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AP CALCULUS THE DERIVATIVE & ITS APPLICATIONS REVIEW 1) Let yx x = 2 . Show that () dy dx xx x =+ 21 2 ln . 2) Find the exact value of sin arctan 1 2 . 3) Consider the graph of y arc x = cot to the right. Find the following limits: lim cot x arc x →∞ and lim cot x arc x →−∞ . 4) The following shows the graph of a function ( ) fx together with graph of its first derivative fx and its second derivative ′′ . Each answer shows the same three graphs, but only one is labelled correctly. Which one? 5) a) Explain why x x =− 3 has an inverse function. b) Find the equation of the tangent line to the inverse of ( ) x x 3 at the point ( ) 30 3 , . 6) Suppose that there exists a function f such that ( ) x 1 2 cos and ( ) f π = 0 for all x . Explain why f has an inverse function gx and find g 0 . 7) Let ye x = 4 arctan ln . Find dy dx . 8) Suppose that the position of a particle at time t seconds is given by the function ( ) yt t t =−+ 3 34 where y is measured in meters. a) Find the average velocity of the particle for [ ] t 03 , . b) Find the average speed of the particle for [ ] t , .
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9) Use linear approximation to approximate 801 3 . . Use the fact that 82 3 = . 10) Find the number c that satisfies the conclusion of Rolle’s Theorem for ( ) fx xx =+ 18 for [] x ∈− 18 0 , . 11) Suppose you know that a certain function ( ) fx has a derivative ( ) which has values in the range () −≤ 45 when x is between 6 and 10. Find the maximum and minimum possible values of f 10 if f 61 7 = . 12) A 12 meter ladder is leaning against a vertical wall. At the moment that the angle between the ladder and the ground (at the bottom of the ladder) is π 6 radians, the bottom of the ladder is sliding away from the wall at a speed of 3 m s . How fast, in radians per second, is the angle between the ladder and the ground changing at that moment? 13) A lighthouse is 0.5 km from a shoreline. If the the lighthouse beam rotates at the rate of 0.25 revolutions per minute, how fast is the lightspot on the shoreline moving (in km/hr) at a point 1 km along the shore from the point nearest the lighthouse? Note : You need to convert revolutions per minute to radians per hour. 14) Find the point on the line 69 xy += that is closest to the point ( ) 10 , . 15) A table with a horizontal square top and four legs of equal length is to be constructed (see diagram to the right). The enclosed volume of the table must be 12 cubic meters. The cost for the tabletop is $4 per square meter while the cost for the metal legs is $3 per meter.
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This note was uploaded on 07/27/2010 for the course MATH MATH 1025 taught by Professor Tanny during the Spring '09 term at York University.

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