Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 81. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | The Average Value of a Function An Application of the Definite Integral 781 6. The amount of a certain chemical in a mixture varies with time. If g t 5e is the number of grams of the chemical at time t what is the average number of grams of the chemical in the mixture on the time interval 0 1 7. The velocity of an object is given by 3 sin t . a What is the object s speed as a function of time b What is the object s net displacement from t 0 to t 2 c How far has the object traveled from t 0 to t 2 d What is the object s average velocity on 0 2 e What is the object s average speed on 0 2 8. The graphs of functions f g h and k are given below. Let I denote the average value of f on 0 4 . Let II denote the average value of g on 0 4 . Let III denote the average value of h on 0 4 . Let IV denote the average value of k on 0 4 . Put I II III and IV in ascending order with or signs between them as appropriate. 9. A bicycle speedometer will give the average velocity of a bicyclist over the time period the bicycle is moving. By pressing a button the bicylist can reset the average velocity counter. Suppose a long-distance cyclist has averaged 14 miles per hour for the first two hours of her trip. She resets the average velocity counter. For the next four hours her average velocity is 18 miles per hour. a What is the cyclist s average velocity for the six-hour trip b How far has she traveled 10. It takes a bicyclist 8 minutes to ride 1 mile uphill and then 2 minutes to ride 1 mile downhill. Explain how we know that the cyclist s average velocity for the hill is 12 miles per hour. 11. The temperature of a hotplate of radius 5 inches varies with the distance from the center of the plate. For the area within 2 inches of the center the average temperature is 100 degrees. For the area between 2 and 5 inches from the center the average temperature is 80 degrees. What is the average temperature of the plate 12. Find the average value of sin 3t on 0 2 . Explain your reasoning. 782
