Lesson 4-2 Mean Value Theorem and Rolle’s Theorem
Quiz • Homework Problem: Related Rates 3-10Gravel is being dumped from a conveyor belt at a rate of 30 ft³/min, and forms a pile in shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10ft high? • Reading questions: • What were the names of the two theorems in section 4.2? • What pre-conditions (hypotheses) do the Theorems have in common?
Objectives • Understand Rolle’s Theorem • Understand Mean Value Theorem
Vocabulary • Existence Theorem – a theorem that guarantees that there exists a number with a certain property, but it doesn’t tell us how to find it.
Theorems Mean Value Theorem: Let f be a function that is a) continuous on the closed interval [a,b] b) differentiable on the open interval (a,b) then there is a number c in (a,b) such that f(b) – f(a) f’(c) = --------------- or equivalently: f(b) – f(a) = f’(c)(b – a) b – a Rolle’s Theorem: Let f be a function that is a) continuous on the closed interval [a,b] b) differentiable on the open interval (a,b) c) f(a) = f(b) then there is a number c in (a,b) such that f’(c) = 0
Mean Value Theorem (MVT) Let f be a function that is a) continuous on the closed interval [a,b] b) differentiable on the open interval (a,b) then there is a number c in (a,b) such that f(b) – f(a) f’(c) = --------------- (instantaneous rate of change, mtangent = average rate of change, msecant) b – a or equivalently: f(b) – f(a) = f’(c)(b – a) P(c,f(c)) y y P1 A(a,f(a)) P2 B(b,f(b)) x x c2 c1 a a c b b For a differentiable function f(x), the slope of the secant line through (a, f(a)) and (b, f(b)) equals the slope of the tangent line at some point c between a and b. In other words, the average rate of change of f(x) over [a,b] equals the instantaneous rate of change at some point c in (a,b).
Example 1 Verify that the mean value theorem (MVT) holds for f(x) = -x² + 6x – 6 on [1,3]. a) continuous on the closed interval [a,b] polynomial b) differentiable on the open interval (a,b) polynomial f’(x) = -2x + 6 f(b) – f(a) = f(3) – f(1) = 3 – (-1) = 4 f(b) – f(a) / (b – a) = 4/2 = 2 f’(x) = 2 = 6 – 2x so -4 = -2x 2 = x
Example 2 Find the number that satisfies the MVT on the given interval or state why the theorem does not apply. f(x) = x2/5 on [0,32] a) continuous on the closed interval [a,b] ok b) differentiable on the open interval (a,b) ok on open f’(x) = (2/5)x-3/5 f(b) – f(a) = f(32) – f(0) = 4 – 0 = 4 f(b) – f(a) / (b – a) = 4/32 = 0.125 f’(x) = 0.125 = (2/5)x-3/5 so x = 6.94891
Example 3 Find the number that satisfies the MVT on the given interval or state why the theorem does not apply. f(x) = x + (1/x) on [1,3] a) continuous on the closed interval [a,b] ok b) differentiable on the open interval (a,b) ok on open f’(x) = 1 – x-2 f(b) – f(a) = f(3) – f(1) = 10/3 – 2 = 4/3 f(b) – f(a) / (b – a) = (7/3)/2 = 2/3 f’(x) = 2/3 = 1 – x-2 so x-2 = 1/3 x² = 3 x=3 = 1.732
Example 4 Find the number that satisfies the MVT on the given interval or state why the theorem does not apply. f(x) = x1/2 + 2(x – 3)1/3 on [1,9] a) continuous on the closed interval [a,b] ok b) differentiable on the open interval (a,b) vertical tan f’(x) = 1/2x-1/2 + 2/3(x – 3)-2/3 f’(x) undefined at x = 3 (vertical tangent) MVT does not apply
Rolle’s Theorem y y y y c x x c c1 c2 x x c a a a a b b b b Let f be a function that is a) continuous on the closed interval [a,b] b) differentiable on the open interval (a,b) c) f(a) = f(b) then there is a number c in (a,b) such that f’(c) = 0 A. C. B. D. Case 1: f(x) = k (constant) [picture A] Case 2: f(x) > f(a) for some x in (a,b) [picture B or C] Extreme value theorem guarantees a max value somewhere in [a,b]. Since f(a) = f(b), then at some c in (a,b) this max must occur. Fermat’s Theorem says that f’(c) =0. Case 3: f(x) < f(a) for some x in (a,b) [picture C or D] Extreme value theorem guarantees a min value somewhere in [a,b]. Since f(a) = f(b), then at some c in (a,b) this min must occur. Fermat’s Theorem says that f’(c) =0.
Example 5 Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. f(x) = x² + 9 on [-3,3] • a) continuous on the closed interval [a,b] polynomial • differentiable on the open interval (a,b) polynomial • f(a) = f(b) f(-3) = 18 f(3) = 18 f’(x) = 2x f’(x) = 0 when x = 0 0 in the interval [-3,3]
Example 6 Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. f(x) = x³ - 2x² - x + 2 on [-1,2] • a) continuous on the closed interval [a,b] polynomial • differentiable on the open interval (a,b) polynomial • f(a) = f(b) f(-1) = 0 f(2) = 0 f’(x) = 3x² - 4x – 1 f’(x) = 0 when x = (7 + 2)/3 = 1.5486 f’(x) = 0 when x = -(7 - 2)/3 = -0.2153 -0.2153 and 1.5486 in the interval [-1,2]
Example 7 Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. f(x) = (x² - 1) / x on [-1,1] • a) continuous on the closed interval [a,b] no at x = 0 • differentiable on the open interval (a,b) no at x = 0 • f(a) = f(b) f(-1) = 0 f(1) = 0
Example 8 Determine whether Rolle’s Theorem’s hypotheses are satisfied &, if so, find a number c for which f’(c) = 0. f(x) = sin x on [0,π] • a) continuous on the closed interval [a,b] ok • differentiable on the open interval (a,b) ok • f(a) = f(b) f(0) = 0 f(π) = 0 f’(x) = cos x f’(x) = 0 (or undefined) when x = π/2 π/2 in the interval [0,π]
Summary & Homework • Summary: • Mean Value and Rolle’s theorems are existance theorems • Each has some preconditions that must be met to be used • Homework: • pg 295-296: 2, 7, 11, 12, 24
