Hndout I - Mthemtics II The im of this hndout is to briefly summrize the most importnt definitions nd theorems, nd to provide some smple exercises. The topics re discussed in detil t the lectures nd seminrs. Students should lso consult the suggested redings... Theory.. Derivtives of rel functions Definition.. Let f : D R be function, where D is n intervl, nd let x D. If the limit f(x) f(x ) lim x x x x exists then we sy tht f is differentible t x. In this cse we write f (x ) = lim x x f(x) f(x ) x x for the derivte of f t x. If f is differentible for ll x D then we sy tht f is differentible. We write f for the derivtive of f. Theorem.. Suppose tht f(x) nd g(x) re differentible functions for ll x R. Then we hve (f(x) + g(x)) = f (x) + g (x), (f(x) g(x)) = f (x) g (x), (cf(x)) = cf (x) (c R), (f(x)g(x)) = f (x)g(x) + f(x)g (x), ( ) f(x) = f (x)g(x) f(x)g (x), (f(g(x))) = f (g(x)) g (x). g(x) g 2 (x) Theorem.2. All the elementry functions re differentible. Further, we hve c = (c R), (x α ) = αx α (α R), ( x ) = x ln() ( > ), (e x ) = e x, (log (x)) = x ln(), (ln(x)) = x, (sin(x)) = cos(x), (cos(x)) = sin(x), (ctg(x)) = sin 2 (x), (tg(x)) = cos 2 (x), (rctg(x)) = + x. 2 Here rctg(x) is the inverse of the function tg(x) : ( π/2, π/2) R.
2.2. Smple exercises. Exercise.. Differentite the following functions: 2(x 5 5 x cos x x), e x sin(x), x 2 + ln x, ln(sin(x)). Exercise.2. Differentite the following functions: ( ) 3 2 x cos(cos(x 2 )) x 2 + x ln( x) +, e x2 sin(3x). ln(x) Throughout this mteril ll functions f re ssumed to be elementry functions, with f : D R, where D is n intervl. 2.. Theory. 2. Anlysis of rel functions Definition 2.. Let f be function nd x D. If there exists n ε > such tht for ll x D (x ε, x + ε) we hve f(x ) f(x), then f hs locl minimum t x, f(x ) < f(x), then f hs strict locl minimum t x, f(x ) f(x), then f hs locl mximum t x, f(x ) > f(x), then f hs strict locl mximum t x. In ny of the bove cses we sy tht f hs locl extremum t x. Definition 2.2. Let f be function. If for ll x, y D with x < y we hve f(x) f(y), then we sy tht f is monotone incresing. If for ll x, y D with x < y we hve f(x) f(y), then we sy tht f is monotone decresing. Definition 2.3. Let f be function. If for ll x, y D nd λ [, ] we hve λf(x) + ( λ)f(y) f(λx + ( λ)y), then we sy tht f is convex on D. If f is convex on D, then we sy tht f is concve on D. Let x D. If for some ε >, f is convex on (x ε, x ) nd concve on (x, x + ε), or vice vers, then x is clled n inflection point of f. Theorem 2.. Suppose tht f hs locl extremum t some x D. Then we hve f (x ) =. Theorem 2.2. Suppose tht x D, nd f (x ) = f (x ) = = f (n) (x ) =, but f (n+) (x ). If n + is even, then f hs locl extremum t x. Further, if f (n+) (x ) < then f hs locl mximum t x, nd if f (n+) (x ) > then f hs locl minimum t x, respectively. On the other hnd, if n + is odd, then f hs no locl extremum t x.
Theorem 2.3. Suppose tht f (x) for ll x D. monotone incresing on D. Theorem 2.4. Suppose tht f (x) for ll x D. monotone decresing on D. 3 Then f is Then f is Theorem 2.5. Suppose tht f (x) for ll x D. Then f is convex on D. Theorem 2.6. Suppose tht f (x) for ll x D. concve on D. 2.2. Smple exercise. Then f is Exercise 2.. Let f : R R be defined by f(x) = x 3 3x. Give the complete nlysis of f, tht is: determine the zeroes of f; clculte the limits lim f(x) nd lim f(x); determine the locl extrem of f; x x determine the intervls where f is monotone; determine the intervls where f is convex/concve, nd give the inflection points of f. 3.. Theory. 3. L Hospitl s rule Theorem 3.. Let f, g be two functions, such tht g(x) (x D). Further, ssume tht either lim f(x) = lim g(x) =, or lim f(x) = x x x x x x ±, lim g(x) = ±, where x D, or x is n endpoint of D x x (possibly x = ± ). Then we hve f(x) lim x x g(x) = lim f (x) x x g (x), if the ltter limit exists. 3.2. Smple exercises. Exercise 3.. Clculte the following limits: sin x lim x x, lim x x e, lim x ln x. x x 4.. Theory. 4. Tylor polynomil, Tylor series Definition 4.. Let f : [, b] R be function differentible n times. Then the Tylor polynomil of order n of f(x) round point x (, b) is given by n f (k) (x ) T n (x) = (x x ) k. k! k= If x = then the bove polynomil is clled the McLurin polynomil of f(x) of order n.
