ws12 - k 5 a Prove that the formula for the derivative of...

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WORKSHEET 12 - Fall 1995 1. Write f ( x ) as a composition of two functions in two diFerent ways. Write f ( x ) as a composition of three functions. DiFerentiate f ( x ). a) f ( x )= p x 2 +1 b) f ( x )= ± sin x x ²± x 2 x +2 ² c) f ( x )=(2 x 2 x ) 5 / 2 2. a) The radius of a balloon is given by the formula r ( t )= t 2 +1 1. (Imagine you are blowing it up, so after one second, the radius is r (1) = 2 1 inches.) Give a formula for the rate of change in the radius with respect to time. b) Give a formula for the rate of change of the volume of the baloon with respect to time. c) How fast is the volume changing when t =2? d) How fast is the volume changing when r =2? 3. ±ind dy dt given that y = 3 x 3 2 x 2 +1 and that x =tan( t 2 ). 4. Thermodynamic experimentation shows that when air expands adiabatically (without losing or gaining heat), the volume V in cubic centimeters and the pressure P in kilopascals (kPa) are approximately related by the following relation: PV k = C where k and C are constants. When the volume of a sample of air was 10 cm 3 and increasing at a rate of 4 cm 3 / min, the pressure was 25 kPa and decreasing at a rate of 14 kPa/min. Determine the value of
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Unformatted text preview: k . 5. a) Prove that the formula for the derivative of an inverse function is ( f − 1 ) ( x ) = 1 f ( f − 1 ( x )) . (Hint: Let g ( x ) = f − 1 ( x ), then f ( g ( x )) = x . DiFerentiate.) b) What is (arccos( x )) ? Write an expression which is free of trig functions. (Hint: Use the formula in part a . To help you simplify, draw a triangle.) c) Let y = tan 2 (arccos( x )). ±ind dy dx using the chain rule. d) ±ind another expression for y which doesn’t mention trig functions. (Hint: Again, draw a triangle.) e) ±ind dy dx from your expression in part d). How does it compare with your answer in part a)? 6. What is f (4) if f (4) = 2 and x 2 f ( x ) = 6 x + ( f ( x )) 3 for x near 4? 7. A diFerentiable function y ( x ) satis²es x 2 cos y + sin y = x and y (1) = 0. What is y (1)?...
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