Theorems and Hypotheses (MVT, IVT, EVT) Page 1 of 11. Theorems
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1 Page 1 of 11 Theorems THEOREM: Intermediate Value Theorem for Continuous Functions (IVT) A function y f( x) a, b takes on every value between f ( a ) and f ( b ). In other words, if y 0 is between f ( a ) and f ( b ), then y0 = f( c) for some c in [ a, b ]. Example: = that is continuous on a closed interval [ ] f ( x ) is continuous. If f (0) = 1 and f () = 4, is there a c on the interval ( 0, ) such that f( c ) = 2.5? Yes, we are guaranteed that f( c ) = 2.5 for some c in the given interval. The function above is continuous over the interval 0 x. Therefore, every y-value between 1 and 4 is guaranteed somewhere on the interval, at least one time. The graph above is just one example of what this function could look like.
2 Page 2 of 11 THEOREM: Mean Value Theorem for Derivatives (MVT) If y f( x) a, b and differentiable at every point of its interior ( a, b ), then there is at least one point c in ( a, b ) at which = is continuous at every point of the closed interval [ ] f ( b) f( a) f ( c) = b a (The instantaneous rate of change equals the average rate of change.) Example: For the given function, ( ) 2 f( x) = x 1, the slope of the segment connecting (0, 1) and (, 4) is 1. If the function is continuous on the closed interval and differentiable on the open interval, there is at least one place where the slope of the tangent line will be 1. This occurs when x =. 2
3 Page of 11 THEOREM: Mean Value Theorem for Integrals If f is continuous on the closed interval [ a, b ], then there exists a number c in [ a, b ] such that 1 b f () c = f() x dx b a a This is also called the average value of the function over the named interval. Example: What is the average value of f ( x) 5 2 = x on [ ] 0, 2? Y (5, 0) (2, 1) X 1 = x dx The average value of the function over the interval ( 5 ). 0 The average value of the function over [ 0, 2 ] is 11. Check your answer for reasonableness. Where does this average value occur on [ 0, 2 ]? x = 4 x =± Choose the answer in the given interval so that 4 x =.
4 Page 4 of 11 THEOREM: Rolle s Theorem If y f( x) a, b and differentiable at every point of its interior ( a, b ), and if f ( a) = f( b), then there is at least one point c in ( a, b ) at which = is continuous at every point of the closed interval [ ] f ( c) = 0 Example: 4 2 Given f( x) x 2x 1 What is that x value? =. Over the interval ( 1, 1), is there an x such that f ( x) = 0? Is the function continuous over the closed interval [ a, b ] and differentiable at every point of its interior ( a, b )? Yes. Does f ( b) = f( a)? Yes. After we verify the hypotheses we can apply the theorem. We take the first derivative and check for critical numbers. Check your answers against the open interval. = 4x 4x = xx ( 1) = 0 x = 1, 0, 1 f ( x) 4x 4x Yes, there is such an x. The answer is x = 0, because this is the only answer within the open interval.
5 Page 5 of 11 THEOREM: Extreme Value Theorem If y f( x) a, b, then f ( x ) has both a maximum value and a minimum value on the interval. We are guaranteed the values exist. In some cases, we have enough information to actually find the values. = is continuous at every point of the closed interval [ ] Example: Over the interval [ 1, 2], find the maximum and minimum values of f x ( ) x 1 =. Evaluate the endpoints and any critical numbers in the interval. These are the candidates for absolute maximum and minimum values. x f ( x ) Max or Min 1 2 Minimum Maximum The absolute minimum value is 2. The absolute maximum value is 7.
6 Page 6 of Let g be a continuous function of the closed interval 1 x. If g( 1) = 10 and g () = 6, which of the following are guaranteed? (A) g (0) = 0 (B) g ( c) = 0 for some c in the interval 1< x < (C) g ( c) = 4 for some c in the interval 1< x < (D) gc ( ) = 4 for some c in the interval 1< x < (E) 10 gc ( ) 6 for all x between 1 and 2. Let g be a continuous function of the closed interval 1 x and differentiable on the open interval 1< x <. If g( 1) = 10 and g () = 6, which of the following are guaranteed? I. g ( c) = 0 for some c in the interval 1< x < II. g ( c) = 4 for some c in the interval 1< x < III. gc ( ) = 4 for some c in the interval 1< x < (A) I only (B) II only (C) III only (D) I and II only (E) II and III only. The function h is continuous on the closed interval [ 2, 5] and differentiable on the open interval ( 2, 5). If f ( 2) = and f (5) = 11, which of the following statements could be false? (A) There exists a c on the interval ( 2, 5) such that f ( c) = 0. (B) There is an absolute maximum value on the interval [ 2, 5]. (C) There exists a c on the interval [ 2, 5] interval [ 2, 5]. such that f () c f() x for all x on the (D) There exists a c on the interval ( 2, 5) such that f( c ) = 0. (E) There exists a c on the interval ( 2, 5) such that f ( c) = 2.
