121616949-math.136 - the remaining one As in the case when...

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122 Chapter 6 Applications of the Derivative . . P Q x y θ Figure 6.10 Swing. (b) Here our two variables are x and θ , so we want to use the same right triangle as in part (a), but this time relate θ to x . Since the hypotenuse is constant (equal to 10), the best way to do this is to use the sine: sin θ = x/ 10. Taking derivatives we obtain (cos θ ) ˙ θ = 0 . 1 ˙ x . At the instant in question ( t = 1 sec), when we have a right triangle with sides 6–8–10, cos θ = 8 / 10 and ˙ x = 6. Thus (8 / 10) ˙ θ = 6 / 10, i.e., ˙ θ = 6 / 8 = 3 / 4 rad/sec, or approximately 43 deg/sec. We have seen that sometimes there are apparently more than two variables that change with time, but in reality there are just two, as the others can be expressed in terms of just two. But sometimes there really are several variables that change with time; as long as you know the rates of change of all but one of them you can find the rate of change of the remaining one. As in the case when there are just two variables, take the derivative
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Unformatted text preview: the remaining one. As in the case when there are just two variables, take the derivative of both sides of the equation relating all of the variables, and then substitute all of the known values and solve for the unknown rate. EXAMPLE 6.18 A road running north to south crosses a road going east to west at the point P . Car A is driving north along the first road, and car B is driving east along the second road. At a particular time car A is 10 kilometers to the north of P and traveling at 80 km/hr, while car B is 15 kilometers to the east of P and traveling at 100 km/hr. How fast is the distance between the two cars changing? Let a ( t ) be the distance of car A north of P at time t , and b ( t ) the distance of car B east of P at time t , and let c ( t ) the distance from car A to car B at time t . By the Pythagorean Theorem, c ( t ) 2 = a ( t ) 2 + b ( t ) 2 . Taking derivatives we get 2 c ( t ) c ± ( t ) = 2 a ( t ) a ± ( t )+2 b ( t ) b ± ( t ),...
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