Instantaneous velocity and its Today we first revisit the notion of instantaneous velocity, and then we discuss how we use its to compute it. Learning Catalytics session: We start with a question about the computation of average velocity over a time interval. Here s a place where you can do a calculation: As we saw last class, calculus comes into play when we go from the notion of average velocity to that of instantaneous velocity. We compute average velocity over successively smaller intervals of time and take the it as the length of the interval goes to zero. The precise mathematical statement for the calculation of instantaneous velocity for the position function s(t) = 6t t 2 at t = 1 is 6t t 2 5 t 1 t 1 We can visualize this iting process as secant lines approaching the tangent line to the graph at the number 1. = 4. Limits Limits were developed to make sense out of expressions where we are basically dividing by zero as we were when we considered the expression at t = 1. In general, we talk about 6t t 2 5 t 1 f(x) x a when we have a function that is defined for all real numbers near the number a except possibly at a, and we want to produce the best candidate for the number f(a). 1
Example. Consider the function f defined by the formula f(x) = sin x x. What is the domain of definition of this function? Now let s graph f. We get () - - This graph suggests that the best value for f(0) is the number 1. We say that sin x x 0 x = 1. Note that this graph does not show us values f(x) for x really close to 0, so it does not prove that the it is 1. Nevertheless, it is convincing evidence. Mastering the precise mathematical definition of a it is a difficult task (see Section 2.7 of our text), and in this course we only need an intuitive understanding of the concept. Therefore, we rely on informal definitions. Your textbook has one at the start of Section 2.2. However, I prefer a different but equivalent one. Informal Definition. We say that L = x a f(x) if, given any table of numbers x that approaches the number a, the corresponding table of values f(x) approaches the number L. 2
Example. Once again, consider f(x) = sin x. Here is one such table for a = 0. x x sin x x 1 0.84147098 0.1 0.99833416 0.01 0.99998333 0.001 0.99999983 0.0001 0.99999999 Note: This table was made using the assumption that the argument x is measured in radians. Here are some general remarks about its. Remarks. 1. Even if the function f is defined at the number a, its value f(a) does not affect the value of the it. 2. The informal definition requires that we check all tables of numbers x approaching the number a. 3. Limits do not always exist. Another Learning Catalytics exercise: 3
One-sided its Sometimes it is useful to talk about its at x = a where x approaches a only from the left (less than a) or only from the right (greater than a). Informal Definition. We say that L = x a + f(x) if, given any table of numbers x that are greater than a and approaches the number a, the corresponding table of values f(x) approaches the number L. Example. For the function graphed earlier, what are the one-sided its as x a for a = 1, 2, 3, 4? One-sided and two-sided its are related by the following theorem. Theorem. Suppose that the function f(x) is defined for all x near a except possibly at a itself. Then the two-sided it of f(x) as x a exists as x a if and only if both one-sided its exist and are equal. That is, x a f(x) = L if and only if f(x) = L = f(x). x a x a + 4
Computing its We often compute its by reducing them to known its via algebraic operations, and we know that its behave as expected when we apply these operations. This comes so naturally that it s not worth repeating the theorems from the textbook that justify this procedure (see Theorems 2.2 and 2.3). Example. Calculate x 1 2x + 3 x 2 + 4. Here s an example that requires some algebraic manipulation in order to calculate the it. x 2 Example. Calculate x 0 1 cos x. 5
Here s the graph that corresponds to the it that we just calculated. - () - - There is one more general theorem about its that we should discuss. The Squeeze Theorem. Suppose that the three functions f, g, and h satisfy g(x) f(x) h(x) for all numbers x near the number a. If x a g(x) = x a h(x) = L, then x a f(x) = L. 6
Example. When we compute the derivative of the sine function, we will need the it sin x x 0 x. We compute this it by squeezing (sin x)/x between cos x and 1/(cos x). (Note that 1/(cos x) is sec x.) - 7