1. Sketch y = x on [16]. a. Bound the area from below by using a rectangle; bound the area from above using a rectangle. Explain graphically why the average value of y = x on [1.6] must be between these two values. b. Find the average value of y = x on [1.6] using our new formula. c. Sketch a rectangle with the average height found in part b (on your original graph) d. What else has the same area as this rectangle? How do you know? 2. Find the average value of y = sinx on [0,2] show your set up and work. The result should have been guessable; explain. 3. Use the position function s(t) = -4.9t + 9.8t +117.6. This represents the height of a rock (in meters) thrown upwards from the top of a building t seconds after it was thrown. a. What is the initial height of the rock? b. Find the time when the rock hits the ground. c. Use the average rate of change formula to find the average velocity of the rock while it was in the air. d. Find the velocity function for the rock. e. Use the average value of a function formula to find the average velocity of the rock while it was in the air. 4. Suppose P(t) = 5.3(1.025) is the population, measured in millions, of a country at year t. a. Calculate P(t) dt. b. What are the units of P(t) dt? c. What is the practical meaning of P(t) at? -10 5. A 3 2 2 3 4 5 6 7 Graph of g'(x) Let g'(x) be a function defined on the closed interval [0,7]. The graph of g', consisting of four line segments is shown above. Let g be the function given by g(x)= f g(t) dt (a) Find g(3), g'(3), and g"(3). (b) Find the average rate of change of g on the interval 0 x 3. (c) For how many values c, where 0 < c < 3, is g'(c) equal to the average rate found in part (b) ? Explain your reasoning. (d) Find the x-coordinate of each point of inflection of the graph of g on the interval 0 < x < 7. Justify your answer.