Rate of Change of a Function
The rate of change of a function is a fundamental concept in calculus, measuring how quickly the function's output value changes concerning its input. It captures the slope or steepness of the function at a particular point, providing valuable insights into its behavior. Calculated using derivatives, this rate illuminates dynamic aspects of functions, such as velocity, acceleration, or growth rates in various real-world scenarios. Understanding the rate of change is essential in fields like physics, economics, and engineering, where analyzing how quantities evolve over time is crucial for making informed decisions and predictions.
Questions
- Can someone help out with the question below?
- What is Rate of Change of a Function?
- The temperature T in #""^oC# of a particular city during a 24 hour period can be modelled by #T = 10 + 8 sin 12 pi t# where t is the time in hours, with t = 0 corresponding to midday. Find the rate at which the temperature is changing at 4pm.?
- How do you determine the rate of change of a function?
- How do you find #\lim _ { x \rightarrow 7} \frac { x ^ { 3} - 343} { x - 7}#?
- How do you graph the derivative of a function when you are given the graph of the function?
- Suppose you are blowing a spherical bubble, filling it in with air at a uniform rate of 400mm^3/s. How fast is the radius of the bubble increasing by the time it is already 20mm long?
- How do you find a function f(x), which, when multiplied by its derivative, gives you #x^3#, and for which #f(0) = 4#?
- A rectangle's base remains 0.5 cm while its height changes at a rate of 1.5 cm/min. At what rate is the area changing, in cm when the height is 1.5 cm?
- A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is #pi/4# rad?
- What is the average rate of change of the function #f(x) = 2x^2 - 3x - 4# for #-3<=x<=-1#?
- What is the purpose of a derivative?
- Suppose #g# is a function whose derivative is #g'(x)=3x^2+1# Is #g# increasing, decreasing, or neither at #x=0#?
- A factory produces bicycles at a rate of 80+0.5t^2-0.7t bicycles per week (t in weeks). How many bicycles were produced from day 15 to 28?
- If #P=215-5Q#, what is price elasticity of demand when #P=15#. (1) #43#; (2) #40#; (3) #-5#; (4) #-0.075#?
- Suppose the population of a town grows according to the equation #y=100t+t^2#, how do you find the rate of growth at time t=100 years?
- The volume of a sphere is changing at a constant rate of #pi/3 \ cm^3s^-1#. How fats is the surface area changing when the volume is #(9pi)/2#?
- The cost function for a product is C(x)=0.8x^2 +120x+110. How to find average cost over [0,600] ?
- How do you solve the AP Calculus 2013 Free Response question #3? http://media.collegeboard.com/digitalServices/pdf/ap/apcentral/ap13_frq_calculus_ab.pdf
- The radius of a sphere is measured at the rate of 2 cm/sec.Find the rate increase of the volume when the radius is 6cm?