Area and Tangent Problem Calculus is motivated by two main problems. The first is the area problem. It is a well known result that the area of a rectangle with length l and width w is given by A = wl. From this fact (or definition) one can readily verifies that the area of a triangle 1 with base b and height h is A = 2bh The formula for area of triangles can be used to find the area of any polygon. We just need to divide the polygon into triangles and find the sum of the area of the triangles. The problem comes when we want to find the area of a figure with curved edges. For example, how do we find the area of a circle? One intuitive idea is to divide the curved figure into rectangles, and estimate the area of the curved figure by the sum of the area of each rectangle. This is just an estimate. What can we do to get a more accurate result? Intuition tells us that if we make the rectangles smaller, the approximation should be bet- ter. on Can we get a still better result? Yes, by making the rectangles still smaller. As we make the rectangles smaller and smaller, the approximation (of the curved area by the rectangles) become better and better. Now, what if we want the exact value? To do this, imagine that we make the width of the rectangles so small that it practically has zero length, then if we are still able to find the area of the rectangles, this should give us an exact value of the curved figure. That is, in the limit case (this word will become meaningful in a while), when the width of the rectangle becomes infinitesimally small, we get the exact value of the area of the curved figure. One thing to keep in mind, too, is that when the width of each rectangle becomes extremely small, the number of rectangles that are used to approximate the figure must necessarily increase. In fact, if the width of the rectangles is to become infinitesimally small, we should expect that we will need to use infinitely many rectangles to fill the curved figure. Another problem of calculus is the tangent problem. We have a curve defined by a function f(x), and we want to find the slope of the line tangent to f at a given point (x0; f(x0)) where x0 is a constant. Intuitively, the line tangent to a curve at a given point is the line that best approximates the curve at that given point. From algebra we know that in order to find the slope of a line, we need two points. We have only one point, namely (x0; f(x0)). We can try to approximate the slope of the line by choosing another point on the curve that is near the point in question, say (x0 + h; f(x0 + h)), where h is a small number. The slope of the line (called the secant line) that contains these two points (x0; f(x0)) and (x0 + h; f(x0 + h)) is given by: f(x0 + h) − f(x0) f(x0 + h) − f(x0) slope of secant line = msec = = (x0 + h) − x0 h The above formula gives an approximation of the slope of the tangent line. Intuitively, if we make the two points P and Q closer to each other, we get a better approximation. Making P and Q closer is the same as making h smaller, or close to zero. At the limit case, when h is arbitrarily close to 0, or when h is infinitesimally small, we should expect the approximation to be exact. What does the slope of the tangent to a function represent? Suppose you are going on a one-direction, 10 hours trip. Let the x-axis represent the time (in hours) since you have started your trip, and the y-axis represent the distance (in miles) you are from your origin. What does the slope of the line between (2; 4) and (10; 100) represent? 100 − 4 The slope of the secant line, m = = 12 represents the average velocity 10 − 2 of your car between time t = 2 and t = 10. That is, on average, you are driving at 12 miles per hour between the 2nd hour to the 10th hour of your travel. In general, if we want to find the average velocity of the car between any two times d2 − d1 t1 and t2, we just need to find the slope m = . But what if we want to t2 − t1 find the velocity of your car at exactly t = 2? This is called the instantaneous velociy of the car at t = 2 and is what the slope of the tangent line represents. In general, the slope between any two points of a function represents the average rate of change of the function between the two given points, while the slope of the tangent line of a function at a given point represents the instantaneous rate of change of the function at that given point. Limits In both the area problem and the tangent problem, a very important concept is common between the two. In the case with the area, we have to take the width of the rectangles very close to zero; in the case of the tangent problem, we also have to take h very close to 0. Both of these give rise to the concept of limit. What do we mean when we say the value of one variable is close to a particular number? What is the behavior of a function f(x) when x approaches a particular number? (Informal) definition of Limit: Let f be a function and a and L constants, We say lim f(x) = L x!a if f(x) can be made arbitrarily close to L by choosing x close to a. (We use the notation x ! a to mean x approaches a) Example: What is lim x + 1 =? x!2 That is, what does the value of the expression x + 1 get close to when x is close to 2? We may try a few values of x that are close to 2 and see if we observe any trend here: x x + 1 2.1 3.1 2.05 3.05 2.01 3.01 2.001 3.001 1.9 2.9 1.95 2.95 1.99 2.99 1.999 2.999 From the table, we can see that the value of the expression x + 1 approaches 3 as x approaches 2. It should not surprise you to find that lim x + 1 = 3 x!2 Note that we could have found the value of the limit by directly substituting the value of 2 into the expression x + 1 to find the value of 3. Occasionally, directly substituting the limiting value into the expression may allow us to find the value of the limit (and we will see under what cercumstances we may do this), but there are cases such method will fail and we need more sophisticated methods. Consider: x2 − 9 lim x!3 x − 3 x2 − 9 If we substitute the value of 3 into the expression we get the indetermi- x − 3 0 nate form . Let us try some values again: 0 x2 − 9 x x − 3 3.1 6.1 3.05 6.05 3.01 6.01 3.001 6.001 2.9 5.9 2.95 5.95 2.99 5.99 2.999 5.999 x2 − 9 It seems that as x approaches 3, approaches 6, so we would guess that x − 3 x2 − 9 lim = 6 x!3 x − 3 This is a correct guess. Notice that in order to evaluate the limit of the function at a point a, we do not need to require the function to be defined at a. In the x2 − 9 above example, the expression is not defined at 3, yet its limit is defined x − 3 at 3. Let's look at another example: jx − 1j lim =? x!1 x − 1 0 Notice that if x = 1, the expression gives the indeterminate form, so we try 0 some values as before: jx − 1j x x − 1 1.1 1 1.05 1 1.01 1 1.001 1 0.9 −1 0.95 −1 0.99 −1 0.999 −1 So what is the limit? Is it 1 or −1? jx−1j Since we cannot find a "trend", i.e., since the value of the expression x−1 does not tend to any particular value, we cannot come to any conclusion as to what jx − 1j the value of the expression should be, we say that the limit lim does not x!1 x − 1 exist The previous example illustrates that limits do not have to exist. In the above example, the value of the expression tends to one value (−1) when x approaches 1 but x is always less than 1 (from the left hand side), and the expression tends to another value (1) when x approaches 1 but x is always greater than 1 (from the right hand side). Since the two values do not agree, we cannot make a proper guess, so the limit does not exist. Trying to find the limit by finding values of f(x) on the left and right hand side of a gives rise to the concept of left and right-handed limits: Definition: We say that lim f(x) = L x!a− if f(x) can be made arbitrarily close to L by choosing x close to a and is always less than a (x is on the left hand side of a).