These areas are depicted below:. The equation for the average value is a statement of the intuitive fact that if we construct a rectangle with the height f avg and width b - a , its area should be the same as the area under curve from a to b. The mean value theorem for integrals states the following: if f is a continuous function on [ a , b ] , there exists at least one c on [ a , b ] such that.
The essential life function of breathing is cyclic. At rest a fit human will complete approximately 10 full respiratory cycles every minute. The maximum rate of air flow into the lungs is about one-half liter per second. Based on these assumptions the rate of air flow into the lungs can be modeled by the function.
The volume in liters of air in the lungs at time in seconds is. The relationship between the rate of air flow and the volume of air in the lungs is clearly shown in the following plot. Observe that the volume is periodic, with period 6 seconds, and that the maximum amount of air in the lungs is L, or approximately 0. The maximum occurs every time ; the minimum value, 0 L, is attained every time. During times of exercise and other stresses respiration increases; the maximum lung capacity is about 3.
The average amount of air in the lungs during one complete respiration cycle is. The average amount of air in the lungs during each respiratory cycle while at rest is a little under L.
Note that the average value is not the maximum rate of air flow. Numerically the two quantities are close, but the values and units are different. For James, we want to calculate. We obtain. Suppose James and Kathy have a rematch, but this time the official stops the contest after only 3 sec. Does this change the outcome? Kathy still wins, but by a much larger margin: James skates 24 ft in 3 sec, but Kathy skates Julie is an avid skydiver.
She has more than jumps under her belt and has mastered the art of making adjustments to her body position in the air to control how fast she falls. Since Julie will be moving falling in a downward direction, we assume the downward direction is positive to simplify our calculations. Julie executes her jumps from an altitude of 12, ft. She continues to accelerate according to this velocity function until she reaches terminal velocity. After she reaches terminal velocity, her speed remains constant until she pulls her ripcord and slows down to land.
Using this information, answer the following questions. These suits have fabric panels between the arms and legs and allow the wearer to glide around in a free fall, much like a flying squirrel. Wingsuit flyers still use parachutes to land; although the vertical velocities are within the margin of safety, horizontal velocities can exceed 70 mph, much too fast to land safely.
If Julie dons a wingsuit before her third jump of the day, and she pulls her ripcord at an altitude of ft, how long does she get to spend gliding around in the air. The Mean Value Theorem for Integrals The Mean Value Theorem for Integrals states that a continuous function on a closed interval takes on its average value at the same point in that interval. Hint Use the procedures from Example to solve the problem. Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas.
Hint Follow the procedures from Example to solve the problem. Hint Use the chain rule to solve the problem. Add up the data and divide by the number of data points:. What about f 1. No matter how many sample points we include, there will always be some missing… Unless we can use the magic of calculus to catch them all. The sampling process should remind you of a Riemann Sum. Midpoint Riemann sum. Then the estimated average is the sum:.
The trick is to multiply and divide by b — a. Above, we only estimated the average to be 2. Find the average energy of the reaction over the range of possible levels of reactant. Chemistry can be fun too! But what does this have to do with calculus?? However, the keyword average tells us that mathematics plays a major role in this problem. Then work out the integration, which involves Integration By Parts in this case. Averages are also called means.
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