1969 AP Calculus BC: Section I

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1 969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric equations, y t t =, y = = only =, y = = only =, y =. What are the coordinates of the inflection point on the graph of y = ( + )arctan? (,) (, ) ( ),,,. The Mean Value Theorem guarantees the eistence of a special point on the graph of y =,. What are the coordinates of this point? between (, ) and ( ) (, ) (, ) (, ), None of the above. 8 d + = 6 5. If + y+ y =, then the value of dy d at = is not defined AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

2 6. What is h h 8 lim? h 969 AP Calculus BC: Section I The limit does not eist. It cannot be determined from the information given. 7. For what value of k will k + have a relative maimum at =? None of these 8. If h ( ) = f ( ) g ( ), f ( ) = g ( ), and g ( ) = f( ), then h ( ) = f ( g ) ( ) ( g ( )) ( f( ) ) ( g ( ) + f( ) ) 9. The area of the closed region bounded by the polar graph of r = + cosθ is given by the integral + cosθdθ + cosθdθ ( + cosθ) dθ + cosθ dθ ( ) + cosθdθ. + d = ln ln + AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

3 969 AP Calculus BC: Section I. The point on the curve none of the above + y = that is nearest the point, occurs where y is. If t F( ) = e dt, then F ( ) = e e e e + e e +. The region bounded by the -ais and the part of the graph of y = cos between = and = is separated into two regions by the line = k. If the area of the region for k is three times the area of the region for k, then k = arcsin arcsin 6. If = + and =, then dy y u du = + ( ) 6 + AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

4 969 AP Calculus BC: Section I 5. If f ( ) and g ( ) eist and f ( ) > g ( ) for all real, then the graph of y = f( ) and the graph of y = g( ) intersect eactly once. intersect no more than once. do not intersect. could intersect more than once. have a common tangent at each point of intersection. 6. If y is a function such that y > for all and y < for all, which of the following could be part of the graph of y = f( )? 7. The graph of 5 y = 5 has a point of inflection at (, ) only (,6 ) only ( ) (, ) and (,6 ) (, ) and (, 56 ), 56 only 8. If f( ) = + for all, then the value of the derivative f ( ) at = is noneistent AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

5 969 AP Calculus BC: Section I 9. A point moves on the -ais in such a way that its velocity at time t ( t > ) is given by At what value of t does v attain its maimum? ln t v =. t There is no maimum value for v. e e e. An equation for a tangent to the graph of arcsin y = at the origin is y = y = = y = y =. At =, which of the following is true of the function f defined by f ( ) = + e? f is increasing. f is decreasing. f is discontinuous. f has a relative minimum. f has a relative maimum.. If f ( ) = dt, which of the following is FALSE? t + f () = f is continuous at for all. f () > f () = f ( ) > AP Calculus Multiple-Choice Question Collection Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

6 . If the graph of y f( ) + = contains the point ( ) 969 AP Calculus BC: Section I dy,, = and f( ) > for all, then f ( ) = d ye e + + e e + e + e y. If sin = e, < <, what is dy in terms of? d tan cot cot tan csc 5. A region in the plane is bounded by the graph of = m, m >. The area of this region y =, the -ais, the line = m, and the line is independent of m. increases as m increases. decreases as m increases. decreases as m increases when m < ; increases as m increases when m >. increases as m increases when m < ; decreases as m increases when m > d is none of the above AP Calculus Multiple-Choice Question Collection 5 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

7 969 AP Calculus BC: Section I dy 7. If tan, d = then y = tan + C sec + C ln sec + C ln cos + C sec tan + C 8. What is e lim tan? The limit does not eist. 9. ( ) d =. ( ) is the Taylor series about zero for which of the following functions? n! n= n n sin cos e e ln( + ). If f ( ) = f( ) and f () =, then f ( ) = e + e e e e. For what values of does the series n + converge? No values of < > All values of. What is the average (mean) value of t t over the interval t? AP Calculus Multiple-Choice Question Collection 6 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

