Contemporary Calculus PREVIEW OF CALCULUS Two Basic Problems Beginning calculus can be thought of as an attempt, a historically successful attempt, to solve two fundamental problems. In this section we will start to examine geometric forms of those two problems and some fairly simple ways to attempt to solve them. At first, the problems themselves may not appear very interesting or useful, andContemporary Calculus the methods for solving them may seem crude, but these simple problems and methods have led to one of the most beautiful, powerful, and useful creations in mathematics: Calculus. First Problem: Finding the Slope of a Tangent Line Suppose we have the graph of a function y = f(x), and we want to find the equation of the line which is tangent to the graph at a particular point P on y=f(x) the graph (Fig. 1). (We will give a precise definition of tangent in Section 1.0. For now, think of the tangent line as the line which touches the curve slope f(8) at the point P and stays close to the graph of y = f(x) near P.) We know - Ill tan that the point P is on the tangent line, so if the x-coordinate of P is x = a, then the y-coordinate of P must be y = f(a) and P = (a, f(a)). The a Fig. 1 only other information we need to find the equation of the tangent line is its slope, Mean- and that is where the difficulty arises. In algebra, we needed two points in order to determine a slope, and so far we only have the point P. Lets simply pick a second point, say Q, on the graph of y = f(x). If the x-coordinate of Q is b (Fig. 2), then the y-coordinate is f(b), so Q = (b, f(b)). The slope of the line through P and Q is noe f(b) - f(a) mpo = In = b-a If we drew the graph of y = f(x) on a wall, put nails at the points P and y y=f() | Q on the graph, and laid a straightedge on the nails, then the straightedge would have slope mpo (Fig. 2). However, the slope mpo can be very different from the value we want, the slope mean of the tangent line. The secant slope key idea is that if the point Q is close to the point P, then the slope mpo f(a) am po is close to the slope we want, Iran . Physically, if we slide the nail at Q along the graph towards the fixed point P, then the slope, f(b) - fa a mpo = b-a of the straightedge gets closer and closer to the Fig. 2 slope, Mean , of the tangent line. If the value of b is very close to a, then the point Q is very close to P, and the value of mpg is very close to the value of man- Rather than defacing walls with graphs and nails, we can calculate mpQ = f(b) - f(a) b- a and examine the values of mog as b gets closer and closer to a. We say that Iran is the limiting value of mpo as b gets very close to a, and we write lim f(b) - f(a) man = b- aContemporary Calculus The slope mean of the tangent line is called the derivative of the function f(x) at the point P, and this part of calculus is called differential calculus. Chapters 2 and 3 begin differential calculus. The slope of the tangent line to the graph of a function will tell us important information about the function and will allow us to solve problems such as: The US Post Office requires that the length plus the girth girth (Fig. 3) of a package not exceed $4 inches. What is the largest volume which can be mailed in a rectangular box?" length An oil tanker was leaking oil, and a 4 inch thick oil slick had formed. When first Fig. 3 measured, the slick had a radius 200 feet and the radius was increasing at a rate of 3 feet per hour. At that time, how fast was the oil leaking from the tanker? Derivatives will even help us solve such "traditional" mathematical problems as finding solutions of equations like x- = 2 + sin(x) and x"+ 5x + x3+3=0. Second Problem: Finding the Area of a Shape Suppose we need to find the area of a leaf (Fig. 4) as part of a study of how much energy a plant gets from sunlight. One method for finding the area would be to trace the shape of the leaf onto a piece of paper and then divide the region into "easy" shapes such as rectangles and mangles whose areas we could calculate. Fix. 4 We could add all of the "easy" areas together to get the area of the leaf. A modification of this method would be to trace the shape onto a piece of graph paper and then count the number of squares completely inside the edge of the leaf to get a lower estimate of the area and count the number of squares that touch the leaf to get an upper estimate of the area. If we repeat this process with smaller Each square is I sq, em Each square is Li4 sq/ em squares, we have to do more counting and adding, but our estimates totally inside = 1 totally inside = 16 partially inside = 18 partially inside = 3 are closer together and closer to the actual area of the leaf. (This I S number = 19 16 5 number = 30 1 squem = area = 19 94.cm I sq.co s area 5 12.5 5q.cm area can also be approximated using a sheet of paper, scissors and an accurate scale. How?) We can calculate the area A between the graph of a function y = f(x) and the x-axis (Fig. 5) by using similar methods. We can divide y y=f(x) the area into strips of width w and determine the lower and upper values of y = f(x) on each swip. Then we can approximate the area of each rectangle and add all of the little areas together to get A,. A an approximation of the exact area. The key idea is that if w is small, then the