The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.


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1 ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion Problem 1. Let f : R R be function. Stte the definition of the derivtive of f t point R. Problem 2. Let f : [, b] R be bounded function. Stte when the Riemnn integrl of f() over [, b], b f(), eists. Wht is the vlue of b f() when it does eist. Problem 3. Suppose tht f : [, b] R is monotone. Show tht b f() eists. Problem 4. Let C = [0, 1] = } n=1 n 3, n n 0, 2} be the Cntor set. Let f : C [0, 1] be defined by f) = n/2 n=1 2 where = n n=1 n 3 is n element of the n Cntor set. This is clled the Cntor ternry function. This function cn be etended so tht f : [0, 1] [0, 1] by mking f be constnt on the intervls tht re complementry to the Cntor set. Compute the Riemnn integrl 1 0 f() of this function f(). Problem 5. Let r i } i=1 be n enumertion of the rtionl numbers in (0, 1). Define f : [0, 1] [0, 1] by the formul f() = r i < 1 for 0 < 1 nd f(0) = 0. Show tht f 2 i is n incresing function. Determine the vlue of 1 0 f(). Problem 6. Determine formul for n i=1 i2. Use this to determine using the Riemnn sum definition of the integrl. Problem 7. Suppose tht f() is piecewise monotone on [, b]. By tht we men tht [, b] = [ 0 =, 1 ] [ 1, 2 ] [ n 1, n = b] with f() monotone on [ i, i+1 ] for 0 i < n. Show tht the Riemnn integrl for f(), b f(), eists. Problem 8. Stte nd prove the Fundmentl Theorem of Clculus. 1
2 2 JAMES KEESLING Problem 9. Suppose tht f() is continuous on [, b]. Show tht the Riemnn integrl for f(), b f(), eists. Problem 10. Eplin how Romberg integrtion works. Be ble to use the TINspire CX CAS progrm to determine the Romberg estimte of integrl. 2. Definition of ln() nd ep() Problem 11. Define ln() = 1 d ln() 1 t dt. Show tht 1. Show tht ln() is strictly monotone incresing. Show tht ln( y) = ln() + ln(y) for ll, y > 0. Show tht lim ln() = nd tht lim 0 + ln() = Problem 12. Define ep() = y where ln(y) =. Show tht ep( + y) = ep() ep(y), lim = 0, nd lim =. Show lso tht d ep() ep(). 3. Pointwise nd uniform convergence Problem 13. Define pointwise convergence nd uniform convergence. Show tht f n () = n } n=1 converges pointwise to the following function. f() = Show tht this convergence is not uniform. 0 0 < 1 1 = 1 Problem 14. Suppose tht f n } n=1 converges uniformly to f() on [, b]. Suppose tht f n () is Riemnn integrble for ll n. Suppose lso tht f() is Riemnn integrble. Show tht b lim n f n () = b f(). Problem 15. Suppose tht f n } n=1 converges uniformly to f() on [, b]. Suppose tht for ech n f n () is continuous on [, b]. Show tht f() is continuous on [, b]. Problem 16. Show tht n=0 n = The geometric series for ll < 1.
3 ADVANCED CALCULUS PRACTICE PROBLEMS 3 5. Derivtives Problem 17. Suppose tht f() is differentible on [, b] nd tht f () > 0 for ll (, b). Show tht f() is strictly incresing on [, b]. Suppose tht f() 0 on [, b]. Show tht there is constnt C such tht f() C on [, b] Problem 18. Define f() in the following wy. 0 = 0 f() = sin ( ) 1 0 Show tht f (0) > 0, but tht f() is not incresing on ny intervl contining 0. Problem 19. Suppose tht f n () converges uniformly to f() on [, b]. Is it true tht f n() converges to f ()? Prove this if it does. Give counteremple if it does not. Problem 20. Suppose tht f : R R is differentible nd tht f () is bounded on R. Show tht f() is uniformly continuous on R. 6. Some emples nd pplictions of integrtion Problem 21. Suppose tht f() is continuous on [, b]. Show tht f() is uniformly continuous on [, b]. Problem 22. Let f() be defined s below. 1 Q f() = 0 / Q This is known s the Dirichlet function. Show tht f() is not Riemnn integrble on [0, 1]. Problem 23. Let f() be define s below. f() = 1 n = p n Q 0 / Q This is known s the popcorn function or Thome s function. Riemnn integrble on [0, 1]. Wht is 1 0 f()? Show tht f() is Problem 24. Stte Cvlieri s Principle. Determine the volume of sphere using this principle. Determine the volume of solid torus using the method.
