8/7/2019 APCalculusBCExamFall2007

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AP Calculus BC | Final Exam Fall 2007 | ShublekaFull Name __________________________

Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what theinfinitely little might be. Russell, Bertrand (1872-1970)In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press

Inc., 1988.

Please write your full name on every page, label all your solutions and present your workneatly.

Please write and sign a statement to indicate that you have followed all the exam rules.

_______________________________________________________________________________________________________________________________________________________________________________________________________________________.

Student Signature: _______________________________________________

8/7/2019 APCalculusBCExamFall2007

2/2

AP Calculus BC | Final Exam Fall 2007 | ShublekaFull Name __________________________

Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what theinfinitely little might be. Russell, Bertrand (1872-1970)In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press

Inc., 1988.

Problem One

The equation 322 =+ yxyx represents a rotated ellipse. Find the points at which this

ellipse crosses the x-axis and show that the tangent lines at these points are parallel. 10ptsProblem TwoA runner sprints around a circular track of radius 100m at a constant speed of 7 m/s. The

runners friend is standing at a distance of 200m from the center of the track. How fast is thedistance between the friends changing when the distance between them is 200m? 10ptsProblem Three

For positive BA, , the force between two atoms is a function of the distance, ,r between

them: 32)(r

B

r

Arf += for 0>r . Find the zeros, asymptotes, and the coordinates of the

critical and inflection points of )(rf then sketch a graph. Illustrating your answers with a

sketch, describe the effect on the graph of )(rf of: (i) increasing B , holding A fixed. (ii)

increasing A , holding B fixed. 10ptsProblem FourInvestigate how extrema and inflection points move when c changes. You should also

identify any transitional values of at which the basic shape of the curve22

1)(

xc

cxxf+

= changes. 10pts

Problem Fivea) Show that of all rectangles with a given area, the one with the smallest perimeter is a

square.b) Show that of all the rectangles with a given perimeter, the one with the greatest area

is a square. 10ptsProblem Six

Find the area of the largest rectangle that can be inscribed in the ellipseb

y

ax

22

1 += . 10 pts

Problem Seven

A number a is called a fixed point of a function )(xf if aaf =)( . Prove that if 1)(' xf

for all real numbers x , then )(xf has at most one fixed point. 10pts

Problem EightIn an automobile race along a straight road, car A passed car B twice. Prove that at sometime during the race their accelerations were equal. 10ptsProblem NineShow that of all the isosceles triangles with a given perimeter, the one with the greatest areais equilateral. 10ptsProblem TenFind an equation of the line passing through the point (3 , 5) that cuts off the least area fromthe first quadrant. 10pts