rectangles are narrow, and the approximate area Aw is very a b Fig. 5: What is the area of region A?y=f(x) close to the actual area A. If we take narrower and narrower rectangles, the approximate areas get closer and closer to the actual area: A = limit A, The process we used is the basis for a technique called integration, and this A b part of calculus is called integral calculus. Integral calculus and integration Fig. 5: What is the arca of region A? will begin in Chapter 4. The process of taking the limit of a sum of "little" quantities will give us important information about the function and will also allow us to solve problems such as: 'Find the length of the graph of y = sin(x) over one period (from x = 0 to x = 2m)." "Find the volume of a torus ("doughnut") of radius 1 inch tones (doughnut) which has a hole of radius 2 inches. (Fig. 6)" "A car starts at rest and has an acceleration of 5 + 3sin(t) feet per second per second in the northerly direction at time t seconds. Where will the car be, relative to its starting position, after 100 seconds?" 2 Fig 6: What is the volume of the torus? A Unifying Process: Limits We used a similar processes to "solve" both the tangent line problem and the area problem. First, we found a way to get an approximate solution, and then we found a way to improve our approximation. Finally, we asked what would happen if we continued improving our approximations "forever", that is, we "took a limit." For the tangent line problem, we let the point Q get closer and closer and closer to P, the limit as b approached a. In the area problem, we let the widths of the rectangles get smaller and smaller, the limit as w approached O. Limiting processes underlie derivatives, integrals, and several other fundamental topics in calculus, and we will examine limits and their properties in Chapter 1. Two Sides Of The Same Coin: Differentiation and Integration Just as the set-up of each of the two basic problems involved a limiting process, the solutions to the two problems are also related. The process of differentiation for solving the tangent line problem and the process of integration for solving the area problem turn out to be "opposites" of each other: each process undoes the effect of the other process. The Fundamental Theorem of Calculus in Chapter 4 will show how this "opposite" effect works.Contemporary Calculus Extensions of the Main Problems The first 5 chapters present the two key ideas of calculus, show "easy" ways to calculate derivatives and integrals, and examine some of their applications. And there is more. In later chapters, new functions will be examined and ways to calculate their derivatives and integrals will be found. The approximation ideas will be extended to use "easy" functions, such as polynomials, to approximate the values of "hard" functions such as sin(x) and e" . And the notions of "tangent lines" and "areas" will be extended to 3-dimensional space as "tangent planes" and "volumes". Success in calculus will require time and effort on your part, but such a beautiful and powerful field is worth that time and effort. PROBLEMS (Solutions to odd numbered problems are given at the back of the book.) Problems 1 -4 involve estimating slopes of tangent lines. 1) Sketch the lines tangent to the curve shown in Fig. 7 at x = 1, 2 and 3. Estimate the slope of each of the tangent lines you drew. Fig.7 2) Fig. 8 shows the weight of a "typical" child from age 0 to age 24 months. (Each of your answers should have the units "Kilograms per month.") (a) What was the average rate of weight gain from 10- Weight (kg) month 0 to month 24? (b) What was the average weight gain from month 9 to month 12? from month 12 to month 15? 2 - - - - -4-- (c) Approximately how fast was the child gaining 10 20 24 weight at age 12 months? at age 3 months? Age (months) Fig.8Contemporary Calculus 3) Fig. 9 shows the temperature of a cup of coffee during a ten minute period. (Each of your answers in (a) - (c) should have the units "degrees per minute.") 150 * What was the average rate of cooling Temperature of a cup of coffee from minute 0 to minute 10? (b) What was the average rate of cooling Temperature ( of ) 100- from minute 7 to minute S? from minute 8 to minute 9? (c) What was the rate of cooling at minute 8? at minute 2? (d) When was the cold milk added to Time (minutes) the coffee? Fig. 9 4) Describe a method for determining the slope at the middle of a steep hill on campus (a) using a ruler, a long piece of string, a glass of water and a loaf of bread. (b) using a protractor, a piece of string and a helium-filled balloon. Problems 5 and 6 involve approximationg areas. 5) Approximate the area of the leaf in Fig. 4 . 65 6) Fig. 10 shows temperatures during the month of November. (a) Approximate the shaded area between the Outside temperature (OF) temperature curve and the 65" line from Nov. 15 to Nov. 25. 30 (b) The area of the "rectangle" is (base )(height) so what are the units of your answer in part (a)? 20 30 (c) November date Approximate the shaded area between the temperature Fig. 10 curve and the 65" line from Nov. 5 to Nov. 30. (d) Who might use or care about these results? 7) Describe a method for determining the volume of a standard incandescent light bulb using a ruler, a tin coffee can, a scale, and a jug of wine