MA123, Chapter 2: Change, and the idea of the derivative (pp. 17-45, Gootman)Chapter Goals:• Under stand average rates of change.• Under stand the ideas leading to instantaneous rates of change.• Under stand the connection between instantaneous rates of change and the derivative.• Know the definition of the derivative at a point.• Use the definition of the derivative to calculate derivatives.• Under stand the connection between a position function, a velocity function, and thederivative.• Under stand the connection between the derivative and the slope of a tangent line.Assignments:Assignment 02 Assignment 03Roughly speaking, Calculus describes h ow quantities change, and uses this description of change to give usextra information about the quantities themselves.◮ Average rate s of change:We are all familiar with the concept of velocity (speed): If you drive adistance of 120 miles in two h ou rs, then your average velocity, or rate of travel, is 120/2 = 60 miles per hour.In other words, the average velocity is equal to the ratio of the distance traveled over the time elapsed:average velocity =distance traveledtime elapsed=∆s∆t.In general, the quantityy2− y1x2− x1=∆y∆xis called the average rate of change of y with respect to x.Note:Often, a change in a quantity q is expressed by the symbol ∆q (you should not think of th is as ∆times q, bu t rather as one quantity!).Note:Finding average rates of change is important in many contexts. For instance, we may be interested inknowing how quickly the air temperature is dropping as a storm approaches, or how fast revenues are increasingfrom the sale of a new product.Note:In this course we use the terms “speed” and “velocity” for the same concept. This is not the casein some other courses. Thus “ins tantaneous speed” and “instantaneous velocity” have the same meaning, and“average speed” and “average velocity” have the same meaning.Example 1:A train travels from city A to city B. It leaves A at 10:00 am and arrives at B at 2:30 pm. Thedistance between the cities is 150 miles. What was the average velocity of the train in miles per hour (mph )?Do you th ink the train was always traveling at the same speed?11Example 2: A train leaves station A at 8:00 am and arrives at station B at 10:00 am. The train stops atstation B for 1 hour and then continues to station C. It arrives at station C at 3:00 pm. The average velocityfrom A to B was 40 mph and the average velocity from B to C was 50 mph. What was the average velocityfrom A to C (including stopping time)?Generally, in computing average rates of change of a quantity y w ith respect to a quantity x, there is a functionthat shows how the values of x an d y are related.◮ Average rates of change of a function:The average rate of change of the function y = f(x)between x = x1and x = x2isaverage rate of change =change in ychange in x=f(x2) − f(x1)x2−x1The average rate of change is the slope of the secant linebetween x = x1and x = x2on the graph of f , that is, theline that passes through (x1, f(x1)) and (x2, f(x2)).xy0f(x1)f(x2)x1x2••y = f(x)f(x2) − f(x1)x2− x1Example 3:Find the average rate of change of g(x) = 2 + 4(x −1) with respect to x as x changes from −2to 5. Could you have pr edicted your answer using your knowledge of linear equations?Example 4:Find the average rate of change of k(t) =√3t + 1 with respect to t as t changes from 1 to 5.12Example 5: A particle is traveling along a straight line. Its position at time t seconds is given bys(t) = 2t2+ 3. Find the average velocity of the particle as t changes from 0 seconds to 4 seconds.Example 6:Let g(x) =1x. Find a value for x such that th e average rate of change of g(x) from 1 to xequals −110.Example 7:Find the average rate of change of k(t) = t3−5 with r espect to t as t changes from 1 to 1 + h.◮ Instantaneous rates of change: The phrase ‘instantaneous rate of change’ seems like an oxymoron,a contradiction in terms like the phrases ‘thunderous silence’ or ‘sweet sorrow’. However, because of yourexperience with traveling and looking at speedometers, both the concept of average velocity and the conceptof velocity at an instant have an intuitive meaning to you. The conn ection between the two concepts is that ifyou compute the average velocity over smaller and smaller time periods you should get numbers that are closerand closer to the speedometer reading at the instant you look at it.Definition: The instantaneous rate of change is defined to be the resu lt of computing the average rate ofchange over smaller and smaller intervals.The following algebraic approach makes this idea more precise.13Algebraic approach: Let s(t) denote, for sake of simp licity, the position of an object at time t. Our goalis to find the instantaneous velocity at a fixed time t = a, say v(a). Let the first value be t1= a, and the secondtime value t2= a + h. The correspond ing positions of the object ares1= s(t1) = s(a) s2= s(t2) = s(a + h),respectively. Thus the average velocity between times t1= a and t2= a + h iss2− s1t2− t1=s(a + h) − s(a)h.To see what happens to this average velocity over smaller an d smaller time intervals we let h get closer andcloser to 0. T his latter process is called finding a limit. Symbolically:v(a) = limh→0s(a + h) − s(a)h.Note:We can discuss the instantaneous rate of change of any function using the method above. When wediscuss the instantaneous rate of change of the position of an object, then we call this change the instantaneousvelocity of the object (or the velocity at an instant). We often shorten this phrase and speak simply of thevelocity of the object. Thus, the velocity of an object is obtained by computin g the average velocity of theobject over smaller and smaller time intervals.Example 8:A particle is traveling along a straight line. Its position at time t is given by s(t) = 5t2+ 3.Find the velocity of the particle when t = 4 seconds.Example 9:A particle is traveling along a straight line. Its position at time t is given by s(t) = 5t2+ 3.Find the velocity of the particle when t = 2 seconds.14The approach we will see now has the tremendous advantage that it yields a formula for the instantaneousvelocity of this object as a function of time t.Example 10:A particle is traveling along a straight line. Its position at time t is given by s(t) = 5t2+ 3.Find the velocity of the particle as a function of t.Note:Even if you have a formula