7 Page 7 of Let g be a function that is differentiable over the interval (2, 9). Given g () = 5, g (6) = 2, and g (8) = 5, which of the following must be true? I. g has at least one horizontal tangent line. II. g has at least 2 zeros. III. For some c in the interval (, 6), f ( c) 7 =. (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III 5. If f( x) = x + 1, then there exists a number c in the interval (0, 1) that satisfies the conclusion of the Mean Value Theorem. Which of the following could be c? (A) 1 (B) 0 (C) 1 (D) 1 (E) 2 6. Which of the following theorems may be applied to the graph below, y x b, b 0 2, 4? = + >, over the interval [ ] Y X I. Mean Value Theorem II. Intermediate Value Theorem III. Extreme Value Theorem (A) I only (B) II only (C) III only (D) II and III only (E) I, II, and III
8 Page 8 of hx ( ) is a differentiable function that contains the points (2, 5) and (5, 4). Which of the following must be true? (A) hx ( ) is increasing over the interval (2, 5). (B) hx ( ) intercepts the x axis at 11. = for some c in the interval ( 5, 4) (C) h( c) 0 (D) h ( c) = for some c in (2, 5) (E) h ( c) = for all x in (2, 5). 8. The Mean Value Theorem may be applied to which of the following functions over the interval named? (A) f ( x) tanx (B) f ( x) (C) f( x) = over [ 0, π ] = x over [ 1, 1] 1 x sin x x x = over [ 1, 2 ] (D) f( x) = over [ π, π ] (E) f ( x) = [ ] over [ 1, ] 9. The average value of the function h on the interval [ c, d ] is 5. c d hx ( ) dx= (A) hd ( ) hc ( ) 5 (B) 5 hd ( ) hc ( ) (C) d c 5 (D) 5d 5c (E) 5c 5d 10GC. The velocity of a particle, in feet per second, moving along the x-axis is given by t vt () 2 te 0, 2? = +. What is the average velocity of the particle over the time interval [ ] (A) 1.00 (B) (C) 7.45 (D) (E)
9 Page 9 of 11 Free Response 1. (No calculator) The table below shows data points for a twice-differentiable function g. x gx ( ) (a) Find the left Riemann sum approximation for length 2. 5 gx ( ) dx using sub-intervals of 1 (b) Estimate the average value of the function ( ) the approximation you found in part (a). gx over the interval [ 1, 5] using (c) Find the average rate of change of the function ( ) gx over the interval [ 1, 5]. (d) Over the interval ( 1, 1), explain why there must be a r such that gr () = 0. (e) Over the interval ( 1, 1), explain why there must be an c such that g ( c) = 5.
10 Page 10 of 11 Free Response 2. (No calculator) t (hours) Pt ()(people) The number of people in line at the Department of Motor Vehicles is modeled by a twicedifferentiable function Pt ( ) for 0 t 12. The time t = 0 corresponds to 7:00 A.M. (a) Use the data above to estimate the rate at which the number of people in line was changing at 10:0 A.M. ( t =.5 ). Indicate units of measure and show the computations that lead to your answer. Explain the meaning of this number in the context of the problem. (b) Using a right Riemann sum with intervals [ 0, 2 ], [ 2, 7 ], and [ 7, 12 ], what is the average number of people waiting in line over the interval 0 t 12? (c) Over the time interval 0 t 12, what is the least number of times that P () t = 0? Justify your answer.
11 Page 11 of 11 Free Response. (No calculator) The functions g and h are differentiable for all real numbers. The table below gives values of these two functions and their first derivatives at certain data points. The function f is given by f( x) = 2 g( h( x)) 17. x gx ( ) hx ( ) g ( x) h ( x) (a) Over the interval ( ) your answer. 0,, must there be a value c such that f() c = 2? Justify (b) Over the interval ( ) your answer. 0,, must there be a value d such that f ( d) = 2? Justify (c) Using the data above, find f (). Show the work that leads to your answer.
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