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Chapter 4 | Applications of Derivatives
4.4 EXERCISES 148. Why do you need continuity to apply the Mean Value Theorem? Construct a counterexample. 149. Why do you need differentiability to apply the Mean Value Theorem? Find a counterexample. 150. When are Rolle’s theorem and the Mean Value Theorem equivalent? 151. If you have a function with a discontinuity, is it still possible to have f ′( c )( b − a ) = f ( b )− f ( a )? Drawsuch an example or prove why not. For the following exercises, determine over what intervals (if any) the Mean Value Theorem applies. Justify your answer. 152. y = sin( πx )
163. f ( x ) =cos(2 πx ) 164. f ( x ) =1+ x + x 2 165. f ( x ) = ( x −1) 10 166. f ( x ) = ( x −1) 9
For the following exercises, show there is no c such that f (1)− f (−1) = f ′( c )(2). Explain why the Mean Value Theorem does not apply over the interval [−1, 1]. 167. f ( x ) = | x − 1 2 | 168. f ( x ) = 1 x 2 169. f ( x ) = | x | 170. f ( x ) =⌊ x ⌋ ( Hint : This is called the floor function and it is defined so that f ( x ) is the largest integer less than or equal to x .) For the following exercises, determine whether the Mean Value Theorem applies for the functions over the given interval [ a , b ]. Justify your answer. 171. y = e x over [0, 1]
153. y = 1 x 3
154. y = 4− x 2 155. y = x 2 −4 156. y = ln(3 x −5)
For the following exercises, graph the functions on a calculator and draw the secant line that connects the endpoints. Estimate the number of points c such that f ′( c )( b − a ) = f ( b )− f ( a ). 157. [T] y =3 x 3 +2 x +1 over [−1, 1]
172. y = ln(2 x +3) over ⎡ ⎣ − 3 2 , 0 ⎤ ⎦ 173. f ( x ) = tan(2 πx ) over [0, 2] 174. y = 9− x 2 over [−3, 3] 175. y = 1 | x +1 | over [0, 3] 176. y = x 3 +2 x +1 over [0, 6] 177. y = x 2 +3 x +2 x over [−1, 1] 178. y = x sin( πx )+1 over [0, 1] 179. y = ln( x +1) over [0, e −1] 180. y = x sin( πx ) over [0, 2]
158. [T] y = tan ⎛ ⎝ π 4 ⎤ ⎦ 159. [T] y = x 2 cos( πx ) over [−2, 2] x ⎞ ⎠ over ⎡ ⎣ − 3 2 , 3 2
[T]
160.
y = x 6 − 3 4
x 5 − 9 8
x 3 + 3 32
x 4 + 15 16
x 2 + 3 16
x + 1 32
over
[−1, 1]
For the following exercises, use the Mean Value Theorem and find all points 0< c <2 such that f (2)− f (0) = f ′( c )(2−0). 161. f ( x ) = x 3 162. f ( x ) = sin( πx )
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