Please neatly print your name (First last): Clculus in Context #3 Math 113 Chapter 3 How does our understanding of derivatives give us insight into air pressure estimates used by climbers? Higher altitudes can present challenges to mountain climbers and hikers. One of those challenges is dealing wnh air pressure changes. A rule of thumb is that air pressure {using inches of mercury) drops about 1 inch for every lOOO-foot increase in height above sea level. 1. Given that the air pressure at sea level is 30 inches of mercury, write a formula for air pressure as a function of height above sea level for this rule of thumb. Call this function the Climbcr's Estimate and use the function notation C (h). 2. The summit of Babad Do'ag (Mount Lemmon) north of Tucson is roughly 9200 feet above sea level. Use the Climber's Estimate function to estimate the air pressure at the summit of Babad Do'ag. Show your work appropriately. 3. At roughly 14500 feet above sea level, the summit of Tumanguya (Mount Whitney) in California is the highest point in the lower 48 states of the US. Use the Climber's Estimate mction to estimate the air pressure at the summit of Tumanguya. Show your work appropriately. 4. At what elevation does the Climber's Estimate predict that the air pressure would be zero? Show your work appropriately. 5. Find the derivative of the Hiker's Estimate function. Explain why the value of the derivative makes sense within the framework of this rule of thumb. Page 177 We might learn in a physics class that the air pressure can be found as a function of altitude h feet above sea level given by the function: P(h) = 30 e-3.23x10 -h where P(h) is measured using inches of mercury. 6. Find the first derivative of the physics function. 7. Find the second derivative of the physics Show your work appropriately and simplify. unction. Show your work appropriately and simplify. 8. Using calculus, explain how the first and second derivatives tell us whether the physics function is increasing or decreasing and whether it is concave up or concave down. Now sketch a reasonable graph of P(h) = 30 e -3.23x10-5.h using the domain [0, 30000]. 3000 30 10. Sketch a reasonable graph of both the Climber's Estimate function and the physics function using the window h=[0, 5000], pressure =[25, 30]. 1 1. Based on your graph, for what altitudes would you suggest that the rule of thumb is sufficient in estimating air pressure? 1000 2000' 3000 4000 5000 Page 178