Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use your own judgment, based on the group of students, to determine the order and selection of questions to work in the session. Be sure to include a variety of types of questions (multiple choice, free response, calculator, and non-calculator) in the time allotted. Multiple Choice Solutions 1. B (1998 AB4) B is false because this special case of the MVT called Rolle s Theorem also requires that f ( a) f( b). A is true by MVT; C and D are true by EVT, E is true by the definition of a definite integral.. C (1988 AB) Rolle s Theorem guarantees at least one value of x between a and b such that f( x).. B (1985 BC1) f ( b) f( a) 7 7 Show f() c using MVT: x 6x ; xx ( ) when x and b a x ; however, only is eligible since there is an endpoint at x. 4. A (1998 AB6) Any value of k less than 1 will require the function to assume the value of 1 at least twice because of the Intermediate Value Theorem on the intervals [, 1] and [1, ], so k is the only option. 5. D (197 BC18 appropriate for AB) D could be false, consider gx ( ) 1 xon [, 1]. A is true by the Extreme Value Theorem. B is true because g is a function. C is true by the Intermediate Value Theorem. E is true because g is continuous. 6. D (199 AB18) f 1 c f sin sin f() c cos ; f() c ; 1 c cos ; c Copyright 14 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
7. D (AP-like) The Mean Value Theorem requires differentiability so only IVT and EVT apply. 8. B (1969 BC appropriate for AB) 1 y x, so y. By the Mean Value Theorem, x 1, so c 1. The point is (1, 1). x 4 9. E (1998 AB91) I and III are true by IVT; II is true by MVT. 1. E (8 AB89/BC89) Since there is no c for which f() c, Rolle s Theorem is violated and f ( k) does not exist on (, ). 11. B ( AB8) B could be false since this is a special case of MVT (Rolle s Theorem) which also requires that f ( a) f( b). A and C are true by IVT; D is true by MVT; E is true by EVT. 1. D (1997 BC81 appropriate for AB) f assumes every value between 1 and on the interval (, 6), so f( c) 1 at least once. Copyright 14 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
1. 1999 AB/BCb Yes; Since, the Mean Value Theorem guarantees that there is a t, < t < 4, such that answer MVT or equivalent 14. 9B ABbc f(6) f( a) when f ( a) f(6). There 6 a are two values of a for which this is true. (c) Yes, a. The function f is differentiable on the interval x 6 and continous on f(6) f() 1 1 x 6. Also,. 6 6 By the Mean Value Theorem, there is a 1 value c, x 6, such that f() c. expression for average rate of change answer with reason answers yes and identifies a justification 15. 8B AB5/BC5d (d) No, the MVT does not guarantee the existence of a value c with the stated properties because is not differentiable for at least one point in average value of answer No with reason Copyright 14 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
16. 7 AB ab (a) h f g f h f g f Sinceh 5 h1 1 1 6 696 6 4 616 7 and h is continuous, by the Intermediate Value Theorem, there exists a value r, 1r, such that hr 5. h1 and h conclusion, using IVT h h 1 7 5 1 1 Since h is continuous and differentiable, by the Mean Value Theorem, there exists a value c, 1 c, such that h c. 5 h h 1 1 conclusion, using MVT 17. 7B AB 6abd (a) The Mean Value Theorem guarantees that there is a value c, with so that conclusion, using MVT uses MVT with (d) Since f is twice-differentiable, is differentiable everywhere, so the Mean Value Theorem applied to on [, 5] guarantees there is a value k, with such that h() and h(5) conclusion, using IVT Since the Intermediate Value Theorem guarantees that there is a value r, with such that Copyright 14 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org
18. AB 6c (c) By the Mean Value Theorem there is a c with c.5 such that f.5 f f c 6 r.5.5 reference to MVT for f (or differentiability of f ) value of r for interval x.5 Copyright 14 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org