Relationship Between Integration and Differentiation

Transcription

1 Relationship Between Integration and Differentiation Fundamental Theorem of Calculus Philippe B. Laval KSU Today Philippe B. Laval (KSU) FTC Today 1 / 16

2 Introduction In the previous sections we defined the Riemann integral of a function by taking two slightly different approaches. We found that the two approaches resulted in the same integral. We then looked at properties of the Riemann integral. However, though in theory we can compute Riemann integrals, in practice we do not have an easy way of doing it. This section will resolve this problem by establishing a relationship between integration and differentiation. Philippe B. Laval (KSU) FTC Today 2 / 16

3 Antiderivatives Definition Let f be a function defined on any interval I. A function F is called an antiderivative of f or a primitive of f if F (x) = f (x) for every x I. Antiderivatives are not unique. In fact, we prove that if two functions are antiderivatives of the same function, they must differ by a constant. Theorem Let f : [a, b] R. Let F and G be antiderivatives of the same function f. Then, F G = C on [a, b] where C is a constant. Philippe B. Laval (KSU) FTC Today 3 / 16

4 The Fundamental Theorem of Calculus: part 1 We are now ready to state the Fundamental Theorem of Calculus. There are two parts which we will state as different theorems. Theorem If f is integrable on [a, b] and F is any antiderivative of f then b a f = F (b) F (a). Note that the quantity F (b) F (a) is usually denoted F (x) b a. Philippe B. Laval (KSU) FTC Today 4 / 16

5 The Fundamental Theorem of Calculus: Proof Sketch of the proof: 1 Verify that F satisfies the conditions of the Mean Value Theorem. Philippe B. Laval (KSU) FTC Today 5 / 16

6 The Fundamental Theorem of Calculus: Proof Sketch of the proof: 1 Verify that F satisfies the conditions of the Mean Value Theorem. 2 Verify that for any partition P = {x i } n i=0 of [a, b] n F (b) F (a) = [F (x i ) F (x i 1 )]. i=1 Philippe B. Laval (KSU) FTC Today 5 / 16

7 The Fundamental Theorem of Calculus: Proof Sketch of the proof: 1 Verify that F satisfies the conditions of the Mean Value Theorem. 2 Verify that for any partition P = {x i } n i=0 of [a, b] n F (b) F (a) = [F (x i ) F (x i 1 )]. i=1 3 Applying the Mean Value Theorem, show F (b) F (a) = for some t i [x i 1, x i ]. n f (t i ) d i i=1 Philippe B. Laval (KSU) FTC Today 5 / 16

8 The Fundamental Theorem of Calculus: Proof Sketch of the proof: 1 Verify that F satisfies the conditions of the Mean Value Theorem. 2 Verify that for any partition P = {x i } n i=0 of [a, b] n F (b) F (a) = [F (x i ) F (x i 1 )]. i=1 3 Applying the Mean Value Theorem, show F (b) F (a) = for some t i [x i 1, x i ]. 4 Conclude by taking the limit on each side as P 0 n f (t i ) d i i=1 Philippe B. Laval (KSU) FTC Today 5 / 16

9 The Fundamental Theorem of Calculus Corollary If f is integrable on [a, b] then b a f = f (b) f (a). Let us make several remarks 1 The Fundamental Theorem of Calculus makes the process of integration much easier. Any antiderivative will work. 2 However, the Fundamental Theorem of Calculus applies only when an antiderivative is available. One may not exist or may be very be hard to find. 3 The next theorem will give conditions which guarantee the existence of an antiderivative. However, knowing an antiderivative exists does not mean we can find it. Also, as we will see in the problems, the fact that a function has a primitive on an interval [a, b] does not mean it is Riemann integrable on that interval. 4 It is also possible for a function to be Riemann integrable but have no antiderivative as illustrated in the exercises. Philippe B. Laval (KSU) FTC Today 6 / 16

10 The Fundamental Theorem of Calculus: Part 2 We are now ready to give a condition a function has to satisfy in order to have an antiderivative. This theorem is often known as the second part of the Fundamental Theorem of Calculus. Theorem Let f be integrable on [a, b]. For x [a, b], define F (x) = Then F is a continuous function on [a, b]. Furthermore, if c (a, b) and f is continuous at c then F is differentiable at c and F (c) = f (c). We will denote f or f (x) dx an antiderivative of f. This integral without limits of integration is often called an indefinite integral. x a f Philippe B. Laval (KSU) FTC Today 7 / 16

11 The Fundamental Theorem of Calculus: Proof Continuity of f on [a, b]: 1 We actually show that F is uniformly continuous. Philippe B. Laval (KSU) FTC Today 8 / 16

12 The Fundamental Theorem of Calculus: Proof Continuity of f on [a, b]: 1 We actually show that F is uniformly continuous. 2 Explain why M > 0 be such that f (x) < M on [a, b]. Philippe B. Laval (KSU) FTC Today 8 / 16

13 The Fundamental Theorem of Calculus: Proof Continuity of f on [a, b]: 1 We actually show that F is uniformly continuous. 2 Explain why M > 0 be such that f (x) < M on [a, b]. 3 Let ɛ > 0 be given, show F (y) F (x) M y x for any x, y [a, b]. Philippe B. Laval (KSU) FTC Today 8 / 16

14 The Fundamental Theorem of Calculus: Proof Continuity of f on [a, b]: 1 We actually show that F is uniformly continuous. 2 Explain why M > 0 be such that f (x) < M on [a, b]. 3 Let ɛ > 0 be given, show F (y) F (x) M y x for any x, y [a, b]. 4 Conclude that f is uniformly continuous. Philippe B. Laval (KSU) FTC Today 8 / 16