8 969 AP Calculus BC: Section I. Which of the following is an equation of a curve that intersects at right angles every curve of the family y = + k (where k takes all real values)? y = y = y = y = y = ln 5. At t = a particle starts at rest and moves along a line in such a way that at time t its acceleration is t feet per second per second. Through how many feet does the particle move during the first seconds? The approimate value of y = + sin at =., obtained from the tangent to the graph at =, is Of the following choices of δ, which is the largest that could be used successfully with an arbitrary ε in an epsilon-delta proof of ( ) lim 5? = δ= ε δ=ε ε δ = ε δ = ε δ= 5 8. If ( ) ( f ) ( ) = +, then f () = ln(8 e ) ln(8 e) ln() 8 9. If y = tan u, u = v, and v= ln, what is the value of dy v d at = e? e e sec e AP Calculus Multiple-Choice Question Collection 7 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

9 n n. If n is a non-negative integer, then = ( ) 969 AP Calculus BC: Section I d d for no n n even, only n odd, only nonzero n, only all n. If f = ( ) 8 for, f( ) = elsewhere, then f ( d ) is a number between and 8 8 and 6 6 and and and. If cos d = f ( ) sin d, then f ( ) = sin + cos+ C sin + C cos sin+ C cos sin+ C ( ) cos sin+ C. Which of the following integrals gives the length of the graph of y = tan between = a and = b, where < a< b<? b a b a b a b a b a + tan + tan d + sec + tan + sec d d d d AP Calculus Multiple-Choice Question Collection 8 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

10 . If f ( ) f ( ) f( ) =, f () =, and f () =, then f () = 969 AP Calculus BC: Section I e e + C) e e 5. The complete interval of convergence of the series k= ( + ) k is k < < < < AP Calculus Multiple-Choice Question Collection 9 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

11 969 Answer Key 969 AB 969 BC. B. C. B. D 5. E 6. B 7. D 8. B 9. C. E. B. A. C. E 5. B 6. B 7. B 8. E 9. C. A. B. E. C. C 5. A 6. C 7. C 8. C 9. A. E. C. B. A. D 5. A 6. B 7. D 8. C 9. D. E. D. D. D. C 5. D. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E AP Calculus Multiple-Choice Question Collection 5 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

12 969 Calculus BC Solutions. C For horizontal asymptotes consider the limit as ± : t y= is an asymptote For vertical asymptotes consider the limit as y ± : t = is an asymptote. E + y = ( + ) tan, y = + tan + ( )() ( )( ) + + y = + = ( + + ) ( + ) y changes sign at = only. The point of inflection is (, ). B y =, y =. By the Mean Value Theorem we have The point is (,). c c = =.. D 8 d d = + = ( ) = E Using implicit differentiation, 6+ y + y+ y y =. Therefore When =, + y+ y = = y + y+ = ( y+ ) y = Therefore + y = and so dy is not defined at d =. y 6 y =. + y f 8 at B This is the derivative of ( ) = = f = 6 = 7 7. D k k With f( ) = +, we need = f ( ) = and so k =. Since f ( ) < for k =, f does have a relative maimum at =. 8. C ( ) h( ) = f( ) f( ) g ( ) g( ) = f( ) g ( ) g ( ) f( ) = f( ) g ( ) AP Calculus Multiple-Choice Question Collection 66 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

13 9. D ( cos ) ( cos ) ( cos ) 969 Calculus BC Solutions A= + θ dθ= d d + θ θ= + θ θ. A d d d ( ) + + = = = tan = = B Let L be the distance from, and,. L = ( ) + dl L = + d ( ) + dl + = = = = d L L L L ( ) + + ( ) dl dl < for all < and > for all >, so the minimum distance occurs at =. d d The nearest point is the origin. t F = e dt F = e.. E By the Fundamental Theorem of Calculus, if ( ) then ( ). C k cos d = cos d; sin k sin sin sin k k = sin k+ = sin k; sin k = k = 6 dy dy d y = + u = = = = du d du. D and, ( ) 5. B The graphs do not need to intersect (eg. f ( ) = e and g( ) = e ). The graphs could intersect (e.g. f ( ) = and g ( ) = ). However, if they do intersect, they will intersect no more than once because f ( ) grows faster than g. ( ) AP Calculus Multiple-Choice Question Collection 67 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

