Instantaneous Velocity We want to shorten the time interval so that it becomes very close to our point. The velocity of an object, with position function s(t) at time t = a, is; V(a) : lim 2 11mm m>o A; hao h Velocity is the slope of the tangent to the graph of s(t) @ P(a, 5(a)) . Eg.l: b) The function s(t) = 8t(t + 2) describes the distance 3, in km that a car travels after a time t, in hours for OSISS. Find the average velocity of the car over the following intervals. i) t=3tot=4 ii) t=3tot=3.1 iii) t=3tot=3.01 Approximate the instantaneous velocity @ t = 3 based on your answers above. Find the velocity @ t = 3 (use definition of derivative) Eg.2: Suppose that a foreign language student has learned N (t) = 20t t2 vocabulary terms after t hours of uninterrupted study. a) How many terms are learned during the 3rd hour? (0 to l is the 15' hour; 1 to 2 is the 2\"(1 hour; 2 to 3 is the 3rd hour) b) What is the rate of change (in terms per hour) at which the student is learning @ t = 2 hr. A function is continuous at x = a, if f(a) is defined at "a" and if lim f (x) = f(a). x->a Eg.1: Discuss the discontinuities of the following function. -2 x , if x 4 4 4 Eg.4: One-sided Limits Strategy lin} Ix 4| 1) x Consider the following; If we look at f(x) as we approach 4 from above and then as we approach 4 from below, they are not the same therefore the limit DNE