AP CALCULUS summer2009 workshopIII

AP CALCULUS summer2009 workshopIII - AP CALCULUS SUMMER...

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AP CALCULUS SUMMER WORKSHOP III Problems : 1) ( 2004 Nova Scotia Math League , Game 2 : Team Question ) The parabola y x = 2 meets the parabola y x x = − + 2 6 4 at two points, P and Q . Find the equation of the line passing through P and Q . 2) ( 2004 Nova Scotia Math League , Game 4 : Team Question ) The integers b and c are chosen so that: i) The quadratic equation 5 0 2 x bx c + + = has two roots, one of which is 2. ii) The quadratic equation 5 0 2 x cx b + + = has two roots, one of which is 3. Determine the values of b and c . 3) ( 2004 Nova Scotia Math League , Game 2 : Team Question ) In Δ ABC , sides AB and AC are 10 units long. Determine the maximum possible area of Δ ABC . 4) Let S = + + + + + 1 2 3 4 5 6 2009 2009 2009 2009 2009 2009 . Determine the last digit of S . Hint : Look for a pattern in each of the addends (i.e. 1 2 3 4 5 2009 2009 2009 2009 2009 , , , , and 6 2009 ). 5) Consider the function ( ) f x x x = 2 2 2 . a) Find the domain of ( ) f x . b) If x 1 , then find the equation of the inverse function, ( ) f x 1 . c) Find the range of ( ) f x . 6) Find the domain of the following functions: a) ( ) f x x x = + 4 b) ( ) f x x x = + 1 3 c) ( ) ( ) f x x = ln 4 2 d) ( ) f x x x x = + + 2 7 10 1 7) What is the range of ( ) f x x = 1 2 2 sin ? Hint : Start with the inequality − ≤ 1 2 1 sin x and perform some algebraic manipulations on it. 8) Data was collected during an experiment in which an enzyme with its substrate was subjected to a medium with a varying pH. The result of the effects of pH on the action of the enzyme on its substrate is shown in the graph
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