14 969 Calculus BC Solutions 6. B y > y is increasing; y < the graph is concave down. Only B meets these conditions. 7. B y 5, y 6 ( ) = = =. The only sign change in y is at =. The only point of inflection is (,6). 8. E There is no derivative at the verte which is located at =. 9. C. A. B dv lnt dv = > for < t < e and < for t > e, thus v has its maimum at t = e. dt t dt y() = and y () = = =. The tangent line is = = y = y =. f ( ) = e, f () =, so f is decreasing f = dt, f = dt = dt < t + t + t + f < so E is false.. E ( ) ( ) ( ). D. C dy e = ydy= e d y = e + C d y = + C C = ; cos y = ln sin, y = = cot sin y = e + y = e + m m = ln = ln ln = ln so the area is independent of m. m m 5. A d ( m) ( m) AP Calculus Multiple-Choice Question Collection 68 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

15 + = = = = 6. C d d ( ) d ( ) 969 Calculus BC Solutions Alternatively, the graph of the region is a right triangle with vertices at (,), (,), and (,). The area is. 7. C sin tan d = d = ln cos + C = ln sec + C cos 8. D Use L Hôpital s Rule: e e lim = lim tan = sec 9. C Make the subsitution = sin θ d = cos θdθ. ( ) 6 6 θ d d 6 sec d tan cos = θ= θ θ= θ = = 8cos θ. D Substitute for in. C ( ) n n n = e to get = e n= n! n= n! dy = y y = ce and = ce c = e; y = e e = e d. B. A n + = where p p =. This is a p-series and is n= n convergent if p> > <. t t dt = t t = + = 8. D y =, so the desired curve satisfies y = y = + C a t = t, v( t) = 8 t + C and v() = C =. The particle is always moving to the 5. A () right, so distance = 8tdt= t =. AP Calculus Multiple-Choice Question Collection 69 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

16 969 Calculus BC Solutions 6. B cos y = + sin, y() =, y () = =. The linear approimation to y is + sin L ( ) = +. L (.) = + (.) =. 7. D This item uses the formal definition of a limit and is no longer part of the AP Course Description. Need to have ( ) ( 5) <ε whenever < <δ. ( ) ( 5) = 6 = <ε if <ε /. Thus we can use any δ<ε /. Of the five choices, the largest satisfying this condition is δ=ε /. 8. A Note f () =. Take the natural logarithm of each side of the equation and then differentiate. f ( ) ( ) ( ) ln f( ) = ( )ln + ; = ( ) ln + f( ) + f () f() = ( ) ln() f () = ln = ln e+ ln = ln 8e ( ) ( ) dy dy du dv 9. D e v, u, y ; ( sec u) ()( )( e ) = = = = = = d du dv d + = = v e. E One solution technique is to evaluate each integral and note that the value is n + for each. n n n. Another technique is to use the substitution ; ( ) ( ) Integrals do not depend on the variable that is used and so f d d d. D ( ) ( ) u = d= u du = u du = 8 + = 8 + = 7 n n u du is the same as d. AP Calculus Multiple-Choice Question Collection 7 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

17 . B Use the technique of antiderivatives by parts to evaluate u = dv = cos d du = d v= sin f ( ) sind= cosd= sin sind+ C f( ) = sin+ C b b b sec sec a a a dy. E ( ) L= + d= + d= + d d 969 Calculus BC Solutions cos d. E y y y =, y () =, y() = ; the characteristic equation is r r =. The solutions are r =, r = so the general solution to the differential equation is with y = ce + c e y = ce + c e. Using the initial conditions we have the system: = c + c and = c + c c =, c =. The solution is f ( ) = e f() = e. 5. E The ratio test shows that the series is convergent for any value of that makes + <. The solutions to + = are the endpoints of the interval of convergence. Test = and = in the series. The resulting series are The interval is. k= ( ) k and k which are both convergent. k k= AP Calculus Multiple-Choice Question Collection 7 Copyright 5 by College Board. All rights reserved. Available at apcentral.collegeboard.com.

C) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist.

C) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist. . The asymptotes of the graph of the parametric equations = t, y = t t + are A) =, y = B) = only C) =, y = D) = only E) =, y =. What are the coordinates of the inflection point on the graph of y = ( +

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Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.

Answer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26. Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.

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