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2.7 Related Rates

2.7 Related Rates. Find. Example : Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?. (We need a formula to relate V and h . ). ( r is a constant.). Steps for Related Rates Problems:. 1. Draw a picture (sketch).

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2.7 Related Rates

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  1. 2.7 Related Rates

  2. Find Example: Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping? (We need a formula to relate V and h. ) (r is a constant.)

  3. Steps for Related Rates Problems: 1. Draw a picture (sketch). 2. Write down known information. 3. Write down what you are looking for. 4. Write an equation to relate the variables. 5. Differentiate both sides with respect to t. 6. Evaluate.

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

  5. 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

  6. 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

  7. 2.8 Linear approximations and differentials

  8. We call the equation of the tangent the linearization of the function. For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.

  9. Linear approximation Recall the equation of the tangent line of f(x) at point ( a, f(a) ) : This is called the linear approximation or tangent line approximation of fat a. The linear function is called linearization of f at a . Examples on the board.

  10. Differentials • The ideas behind linear approximations are sometimes formulated in the notation of differentials. • If y=f(x), where f is a differentiable function, then • the differentialdx is an independent variable, • the differentialdy is a dependent variable and is defined in terms of dx by the equation • The next example illustrates the use of differentials in estimating the errors that occur because of approximate measurements.

  11. Example: The radius of a circle was measured to be 10 ft with a possible errorat most 0.1 ft. What is the maximum error in using this value of the radius to compute the area of the circle? error in r error in A maximum error in A

  12. Example (cont.) • Relative error in the area: that is, twice the relative error in the radius. • In our case: • This corresponds to percentage error of 2%

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