Rates of Change
Rates of change represent the fundamental concept in calculus, capturing how one quantity changes concerning another. Whether analyzing the speed of a moving object, the growth of a population, or the slope of a curve, rates of change provide insights into dynamic systems. In calculus, they are quantified through derivatives and differential equations, offering a precise mathematical framework to understand and predict varying phenomena. Understanding rates of change enables us to model and interpret real-world scenarios, from predicting financial trends to optimizing engineering designs, making it a cornerstone of mathematical analysis and problem-solving.
Questions
- Tinseltown wants to know if their population is in danger. Their current population is 12,000 people but in 2001 it was 15,321. What is their growth rate?
- How do you find the average rate of change of a function ?
- For every 1/2 mile that Sira jogs, she walks 3/4 mile. What unit rate gives the number of miles Sira jogs for every mile she walks?
- A deep-sea exploring ship is pulling up a diver at the rate of 25 feet per minute The diver is 200 feet below sea level. How deep was the diver 10 minutes ago?
- Pete worked 7 hours and charged 390. Rosalee worked 8 hours and charged 430. If Pete's charge is a linear function of the number of hours worked, find the formula for Pete's rate, and how much he would charge for working 1010 hours for Fred?
- A bird flies 2/3 of a mile per minute. How many miles per hour is it flying?
- Let #f(x) = (2x)/(x-1)#. What is #f(a), f(a+h)# and the quantity #(f(a+h) - f(a))/h#, where #h# is not 0?
- Given the function g(x)=x^2-9x+16g(x)=x 2 −9x+16, determine the average rate of change of the function over the interval 1\le x \le 61≤x≤6. help?
- A kayak can travel 24 miles downstream in 3 hours, while it would take 12 hours to make the same trip upstream. Find the speed of the kayak in still water, as well as the speed of the current?
- To arrive on schedule a train is to cover a distance of 60km at 72km/hr. If it starts 10minute late, at what speed must it move to arrive on schedule.?
- Carrie attends Grant University. She works 22 hours each week at Freight Unlimited and makes $12 per hour. How much money will she earn during an 18-week semester?
- A pond at a hotel holds 4,290 gallons of water. The groundskeeper drains the pond at a rate of 78 gallons of water per hour. How long will it take to drain the pond?
- An airplane rises vertically 1000 feet over a horizontal distance of 1 mile. What is the angle elevation of the airplane's path?
- At a dairy plant, a milk tank can be filled in 4 hours (using the in-valve). Using the out-valve, the tank can be emptied in 8 hours. If both valves are open and milk is being pumped into the tank, how long will it take to fill the tank?
- What is #y# when #x=10#, if #y=8# when #x=20#?
- A submarine descends below the ocean's surface at a of rate 75 feet per minute. How many feet below the ocean's surface will the submarine be in 12 minutes?
- What is the rate of change if Jenna earns $60 every #2 1/2# weeks?
- A pencil company used 60 grams of rubber to make 10 pencils, which is a rate of grams per pencil?
- A bacteria doubles its population in 8 hours. At this rate how many hours would it take the population of the bacteria to triple?
- How many weeks will it be before Jake will be jogging 61 minutes per day?