Exercise 4.1.8 Sketch the parabola y = x. For what values of x on the parabola is the slope of the tangent line positive? Negative? What do you notice about the graph at the point(s) where the sign of the slope changes from positive to negative and vice versa? Exercise 4.1.9 An object is traveling in a straight line so that its position (that is, distance from some fixed point) is given by this table: time (seconds) 01 2 3 distance (metres) 0 10 25 60 Find the average speed of the object during the following time intervals: [0, 1], [0, 2], [0,3], [1, 2], [1, 3], [2,3]. If you had to guess the speed at t = 2 just on the basis of these, what would you guess? Exercise 4.1.11 If an object is dropped from an 80-metre high window, its height y above the ground at time t seconds is given by the formula y = f(t) = 80-4.9t2. (Here we are neglecting air resistance; the graph of this function was shown in Figure 1.1.) Find the average velocity of the falling object between (a) I sec and 1.1 sec, (b) 1 sec and 1.01 sec, (c) 1 sec and 1.001 sec. Now use algebra to find a simple formula for the average velocity of the falling object between I sec and 1 + At sec. Determine what happens to this average velocity as At approaches 0. That is the instantaneous velocity at time t = 1 second (it will be negative, because the object is falling)