15 The Fundamental Theorem of Calculus: Proof Differentiability of F at c (a, b) and F (c) = f (c): F (x) F (c) 1 Explain why it is enough to show lim = f (c). x c x c Philippe B. Laval (KSU) FTC Today 9 / 16

16 The Fundamental Theorem of Calculus: Proof Differentiability of F at c (a, b) and F (c) = f (c): F (x) F (c) 1 Explain why it is enough to show lim = f (c). x c x c 2 Let ɛ > 0 be given. Explain why δ > 0 : x [a, b] with 0 < x c < δ we have f (x) f (c) < ɛ. Philippe B. Laval (KSU) FTC Today 9 / 16

17 The Fundamental Theorem of Calculus: Proof Differentiability of F at c (a, b) and F (c) = f (c): F (x) F (c) 1 Explain why it is enough to show lim = f (c). x c x c 2 Let ɛ > 0 be given. Explain why δ > 0 : x [a, b] with 0 < x c < δ we have f (x) f (c) < ɛ. 3 Prove that for such x s, F (x) F (c) f (c) x c < ɛ. Philippe B. Laval (KSU) FTC Today 9 / 16

18 The Fundamental Theorem of Calculus: Proof Differentiability of F at c (a, b) and F (c) = f (c): F (x) F (c) 1 Explain why it is enough to show lim = f (c). x c x c 2 Let ɛ > 0 be given. Explain why δ > 0 : x [a, b] with 0 < x c < δ we have f (x) f (c) < ɛ. 3 Prove that for such x s, F (x) F (c) f (c) x c < ɛ. 4 Conclude. Philippe B. Laval (KSU) FTC Today 9 / 16

19 The Fundamental Theorem of Calculus: Remarks Both parts of the Fundamental Theorem of Calculus assume we have an integrable function f on [a, b]. Define F (x) = x a f (t) dt. 1 F (x) is always continuous. 2 If in addition f is continuous, we know it is integrable and the Fundamental Theorem of Calculus applies. In this case, F (x) is continuous and differentiable and F (x) = f (x) that is F is an antiderivative of f. It is why we often say that integration and differentiation are reverse processes in the sense that the derivative of an antiderivative of a function is the function itself and the integral of the derivative of a function is the function itself. 3 However, when f is not continuous, things do not work as well. There are cases when F is not differentiable. It can be even worse, there are cases when F (x) f (x) in other words F is not an antiderivative of f. Philippe B. Laval (KSU) FTC Today 10 / 16

20 The Fundamental Theorem of Calculus: Examples Example [ Consider f (x) = cos x on π 2, π ]. Show that F (x) as defined in the 2 theorem is continuous, differentiable and F (c) = f (c). Example { 1 if 1 x 0 Consider f (x) =. In the exercises, you are asked 1 if 0 < x 1 to prove that f is integrable but has no antiderivative on [ 1, 1]. Here, we prove that F (x) as defined in the theorem is continuous but not differentiable. Philippe B. Laval (KSU) FTC Today 11 / 16

21 The Mean Value Theorem Theorem Let f be continuous on [a, b]. Then, there exists c [a, b] such that f (c) = 1 b f b a a Definition 1 b b a a f is called the average (mean) value of f over [a, b]. Philippe B. Laval (KSU) FTC Today 12 / 16

22 The Mean Value Theorem: Proof 1 Explain why m = inf {f (x) : x [a, b]} and M = sup {f (x) : x [a, b]} exist. Philippe B. Laval (KSU) FTC Today 13 / 16

23 The Mean Value Theorem: Proof 1 Explain why m = inf {f (x) : x [a, b]} and M = sup {f (x) : x [a, b]} exist. 2 Explain why there exists α and β between a and b such that m = f (α) and M = f (β). Philippe B. Laval (KSU) FTC Today 13 / 16

24 The Mean Value Theorem: Proof 1 Explain why m = inf {f (x) : x [a, b]} and M = sup {f (x) : x [a, b]} exist. 2 Explain why there exists α and β between a and b such that m = f (α) and M = f (β). 3 Explain why m f (x) M and m 1 b b a a f M. Philippe B. Laval (KSU) FTC Today 13 / 16

25 The Mean Value Theorem: Proof 1 Explain why m = inf {f (x) : x [a, b]} and M = sup {f (x) : x [a, b]} exist. 2 Explain why there exists α and β between a and b such that m = f (α) and M = f (β). 3 Explain why m f (x) M and m 1 b b a a f M. 4 Apply the Intermediate Value Theorem to [α, β] to conclude. Philippe B. Laval (KSU) FTC Today 13 / 16

26 Integration by Parts Theorem If f and g are differentiable functions on [a, b] such that the derivatives f and g are both integrable on [a, b] then b a b fg = f (b) g (b) f (a) g (a) f g a Philippe B. Laval (KSU) FTC Today 14 / 16

27 Substitution Theorem Let g be a differentiable function on [a, b] such that g is integrable on [a, b]. If f is continuous on the range of f then b a f (g (t)) g (t) dt = g (b) g (a) f (x) dx This is the substitution formula Calculus students know and use often. If we let x = g (t) then dx = g (t) dt. Also, note that the limits of integration must also be updated when doing substitution. Example Find π 2 0 x sin ( x 2) dx Philippe B. Laval (KSU) FTC Today 15 / 16

28 Exercises See the problems at the end of my notes on the Relationship Between Integration and Differentiation. Philippe B. Laval (KSU) FTC Today 16 / 16