The Mean Value Theorem A student at the University of Connecticut - PDF document

The Mean Value Theorem A student at the University of Connecticut happens to be travelling to Boston. He enters the Massachussetts Turnpike at the Sturbridge Village entrance at 9:15 in the morning. Since he uses Fast Lane, he never actually picksup a ticket, but sensorsrecord that hegoesthrough the toll lane at Newton at 9:53, just 38 minutes later. He soon receives a traffic summons in the mail indicating that he vi- olated the posted speed limit and decides to appeal, since there is no indication he was caught by radar. Appearing in court, theprosecutor arguesthat hetravelled 44 . 8 miles in 38 minutesand thereforetravelled at an averagespeed of 70 . 74 miles per hour, more than 5 miles above the highest posted speed limit of 65 miles per hour. The student argues that, although he may have averaged more than 65 miles per hour, there is no evidence that he actually ever travelled at an instantaneous speed of more than 65 miles per hour. The judge considers the arguments and quickly rejects the student’s plea, noting that the Mean Value Theorem implies that since his aver- agespeed was 70 . 74 miles per hour, therehad to beat least oneinstant during which his instantaneous speed was 70 . 74 miles per hour. Rolle’s Theorem Theorem 1 (Rolle’s Theorem) . Suppose a function f is continuous on the closed interval [ a, b ] , differentiable on the open interval ( a, b ) and f ( a ) = f ( b ) . Then there is a number c ∈ ( a, b ) such that f ′ ( c ) = 0 . Geometrically, this says that if there are two points on a smooth curve takes at the same height, there must be a point in between where the tangent is horizontal. Rolle’s Theorem is a special case of the Mean Value Theorem . Proof: Suppose f satisfies the hypotheses of Rolle’s Theorem. By the Extreme Value Theorem for Continuous Function , there must be some point in [ a, b ] at which f attains a minimum and some point at which f attains a maximum. One possibility is that f is constant on the entire interval, in which case f ′ is identically 0 on ( a, b ) and the conclusion is clearly true. So let’s consider the other possibility, that f is not constant. Then either the minimum or the maximum must occur at some point c ∈ ( a, b ), that is, at some point other than the endpoints. 1

2 Wewill show that f ′ ( c ) = 0 if f hasa maximum at c . Similar reasoning would show f ′ ( c ) = 0 if f had a minimum at c , showing the conclusion is true and completing the proof. f ( x ) − f ( c ) We know f ′ ( c ) = lim x → c . x − c Since this ordinary limit exists, it follows that both the left hand limit and the right hand limit exist and are both equal to f ′ ( c ). We’ll con- sider them separately. The Left Hand Limit: f ′ ( c ) = lim x → c − f ( x ) − f ( c ) . x − c Since f hasa maximum at c , if x < c , then f ( x ) ≤ f ( c ), so f ( x ) − f ( c ) ≤ 0. But, if x < c , it’s also true that x − c < 0 and it follows that f ( x ) − f ( c ) We see lim x → c − f ( x ) − f ( c ) ≥ 0. is the limit of non- x − c x − c negative numbers and therefore can’t be negative. It follows that f ′ ( c ) ≥ 0. The line of reasoning for the right hand limit is similar. The Right Hand Limit: f ′ ( c ) = lim x → c + f ( x ) − f ( c ) . x − c Since f hasa maximum at c , if x > c , then f ( x ) ≤ f ( c ), so f ( x ) − f ( c ) ≤ 0. But, if x > c , it’s also true that x − c > 0 and it follows that f ( x ) − f ( c ) ≤ 0. We see lim x → c + f ( x ) − f ( c ) is the limit of numbers x − c x − c less than or equal to 0 and therefore can’t be positive. It follows that f ′ ( c ) ≤ 0. Since f ′ ( c ) ≥ 0 and f ′ ( c ) ≤ 0, it follows that f ′ ( c ) = 0 QED Consequences of Rolle’s Theorem Besides being a special case of the Mean Value Theorem and being a step in the path to proving the Mean Value Theorem, Rolle’s Theorem has some interesting applications of its own. Corollary 2. A polynomial equation of degree n has at most n solu- tions. This is equivalent to the statement that a polynomial of degree n has at most n zeros. Sincea polynomial may bedifferentiated asmany timesasnecessary, with each derivativebeing a polynomial of lower degree, oneimmediate consequence of Rolle’s Theorem is that the derivative of a polynomial has at least one zero between each pair of distinct zeros of the original polynomial.

