Chapter 1 LINES
1.1 LINES
A line is the easiest mathematical structure to describe. You need to know only two things about a line to describe it. For example: i) ii) iii) i) the y-intercept, b, and the slope, m; one point on the line and the slope; two po
Chapter 8 CONTINUOUS FORCE
8.1 INTRODUCTION
The last section was devoted to the computation of shear forces and bending moments for
weightless beams with concentrated forces. In this chapter we extend these ideas to beams with
continuous forces in which t
Chapter 14 CURVE SKETCHING AND MAX- MIN II
14.1 INTRODUCTION
In Chapter 11, we developed a procedure for graphing polynomial functions. In this Chapter we take this one step further and expand the procedure to include functions of any kind. In Chapter 12
Chapter 15 FINDING THE AREA UNDER A CURVE
15.1 FINDING THE AREA UNDER A CURVE
In large part the mathematical theory of architectural structures is based on two observations:
i) The derivative or slope of the bending moment curve is the shear force, i.e.,
Chapter 16 FINDING AREAS II
16.1 FINDING THE AREA BY USING SHEAR FORCE
In Chapter 15, Part I we found the area under the graph of the V (x) function given the M (x) function. Now we specify the V(x) directly and then find the signed area under its graph b
Chapter 12 MAXIMA AND MINIMA
12.1 INTRODUCTION
Derivatives can be used to find the value in the domain of a function at which the
function takes on its largest or smallest value. If the function is smooth, i.e., it is
continuous everywhere and is differen
Chapter 13 PRODUCT, QUTIENT, CHAIN RULE AND TRIG FUNTIONS
13.1 NEW FUNCTIONS FROM OLD ONES
Given two functions f(x) and g(x) we can define three new functions in terms of these old ones:
f ( x)
f
(fg)(x) = f(x)g(x) ; ( )( x)
; and ( f g )( x) f ( g ( x) (
Chapter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS
17.1 THE SLOPE METHOD
The following theorem provides the justification for Method 3, the slope method. Theorem 1: then If V (x) is continuous at x and a continuous force of density
V ( x)dx M ( x) c
(
Chapter 7 COMPUTATION OF SHEAR FORCE
7.1 INTRODUCTION
Architectural structures and calculus are a perfect match. I hope to show you how they are connected. First let's consider the basics of structures applied to a simple beam supported on one end by a hi
Chapter 9 THE DERIVATIVE
9.1 THE ORIGINS OF CALCULUS
All great discoveries have their origins with an important problem that needs to be solved. The problem generally has a long history with other great scientists contributing to its solution. The history
Chapter 10 PROPERTIES OF DERIVATIVES
10.1 INTRODUCTION
In the last chapter you were introduced to a complex procedure that could be used to find the derivative of any function that by a method I refer to as the Derivative Machine. Although this method is
Chapter 11 SKETCHING CURVES 1
11.1 INTRODUCTION
In the early grades you drew the graph of a function y = f(x) by making a table of x and y. To get an idea of how the function looked you needed to plot many points. Even doing this, it is possible that you
Chapter 6 BASICS OF STRUCTURES
6.1 INTRODUCTION
The disciplines of Physics, engineering, architecture, and mathematics come together in the study of structure. The physicist thinks in the language of equilibrium of forces and applications of Newton's thre
Chapter 5 DISTANCE AND ACCUMULATED CHANGE
5.1 DISTANCE
a. Constant velocity
Lets take another look at Marys trip to the lake on her bicycle (see Section 4.1) in Fig. 1.
s = s1 = 2
H
t = t1 = 0
s=4
S
TH
s=0
L
s = s2 = 5
t = t2
Fig.1
This time she starts fr
LESSON 2 FUNCTIONS
2.1 FUNCTIONS
A function f is a relationship between an input and an output and a set of instructions as to how to obtain the output from a given input. For any input, there must be a unique output (only one.) If we call the input x and
Chapter 3 THE AREA UNDER A CURVE
3.1 FINDING THE AREA UNDER A CURVE
We wish to find the area above the x-axis and beneath the graph of a function y = f(x) between x
= a and x = b, where f ( x) 0 . In the study of calculus the notation that is used for thi
Chapter 4 RATE OF CHANGE
4.1 RATE OF CHANGE
Fig. 1 describes a trip that Mary is taking on her bike. The origin of the coordinate system is
taken to be s = 0 at the location of Town Hall (TH). She starts her trip at her house (H) located at
position s1 =
LAB EXERCISES MATHEMATICS LABORATORY EXERCISES
Lab 1: Buffons Needle Problem
Draw five equidistant parallel lines on a piece of cardboard where the distance between lines is
D. From a standing position, drop N needles of length L on the floor. Record the
Chapter 19 DEFLECTION OF BEAMS
19.1 THREE EXPERIMENTS
Let's do three experiments. a) Take a sheet of paper and stretch it between two desks (see Fig. 1). The paper can hold only a few ounces of weight before it deflects and falls to the floor. By altering
Chapter 20 DETERMINING MAXIMUM STRESS ON A BEAM
20.1 MAXIMUM STRESS ON A BEAM
If a beam is subjected to a stress beyond its capabilities, it will rupture. By stress we
mean an internal force per square foot of the beam cross section. Clearly steel can end