Bicol University
College of Engineering
Legazpi City
1.
1.1.
MATH 15:
Numerical Solutions to CE Problems
CHAPTER 1: Review of Mathematical Foundation
Lesson 1: Physical Meaning of Derivatives and Integrals
1.1.1. What is this Lesson About?
This course will mainly focus in using numerical methods instead of using complicated
analytical methods. However, we need to first review the basic mathematical foundation before
we start with our discussions with numerical methods to fully comprehend and appreciate this
course. Analytical method is the process you have been using since your Calculus 1
(Differential Calculus) and Calculus 2 (Integral Calculus) course where you differentiate and
integrate the functions directly. I am sure that there are problems in calculus that you find
difficult in solving using analytical method. This course will help you overcome the fear in
solving difficult problems. Rather than solving the problem in analytical method, we will be
using alternative methods called “Numerical Methods”.
1.1.2. What will you Learn?
After this lesson the student should be able to:
1.
2.
3.
Review of the basic principles in differential and integral calculus.
Review the relationship between the derivative of a function as a function and
the notion of the derivative as the slope of the tangent line to a function at a
point.
Review the use of integral in solving the area between two curves.
1.1.3. Calculus
Calculus is a branch of mathematics that helps us understand changes between values
that are related by a function. For example, if you had one formula telling how much money
you got every day, calculus would help you understand related formulas like how much money
you have in total, and whether you are getting more money or less than you used to. All these
formulas are functions of time, and so that is one way to think of calculus - studying functions
of time.
1.1.4. Differential Calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at
which quantities change. It is one of the two traditional divisions of calculus, the other being
integral calculus. The primary objects of study in differential calculus are the derivative of a
function. The derivative of a function at a chosen input value describes the rate of change of
the function near that input value.
The derivative of at a point is defined as
lim
→
ℎ ∆ lim
∆→ ∆
ℎ
(1.1)
where ℎ ∆, ∆ ℎ ∆ provided the limits exists.
Prepared By: Alvin B. Luzong
15 | P a g e
Bicol University
College of Engineering
Legazpi City
MATH 15:
Numerical Solutions to CE Problems
The differential of is defined by
(1.2)
where ∆.
The process of finding derivatives is called differentiation. By taking derivatives of
′ we can find second, third and higher order derivatives, denoted by
′
′′, ′ ′′′, etc.
Geometrically the derivative of a function at a point represents the slope of the
tangent line drawn to the curve at the point as shown in Figure 1.1.
Figure 1.1. Graph of a function, drawn in black, and two tangent line to that
function, drawn in red and blue. The slope of the tangent line equals the
derivative of the function at the marked point.
1.1.5. Let's Try This Exercise 1.1
Find the slope of the curve at the given x-axis coordinate (Show your solutions):
1. 7 4 , at 1
2. 5 4, at 1
3. !, at 2
#
, at 0.6
$#
#$ #
, at #
# #$, at 0.77
√
4. 5.
6.
#
7. )1 √1 , at !
8. 5 # , at 0.1
9. * √ , at *
0
10. sin ./ , at !
1.1.6. Integral Calculus
Integral Calculus is the branch of calculus which deals with functions to be integrated.
Integration is the reverse process of differentiation. The function to be integrated is referred to
as the integrand while the result of an integration is called integral. Integral Calculus can be
Prepared By: Alvin B. Luzong
16 | P a g e
Bicol University
College of Engineering
Legazpi City
MATH 15:
Numerical Solutions to CE Problems
used to find areas, location of the centroid, arc lengths, surface area of a solid, volume of a
solid and many useful things.
If , then we call an indefinite integral or anti-derivative of and denote it
by
1 (1.3)
1 lim ℎ62 2 ℎ 2 2ℎ ⋯ 2 / 1ℎ8
(1.4)
Since the derivative of a constant is zero, all indefinite integrals of can differ only
by a constant.
The definite integral of between 2 and 3 is defined as
4
→
5
provided this limit exists. Geometrically if 9 0, this represents the area under the curve
bounded by the x axis and the ordinates at 2 and 3 as shown in Figure 1.2.
The integral will exist if is continuous in 2 : : 3.
Definite and indefinite integrals are related in this theorem. If ;, then
4
4
3
; ; ;3 ;2
2
5
5 The process of finding integrals is called integration.
1 1
Figure 1.2. Calculating the areas bounded by the function f(x) with the given
limits from a to b
1.1.7. Let's Try This Exercise 1.2
Solve the following problems and show your solutions (graph the equations):
1.
2.
3.
4.
5.
6.
7.
8.
Find the area bounded by the curve 9 and the x-axis.
Find the area bounded by the curve 3 3 0 and the line 4.
Find the area bounded by the curve 1 , the y-axis and the line 2.
Find each of the two areas bounded by the curves 4 and 2.
Find the area between the two curves 2 4 0 and 2.
Find the area bounded by the curve ln and the lines *, 2* and x-axis.
Find the area under the first arch of the curve sin .
Find the area bounded by the parabolas 4 and 12 36.
#
9. Find the area of the region bounded by the curves !, the x-axis, 1, 4.
10. Find the area bounded by the curve Prepared By: Alvin B. Luzong
# and the lines 0, , 2.
17 | P a g e
0
