Mean Value Theorem and Inequalities

Lecture 14 18.01 Fall 2006 Lecture 14: Mean Value Theorem and Inequalities Mean-Value Theorem The Mean-Value Theorem (MVT) is the underpinning of calculus. It says: If f is differentiable on a < x < b, and continuous on a ≤ x ≤ b, then f (b) − f (a) = f � (c) (for some c, a < c < b) b−a Here, f (b) − f (a) is the slope of a secant line, while f � (c) is the slope of a tangent line. b−a secant line slope f’(c) b a Figure 1: c Illustration of the Mean Value Theorem. Geometric Proof: Take (dotted) lines parallel to the secant line, as in Fig. 1 and shift them up from below the graph until one of them first touches the graph. Alternatively, one may have to start with a dotted line above the graph and move it down until it touches. If the function isn’t differentiable, this approach goes wrong. For instance, it breaks down for the function f (x) = |x|. The dotted line always touches the graph first at x = 0, no matter what its slope is, and f � (0) is undefined (see Fig. 2). 1 Lecture 14 18.01 Fall 2006 Figure 2: Graph of y = |x|, with secant line. (MVT goes wrong.) Interpretation of the Mean Value Theorem You travel from Boston to Chicago (which we’ll assume is a 1,000 mile trip) in exactly 3 hours. At 1000 some time in between the two cities, you must have been going at exactly mph. 3 f (t) = position, measured as the distance from Boston. f (3) = 1000, f (0) = 0, a = 0, and b = 3. 1000 f (b) − f (a) = = f � (c) 3 3 where f � (c) is your speed at some time, c. Versions of the Mean Value Theorem There is a second way of writing the MVT: f (b) − f (a) = f � (c)(b − a) f (b) = f (a) + f � (c)(b − a) (for some c, a < c < b) There is also a third way of writing the MVT: change the name of b to x. f (x) = f (a) + f � (c)(x − a) for some c, a < c < x The theorem does not say what c is. It depends on f , a, and x. This version of the MVT should be compared with linear approximation (see Fig. 3). f (x) ≈ f (a) + f � (a)(x − a) 2 x near a Lecture 14 18.01 Fall 2006 The tangent line in the linear approximation has a definite slope f � (a). by contrast formula is an exact formula. It conceals its lack of specificity in the slope f � (c), which could be the slope of f at any point between a and x. (x,f(x)) (a,f(a)) error y=f(a) + f’(a)(x-a) Figure 3: MVT vs. Linear Approximation. Uses of the Mean Value Theorem. Key conclusions: (The conclusions from the MVT are theoretical) 1. If f � (x) > 0, then f is increasing. 2. If f � (x) < 0, then f is decreasing. 3. If f � (x) = 0 all x, then f is constant. Definition of increasing/decreasing: Increasing means a < b ⇒ f (a) < f (b). Decreasing means a < b =⇒ f (a) < f (b). Proofs: Proof of 1: a < f (b) = b f (a) + f � (c)(b − a) Because f � (c) and (b − a) are both positive, f (b) = f (a) + f � (c)(b − a) > f (a) (The proof of 2 is omitted because it is similar to the proof of 1) Proof of 3: f (b) = f (a) + f � (c)(b − a) = f (a) + 0(b − a) = f (a) Conclusions 1,2, and 3 seem obvious, but let me persuade you that they are not. Think back to the definition of the derivative. It involves infinitesimals. It’s not a sure thing that these infinitesimals have anything to do with the non-infinitesimal behavior of the function. 3 Lecture 14 18.01 Fall 2006 Inequalities The fundamental property f � > 0 =⇒ f is increasing can be used to deduce many other inequali­ ties. Example. ex 1. ex > 0 2. ex > 1 for x > 0 3. ex > 1 + x Proofs. We will take property 1 (ex > 0) for granted. Proofs of the other two properties follow: Proof of 2: Define f1 (x) = ex − 1. Then, f1 (0) = e0 − 1 = 0, and f1� (x) = ex > 0. (This last assertion is from step 1). Hence, f1 (x) is increasing, so f (x) > f (0) for x > 0. That is: ex > 1 for x > 0 . Proof of 3: Let f2 (x) = ex − (1 + x). f2� (x) = ex − 1 = f1 (x) > 0 (if x > 0). Hence, f2 (x) > 0 for x > 0. In other words, ex > 1 + x Similarly, ex > 1 + x + ex > 1 + x + x2 x2 (proved using f3 (x) = ex − (1 + x + )). One can keep on going: 2 2 x2 x3 + for x > 0. Eventually, it turns out that 2 3! ex = 1 + x + x2 x3 + + ··· 2 3! (an infinite sum) We will be discussing this when we get to Taylor series near the end of the course. 4