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Related Rates

Related Rates. Review Problem: Use implicit differentiation to find If. What is a Related Rate?.

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Related Rates

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  1. Related Rates

  2. Review Problem: Use implicit differentiation to find If

  3. What is a Related Rate? When working related-rate problems, instead of finding a derivative of an equation y with respect to the variable x, you are finding the derivative of equations with respect to time, a hidden variable t. This means that variables are tied together in their relationship with time.

  4. Rates of Change • You have studied the derivative as the slope of a tangent line. Now lets look at it as a rate of change of one variable with respect to another. • It’s common to use rate of change too describe motion of an object and until now we dealt only with motion in a straight line. But motion is not always in a straight line, more often it is along a curved path. We will study both in this section. • Remember that average change is written as:

  5. Falling Objects If a billiard ball is dropped from a height of 100 feet, its height s at a time t is given by the position function: S= -16t2 + 100 Where s is measured in feet and t is measured in seconds. Find the average velocity over the interval [1,2]. This procedure uses pre-calculus techniques.

  6. Velocity of a Falling Object The general formula for a free-falling object is the following: s(t) = (½)gt2 + v0t + s0 where • v0 is the initial velocity of the object • s0 is the initial position of the object • g is the acceleration due to gravity. g is either -9.8 m/s or-32 ft/s

  7. Rates of Change with Calculus Now that we’ve looked at average changes in rates, lets look at instantaneous rates of change. Suppose we want to find out the exact velocity of an object dropping from a given height at any given time. If we have the position function (location of the object along the straight path over time) then we can easily find the velocity by using calculus. We just take the derivative.

  8. Using the Derivative to Find Velocity At time t=0, a diver jumps from a platform diving board that is 32 feet above the water. The position of the diver is given by s(t)= -16t2 + 16t + 32 where s is measured in feet and t is measured in seconds • When does the diver hit the water? • What is the diver’s velocity at impact?

  9. Using the Derivative to Find Velocity – continued… • To find the time t when the diver hits the water, let s = 0 and solve for t. -16t2 + 16t + 32 = 0 -16(t+1)(t-2)=0 so t=-1 or t=2. since we can’t go back in time, we’ll use t=2

  10. Using the Derivative to Find Velocity – continued… 2. The velocity at time t is given by the derivative s’(t) = -32t + 16 so the velocity at time t=2 is: s’(2) = -32(2) + 16 =-48 ft/s

  11. Vertical Motion Practice Problems • A silver dollar is dropped from the top of a building that is 1362 feet tall. • Determine the position and velocity functions for the coin. s(t)= -16t2 + 1362 {initial velocity is 0 since it was just dropped} and we have v(t) = s’(t) = -32t • Determine the average velocity on the interval [1,2] Using the formula from a previous slide: [s(2)-s(1)]/2-1 = 1298 -1346 = -48 ft/sec • Find the instantaneous velocities when t=1 and t=2. v(t) = s’(t) = -32t  v(1)= -32ft/sec and v(2)= -64ft/sec • Find the time required for the coin to reach ground level. s(t)= -16t2 + 1362 = 0  t = 9.226 sec • Find the velocity of the coin at impact. v(9.226)= -295.242 ft/sec

  12. For your notes:

  13. The sphere is growing at a rate of . Note: This is an exact answer, not an approximation like we got with the differential problems. Consider a sphere of radius 10cm. Now, suppose that the radius is changing at an instantaneous rate of 0.1 cm/sec. (Possible if the sphere is a soap bubble or a balloon.)

  14. Find Water is draining from a cylindrical tank at 3 liters/second. Where the radius = 3cm. How fast is the surface dropping? Convert:

  15. Steps for Related Rates Problems: 1. Draw a picture (sketch). 2. Write down known information. 3. Write down what you are looking for. Write an equation to relate the variables. Plug in the constants 5. Differentiate both sides with respect to t. Evaluate. Check to make sure you answered the correct question.

  16. Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

  17. Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

  18. Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later? B A

  19. Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later? B A p

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