4 4 JAMES KEESLING Problem 25. Use Cvlieri s method to determine the re in n ellipse with eqution 2 + y2 = 1. 2 b 2 Problem 26. Give the definition of the centroid of n re A. Wht is the centroid of the re in circle? Wht is the centroid of the re in n ellipse hving eqution 2 + y2 = 1? 2 b 2 Wht is the centroid of hlfcircle? Problem 27. Give the definition of the centroid of n rc L. Wht is the centroid of the circumference of circle? Wht is the centroid of the circumference of hlf circle? Problem 28. Determine the volume of cone whose bse hs re A nd whose verte is distnce h from the plne of the bse. Problem 29. Stte Pppus Theorem. Prove Pppus Theorem. Use Pppus Theorem to determine the volume of torus determined by rotting circle of rdius b bout the y is where the center of the circle is on the is t distnce from the origin.. Problem 30. Use Pppus Theorem to determine the surfce re of torus given by rotting circle of rdius b bout the y is where the center of the circle is on the is t distnce from the origin. Problem 31. Give the definition of the centroid of solid figure V. Determine the centroid of right circulr cone. Determine the centroid of cone with bse A whose verte is distnce h from the plne of the bse. Problem 32. Let V be right circulr cone with height h nd rdius t the bse r. Suppose tht the cone hs density d reltive to the density of wter with 0 < d < 1. Determine when the cone will flot stbly with the verte downwrd. Similrly, let V be solid hemisphere of rdius r nd density δ > 0 less thn tht of wter. Show tht it will flot stbly when its flt surfce is prllel to the surfce of the wter. Problem 33. Determine the rclength of the grph of y = 2 over [0, ]. Determine the centroid of this rc. Determine the surfce re rotting the figure round the is nd round the y is. Problem 34. Determine the rclength of ctenry hving the following eqution ( ) ( ( ep ) ( )) + ep y = cosh = 2
5 ADVANCED CALCULUS PRACTICE PROBLEMS 5 over the intervl [0, b]. Problem 35. integrl. Determine the circumference of circle of rdius using the rclength Problem 36. Determine the centroid of the perimeter of the upper hlf of circle given by 2 + y 2 = 2. Determine the centroid of the upper hlf of the re of circle hving this eqution. 7. Power series Problem 37. Show tht the following power series converge to the given functions on the intervl ( 1, 1). Wht hppens t = 1 in ech of these cses? ln(1 + ) = ( 1) n=1 n+1 n n rctn() = n=0 ( 1) n 2n+1 2n + 1 Problem 38. Show tht lim n n n = 1. Show tht lim n n n + 1 = 1. Use this to show tht for ll 1 < < 1. d 1 1 = d n = (n + 1) n n=0 n=0 Problem 39. Let f() = n=0 n n. Show tht the sum converges for ll < R where lim sup n = α = 1 n R. We sy tht R is the rdius of convergence for the power series. Problem 40. Suppose tht R is the rdius of convergence for the power series f() = n=0 n n. Show tht if g() = n=0 (n + 1) n+1 n, then the rdius of convergence for g() is lso R. Show tht d f() = g() for ll R < < R. Problem 41. Suppose tht R is the rdius of convergence for the power series f() = n=0 n n. Show tht f() is infinitely differentible for ll R < < R. Show tht n = f (n) () =0 n! for ll n = 0, 1, 2,....
6 6 JAMES KEESLING 8. Differentil equtions Problem 42. Consider differentil eqution of the form dt = f(t, ) (t 0) = 0. This is equivlent to the following eqution. t (t) = 0 + f(τ, (τ))dτ τ=t 0 The solution will be function (t) such tht (t 0 ) = 0 nd which stisfies the equiton dt = f(t, (t)) identiclly. Use Picrd Itertion to pproimte the solution to the following differentil equtions. Use five itertions. dt = t (0) = 2 dt = sin(t) (0) = 1 Problem 43. Determine polynomil of degree 5 tht pproimtes solution to the following differentil eqution on intervl centered t t = 1. = sin(t) (1) = 2 dt For the sme differentil eqution pproimte the solution t set of grid points using the Tylor Method of degree 5 nd using stepsize h = 1/10 nd n = 10. This estimtes the solution t t = 0, 1 10, 2 10,, 1. How ccurte is this numericl estimte of the solution? Problem 44. Stte nd prove the Contrction Mpping Theorem. This is lso known s the Bnch Fied Point Theorem. Problem 45. Show tht the following differentil eqution does not hve unique solution. dt = (0) = 0 Problem 46. Consider the liner system of differentil equtions given by the following equtions where dt = A (0) = C
7 ADVANCED CALCULUS PRACTICE PROBLEMS 7 (t) = 1 (t) 2 (t). n (t) C = C 1 C 2. C n n n A =..... n1 n2... nn As shown in clss, the solution of this liner system of equtions is (t) = ep(t A) C. Solve this differentil eqution for the following conditions. A = C = Solve on [0, 1] with h = 1/20 nd n = 20.
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