English
equivalent of the Greek word Geometron. Geo means Earth and metron
means Measurement. According to
historians, the geometrical ideas
Chapter 13
Symmetry
13.1 Introduction
Symmetry is quite a common term used in day to day life. When we see
certain figures with evenly balanced proportions, we say, They are
symmetrical.
Tajmahal (U.P.)
Thiruvannamalai (Tamil Nadu)
These pictures of archi
Chapter 9
Data Handling
9.1 Introduction
You must have observed your teacher recording the attendance of students in
your class everyday, or recording marks obtained by you after every test or
examination. Similarly, you must have also seen a cricket scor
Chapter 6
Integers
6.1 Introduction
Sunitas mother has 8 bananas. Sunita has to
go for a picnic with her friends. She wants to
carry 10 bananas with her. Can her mother
give 10 bananas to her? She does not have
enough, so she borrows 2 bananas from her
ne
MATHEMATICS
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188
CHAPTER 11
is
CONSTRUCTIONS
bl
11.1 Introduction
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In earlier chapters, the diagrams, which were necessary to prove a theorem or solving
exercises were not necessarily precise. They were drawn only to give you
113
The Triangle and
its Properties
6.1 INTRODUCTION
A triangle, you have seen, is a simple closed curve made of three line
segments. It has three vertices, three sides and three angles.
Here is ABC (Fig 6.1). It has
Fig 6.1
Sides:
AB , BC , CA
Angles:
BA
Practical
Geometry
10.1 INTRODUCTION
You are familiar with a number of shapes. You learnt how to draw some of them in the earlier
classes. For example, you can draw a line segment of given length, a line perpendicular to a
given line segment, an angle, an
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CHAPTER 7
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7.1 Introduction
is
TRIANGLES
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You have studied about triangles and their various properties in your earlier classes.
You know that a closed figure formed by three intersecting lines is called a triangle.
(Tri m
Lines and
Angles
5.1 INTRODUCTION
You already know how to identify different lines, line segments and angles in a given
shape. Can you identify the different line segments and angles formed in the following
figures? (Fig 5.1)
(i)
93
Chapter 5
LINES AND AN
EXPONENTS
AND
POWERS
193
12
CHAPTER
Exponents and Powers
12.1 Introduction
Do you know?
Mass of earth is 5,970,000,000,000, 000, 000, 000, 000 kg. We have
already learnt in earlier class how to write such large numbers more
conveniently using exponents, a
Chapter 2
Whole
Numbers
2.1 Introduction
As we know, we use 1, 2, 3, 4,. when we begin to count. They come naturally
when we start counting. Hence, mathematicians call the counting numbers as
Natural numbers.
Predecessor and successor
Given any natural nu
Congruence of
Triangles
7.1 INTRODUCTION
You are now ready to learn a very important geometrical idea, Congruence. In particular,
you will study a lot about congruence of triangles.
To understand what congruence is, we turn to some activities.
DO THIS
Tak
Chapter 10
Mensuration
10.1 Introduction
When we talk about some plane figures as shown below we think of their
regions and their boundaries. We need some measures to compare them. We
look into these now.
10.2 Perimeter
Look at the following figures (Fig.
Algebraic
Expressions
12.1 INTRODUCTION
We have already come across simple algebraic expressions like x + 3, y 5, 4x + 5,
10y 5 and so on. In Class VI, we have seen how these expressions are useful in formulating
puzzles and problems. We have also seen ex
Fractions and
Decimals
2.1 INTRODUCTION
You have learnt fractions and decimals in earlier classes. The study of fractions included
proper, improper and mixed fractions as well as their addition and subtraction. We also
studied comparison of fractions, equ
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CHAPTER 6
LINES AND ANGLES
6.1 Introduction
In Chapter 5, you have studied that a minimum of two points are required to draw a
line. You have also studied some axioms and, with the help of these axioms, you
proved some ot