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DO NOW

DO NOW. Approximate 3 √26 by using an appropriate linearization. Show the computation that leads to your conclusion. The radius of a circle increased from 2.00 to 2.02 m. Estimate the resulting change in area. Estimate as a percentage of the circle’s original area. 4.6 – Related Rates.

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DO NOW

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  1. DO NOW • Approximate 3√26 by using an appropriate linearization. Show the computation that leads to your conclusion. • The radius of a circle increased from 2.00 to 2.02 m. • Estimate the resulting change in area. • Estimate as a percentage of the circle’s original area.

  2. 4.6 – Related Rates

  3. What are related rates? • If…. • Two variables are connected by an equation, • And at a given moment one of them is changing at a known rate. Then… using the Chain Rule we can find • The rate at which the other variable is changing. To solve a problem like this is to relate two rates of change  related rates problem. Solution: • Find an equation that related the quantities; • Use the Chain Rule to differentiate both sides with respect to time.

  4. Example 1 • Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?

  5. Example 2 • A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the tope of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?

  6. Example 3 • A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep.

  7. Strategy • Read the problem carefully. • Draw a diagram if possible. • Introduce notation. Assign symbols to all quantities that are functions of time. • Express the given information and the required rate in terms of derivatives. • Write an equation that relates the various quantities of the problem. - If necessary, use the geometry of the situation to eliminate one the variables by substitution (as in the preceding example). • Use the Chain Rule to differentiate both sides of the equation with respect to t. • Substitute the given information into the resulting equation and solve for the unknown rate. - Make sure that Step 7 follows Step 6! A common error is to substitute the given numerical information too early.

  8. Example 4 • Car A is traveling west at 50 mi/h and car B is traveling north at 60 mi/h. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 mi and car B is 0.4 mil from the intersection?

  9. Example 5 • Lady Gaga walks along a straight path on the red carpet at a speed 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on Lady Gaga. At what rate is the searchlight rotating when Lady Gaga is 15 ft from the point on the path closest to the searchlight?

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