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 yx = 2 meets the parabola yxx =− + 2 64 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 50 2 x bx c ++ = has two roots, one of which is 2. ii) The quadratic equation 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 =++++ + 123456 2009 2009 2009 2009 2009 2009 . Determine the last digit of S . Hint : Look for a pattern in each of the addends (i.e. 12345 2009 2009 2009 2009 2009 ,,,, and 6 2009 ). 5) Consider the function () fx xx = 2 2 2 . a) Find the domain of . b) If x 1, then find the equation of the inverse function, ( ) 1 . c) Find the range of . 6) Find the domain of the following functions: a) x x =+ 4 b) x x = + 1 3 c) x =− ln 4 2 d) x = −+ + 2 71 0 1 7) What is the range of x = 1 22 sin ? Hint : Start with the inequality −≤ 12 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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below. The units of the rate of enzyme action are mmol L s and pH is dimensionless. a) If this data were to be fitted to a relation, would this relation be a function ? Justify your answer. b) Does this relation have an inverse ? Using the graph, illustrate your point. c) What is the domain and range of this relation? d) At what pH does the enzyme exhibit maximum rate of enzyme action? (This value is referred to as the optimum pH value.) If we were to draw tangent lines to the graph (these are lines that locally touch the graph at a singular point), then what would the slope of the tangent line be at the optimum pH? e) What can be said about the rate of enzyme action as the pH increases beyond the optimum pH (i.e. it becomes more alkaline or basic)? What can be said of the slopes of the tangent lines within this respective pH range? 9) Solve for x : () ( ) ( )( ) ( )( ) log log log log log log log log log 235 23 25 35 xxx xx =++ Hint : Use the change of base formula for logarithms. 10) Solve for x : 469 x += Hint : Divide both sides of the equation by 9 x and then use a substitution to clean up the algebra.
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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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AP CALCULUS summer2009 workshopIII - AP CALCULUS SUMMER...

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