4 Theorem 4.. Let f : [, b] R be function differentible n + times. Then we hve R n (x) = f (n+) (ξ) (n + )! (x x ) n+, where ξ is number between x nd x, depending on x. Definition 4.2. Let f : [, b] R be function differentible infinitely mny times. Then the Tylor series of f(x) round point x (, b) is given by f (k) (x ) T (x) = (x x ) k. k! k= If x = then the bove series is clled the McLurin series of f(x) of order n. 4.2. Smple exercises. Exercise 4.. Give the Tylor-series of the functions 5.. Theory. x 3 + 2x 2 x + 3, e x, sin x, cos x. 5. Primitive functions, indefinite integrl Definition 5.. Suppose tht we hve F (x) = f(x). Then we sy tht F (x) is primitive function of f(x). Theorem 5.. Let F (x) be primitive function of f(x). Then G(x) is primitive function of f(x) if nd only if G(x) = F (x) + c is vlid with some c R. Definition 5.2. The set of primitive functions of f(x) is clled the indefinite integrl of f(x). Nottion: f(x)dx = F (x) + c (c R), where F (x) is primitive function of f(x). Theorem 5.2. We hve (f(x) ± g(x))dx = f(x)dx ± g(x)dx nd λf(x)dx = λ f(x)dx (λ R). Theorem 5.3. We hve x α dx = xα+ + c (α ), α + cos xdx = sin x + c, e x dx = e x + c, dx = ln x + c, x sin xdx = cos x + c, dx = rctgx + c. + x2
5 Theorem 5.4. We hve f (x)f(x) α dx = f(x)α+ α + nd f (x) dx = lnf(x) + c. f(x) + c (α ) Theorem 5.5. (Prtil integrtion (integrtion by prts)) We hve f (x)g(x)dx = f(x)g(x) f(x)g (x)dx. Theorem 5.6. (Prtil frctions) Suppose tht x 2 + x + b hs two distinct rel roots, α nd β. Then there exist rel numbers A nd B such tht x 2 + x + b = (x α)(x β) = A (x α) + B (x β). Thus we hve dx = A ln(x α) + B ln(x β) + c. x 2 + x + b Theorem 5.7. (Integrl with substitution) We hve ( (g f(x)dx = f(g(t))g (t)dt) (x) ), where g(x) is n invertible function. 5.2. Smple exercises. Exercise 5.. Determine the following indefinite integrls: ( ) x + dx, (e x 3 + 2 cos x)dx, x + x dx. 2 Exercise 5.2. Determine the following indefinite integrls: cos x sin 3 x xdx, ( + x 2 ) dx, e x ctgxdx, 4 2 e. x Exercise 5.3. Determine the following indefinite integrls: x cos xdx, x ln xdx, x 3 sin xdx. Exercise 5.4. Determine the following indefinite integrls: x + x + 2 x 2 +, x 2 dx, 2x x 2 x 2 dx. Exercise 5.5. Determine the following indefinite integrls:, e 2x xe x + 2 + ( x + 2) 3 e x dx, x dx.
6 6.. Theory. 6. Riemnn (definite) integrl Definition 6.. Suppose tht f is continuous on D = [, b], with, b R, < b. For ny positive integer n put δ n := b nd x (n) 2 n i := +iδ n for i =,,..., 2 n. Further, let t (n) i [x (n) i, x (n) i+ ] such tht f(t(n) i ) f(x) for ll x [x (n) i, x (n) i+ ] (i =,,..., 2n ). Put S n := 2 n i= f(t (n) i )δ n. Then the Riemnn integrl of f(x) between nd b is defined s b f(x)dx = lim n S n. Definition 6.2. We use the following nottion: b f(x)dx = f(x)dx, f(x)dx =. Theorem 6.. We hve b b b b (f(x) ± g(x))dx = f(x)dx ± g(x)dx nd b b λf(x)dx = λ f(x)dx (λ R). Theorem 6.2. (Newton-Leibniz formul) Let F (x) be primitive function of f(x). Then we hve b f(x)dx = [F (x)] b = F (b) F (). Theorem 6.3. (Prtil integrtion (integrtion by prts) for definite integrls) We hve b f (x)g(x)dx = [f(x)g(x)] b b f(x)g (x)dx.
