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**Unformatted text preview: **L23 – Mean Value Theorem Rolle’s Theorem – Let f be a function that satisﬁes the following properties: 1. 2. 3. The there is a c in ( a, b ) such that 1 Graphically 2 Show that f (x) = x3 + 3x − 2 has exactly one real root. 3 Find the value of c implied by Rolle’s Theorem for f (x) = (2x − x2 ) 3 on [ 0, 2 ] . 2 4 Find the value of c implied by Rolle’s Theorem for g (x) = cos(πx) on [ 0, 2 ] . 5 Mean Value Theorem – Let f be a function that satisﬁes 1. 2. Then there is a c in ( a, b ) such that 6 Equation of the secant line to f passing through ( a, f (a) ) and ( b, f (b) ) has slope: Equation – Consider h(x) = f (x) − y h(x) is 1. 2. 7 h(a) = h(b) = By Rolle’s Theorem – 8 Find the value of c implied by the Mean Value Theorem for f (x) = x3 − x2 − 2x on [ −1, 1 ] . 9 The position of an object dropped from 650 ft is s(t) = 650 − 16t2 , where t is in seconds. Find the average velocity on [ 0, 5 ] seconds. Use the Mean Value Theorem to verify that at some time in the ﬁrst ﬁve seconds, average velocity = instantaneous velocity. 10 If f (x) = on [ 0, 3 ] . 13 x + 2x ﬁnd the value of c guaranteed by the Mean Value Theorem 3 11 Use the Mean Value Theorem to show that | cos(x) − cos(y ) | ≤ | x − y | . 12 Suppose f (0) = 4 and f (x) ≥ −2 for all values of x, how small can f (3) be? 13 Theorem – If f (x) = 0 for all x in an interval ( a, b ), then Corollary – If f (x) = g (x) for all x in an interval ( a, b ), then f − g is i.e. f (x) = 14 ...

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