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