3 The derivative of a linear polynomial is a non-zero constant, having no zeros, so a linear polynomial can’t have more than 1 zero. Thederivativeof a quadratic polynomial islinear, having no morethan 1 zero, so a quadratic can’t have more than 2 zeros. The derivative of a cubic polynomial is quadratic, having no morethan 2 zeros, so the cubic can’t have more than 3 zeros. This clearly goes on forever. The argument can be made rigorous through the use of Mathematical Induction . The Mean Value Theorem Theorem 3 (The Mean Value Theorem) . Suppose a function f is continuous on the closed interval [ a, b ] and differentiable on the open interval ( a, b ) . Then there is a number c ∈ ( a, b ) such that f ( b ) − f ( a ) = f ′ ( c )( b − a ) or, equivalently, f ′ ( c ) = f ( b ) − f ( a ) . b − a Geometrically, the Mean Value Theorem says that if there is a smooth curve connecting two points, there must be some point in between at which the tangent line is parallel to the line connecting those two points. Analytically, the Mean Value Theorem says the rate of change of a differentiable function must, at some point, take on its average, or mean value. The proof of the Mean Value Theorem depends on the fact that the particular point at which the tangent line is parallel to the line connecting the endpoints also happens to be the point at which the curve is furthers away from that line. The proof essentially consists of applying Rolle’s Theorem to the function measuring the distance between the line and the curve. Since the line goes between the points ( a, f ( a )) and ( b, f ( b )), its slope will be f ( b ) − f ( a ) and its equation may be written, in point- b − a slope form, y − f ( a ) = f ( b ) − f ( a ) · ( x − a ). b − a Solving for y , we may write the equation in the form y = f ( a ) + f ( b ) − f ( a ) · ( x − a ). b − a Sincethesecond coordinateof a point on thecurvewith first coordinate x is f ( x ), thevertical distancebetween thelineand thecurvewill equal

4 � f ( a ) + f ( b ) − f ( a ) � f ( x ) − · ( x − a ) = b − a f ( x ) − f ( a ) − f ( b ) − f ( a ) · ( x − a ). b − a We are now prepared to prove the Mean Value Theorem . Proof: Let φ ( x ) = f ( x ) − f ( a ) − f ( b ) − f ( a ) · ( x − a ). b − a It is easy to see that φ satisfies the hypotheses of Rolle’s Theorem on the interval [ a, b ]. Certainly, the fact that φ is both continuous and differentiable on [ a, b ] follows immediately from the fact that f is. In addition, φ ( a ) = f ( a ) − f ( a ) − f ( b ) − f ( a ) · ( a − a ) = 0 b − a and φ ( b ) = f ( b ) − f ( a ) − f ( b ) − f ( a ) · ( b − a ) = f ( b ) − f ( a ) − [ f ( b ) − f ( a )] = 0. b − a Thus, there must be some c ∈ ( a, b ) such that φ ′ ( c ) = 0. We first obtain φ ′ ( c ) as follows. φ ′ ( x ) = f ′ ( x ) − f ( b ) − f ( a ) b − a φ ′ ( c ) = f ′ ( c ) − f ( b ) − f ( a ) b − a Since φ ′ ( c ) = 0, it follows that f ′ ( c ) − f ( b ) − f ( a ) = 0. b − a f ′ ( c ) = f ( b ) − f ( a ) b − a f ′ ( c )( b − a ) = f ( b ) − f ( a ) QED Consequences of the Mean Value Theorem Perhapsthemost important consequenceof the Mean Value Theorem is that it gives precise meaning to the most important single concept in elementary Calculus, The Derivative Measures Rate of Change. Theorem 4. a. If the derivative of a function is positive at all points on an interval, then the function is increasing on that interval. b. If the derivative of a function is negative at all points on an interval, then the function is decreasing on that interval.