1 / 8

5-6 related rates

Learn how to solve rate problems in calculus involving differentiable functions of time, and find equations that relate corresponding rates. Explore examples and develop strategies for problem-solving.

sheryla
Download Presentation

5-6 related rates

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 5-6related rates

  2. *Any question involving 2 or more variables that are differentiable functions of time t can be used to find an equation that relates their corresponding rates. Ex 1a) Assume that the radius r of a sphere is a differentiable function of t and let V be the volume of the sphere. Find an equation that relates dV/dt and dr/dt. b) Assume that the radius r and height h of a cone are differentiable functions of t and let V be the volume of the cone. Find an equation that relates dV/dt, dr/dt, and dh/dt.

  3. *Calculus deals with variables that change over time. Sometimes we refer to calculus as the “algebra of change.” • STRATEGY • Understand the problem • Variables whose rates we know • Variables whose rates we want • Develop a math model to relate the variables • - picture - geometry 

  4. STRATEGYcont… Write an equation - sometimes we rearrange the equation to get what we want alone - sometimes we must eliminate one of the variables (4) Differentiate both sides implicitly wrtt Substitute the known values & solve Interpret the solution (appropriate units!)

  5. Ex 2) A hot-air balloon rising straight up from a level field is tracked by a range finder 500 feet from the lift-off point. At the moment the range finder’s elevation angle is /4, the angle is increasing at the rate of 0.14 radians per minute. How fast is the balloon rising at that moment? h  500 want:

  6. Ex 3) A police cruiser, approaching a right-angled intersection from the north, is chasing a car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car? c a b a = .6 b = .8 c = 1.0 want

  7. Ex 4) Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep? 5 10 (r needs to go away) 6

  8. homework Pg. 255 # 1–15, 29, 31

More Related