121616949-math.147 - 6.5 The Mean Value Theorem 133...

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6.5 The Mean Value Theorem 133 Exercises 1. Let f ( x ) = x 4 . If a = 1 and dx = Δ x = 1 / 2, what are Δ y and dy ? 2. Let f ( x ) = x . If a = 1 and dx = Δ x = 1 / 10, what are Δ y and dy ? 3. Let f ( x ) = sin(2 x ). If a = π and dx = Δ x = π/ 100, what are Δ y and dy ? 4. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.02 cm thick to a sphere with diameter 40 meters. (Recall that the volume of a sphere of radius r is V = (4 / 3) πr 3 . Notice that you are given that dr = 0 . 02.) 5. Show in detail that the linear approximation of sin x at x = 0 is L ( x ) = x and the linear approximation of cos x at x = 0 is L ( x ) = 1. Here are two interesting questions involving derivatives: 1. Suppose two different functions have the same derivative; what can you say about the relationship between the two functions? 2. Suppose you drive a car from toll booth on a toll road to another toll booth at an average speed of 70 miles per hour. What can be concluded about your actual speed during the trip? In particular, did you exceed the 65 mile per hour speed
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