Theorem 6.4. (Integrl with substitution for definite integrls) We hve b f(x)dx = β α f(g(t))g (t)dt, where g(x) is n invertible function, nd α = g () nd β = g (b). 6.2. Smple exercises. Exercise 6.. Clculte the following integrls: 2 ( ) x + dx, x π (e x + 2 cos x)dx, Exercise 6.2. Clculte the following integrls: π cos x sin 3 xdx, 2 2 x ( + x 2 ) 4 dx, π/2 π/4 Exercise 6.3. Clculte the following integrls: 2π x cos xdx, 3 xe x dx, π/2 Exercise 6.4. Clculte the following integrls: x + x 2 +, 4 x + 2 x 2 dx, Exercise 6.5. Clculte the following integrls: 2 7.. Theory. 2, x + 2 + ( x + 2) 3 8 5 e 2x e x dx, ctgxdx, x 3 sin xdx. 3 + x 2 dx. 2x x 2 x 2 dx. 2 7. Improprius (improper) integrl e x 2 e x. xe x dx. Definition 7.. Let f be bounded on the intervl [, ). Then the improprius integrl of f(x) between nd is defined by if the limit exists. b f(x)dx = lim f(x)dx, b 7
8 Definition 7.2. Let f be bounded on the intervl (, b]. Then the improprius integrl of f(x) between nd b is defined by if the limit exists. b f(x)dx = b lim f(x)dx, Definition 7.3. Let f : R R be bounded. Then the improprius integrl of f(x) between nd is defined by f(x)dx = c f(x)dx + f(x)dx with n rbitrry c R, if the ltter improprius integrls exist. Definition 7.4. Let f[, b] R such tht f is bounded on [, c] for ny c with < c < b. Then the improprius integrl of f(x) between nd b is defined by c if the limit exists. b c f(x)dx = lim f(x)dx, c b Definition 7.5. Let f : [, b] R such tht f is bounded on [c, b] for ny c with < c < b. Then the improprius integrl of f(x) between nd b is defined by if the limit exists. b b f(x)dx = lim f(x)dx, c c 7.2. Smple exercises. Exercise 7.. Clculte the following improprius integrls: 2 x 3 dx, 3 x dx. 8.. Theory. 8. Applictions of the integrl
Theorem 8.. Let f, g : [, b] R be two functions, such tht f(x) g(x) for x b. Then the re of the domin enclosed by f(x) nd g(x) over [, b] is given by b (f(x) g(x))dx. Theorem 8.2. Let T be domin in R 3 (i.e. solid), nd let A(x) denote the re of the intersection of T nd the plin {(u, v, w) : u = x} for x R. Then the volume of T is given by if the bove integrl exists. V (T ) = A(x)dx, Theorem 8.3. Let f : [, b] R be function. Then the length of the curve of f(x) over [, b] is given by if the integrl exists. L(f) = b + (f (x)) 2 dx Theorem 8.4. Let f : [, b] R be function nd let T be the domin (solid) obtined by rotting the curve of f round he intervl [, b] in R 3. Then the volume of T nd the surfce of T re given by V (T ) = b πf 2 (x)dx 9 nd b S(T ) = 2πf(x) + (f (x)) 2 dx respectively, if the integrls exist. Definition 8.. Let f : [, 2π] R be function. Then the Fourierseries of f(x) is given by + ( k cos kx + b k sin kx), with k= = 2π 2π f(x)dx
nd k = π 2π f(x) cos kxdx, b k = π 2π f(x) sin kxdx (k =, 2,... ). 8.2. Smple exercises. Exercise 8.. Clculte the re enclosed by the curves of the functions f(x) = x 2 nd g(x) = x + 2 over the intervl [, 2]. Exercise 8.2. Clculte the length of the curve of f(x) = 2x + 3 over the intervl [, 3], nd the volume nd surfce of the solid T obtined by rotting f(x) over this intervl. Exercise 8.3. Give the Fourier trnsform of the function f(x) = x.