Exercise 3.3. MA 111: Prepared by Dr. Archara Pacheenburawana 26

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1 MA : Prepared b Dr. Archara Pacheenburawana 6 Eercise.. For each of the numbers a, b, c, d, e, r, s, and t, state whether the function whose graphisshown hasanabsolutemaimum orminimum, a localmaimum orminimum, or neither a maimum nor a minimum. (a) a b c d e r s t (b) a b c d e r s t. Use the graph to state the absolute and local maimum and minimum values of the function. (a) = f() (b) = f()

2 MA : Prepared b Dr. Archara Pacheenburawana 7. Find all critical numbers and the local etreme values of the following functions. (a) f() = +5 (b) f() = + (c) f() = + (d) f() = 4 + (e) f() = /4 4 /4 (f) f() = 4 (g) f() = sincos, [,π] (h) f() = + (i) f() = (j) f() = + (e +e ) (k) f() = 4/ +4 / + / (l) f() = + (m) f() = e (n) f() = sin, [,π] 4. Find the absolute maimum and absolute minimum values of f on the given interval. (a) f() = +5, [,] (b) f() = + +4, [,] (c) f() = 4 4 +, [,] (d) ) f() =, [,] (e) f() = +, [,] (f) f() = sin+cos, [,π/] (g) f() = e, [,] (h) f() = ln, [,4] (i) f() =, [,] (j) f() =, [,7] Answer to Eercise.. (a) Absolute maimum at b; local maima at b, e, and r; absolute minimum at d; local minima at d and s (b) Absolute maimum at e; local maima at e and s; absolute minimum at t; local minima at b, c, d, r, and t. (a) Absolute maimum f(4) = 4; absolute minimum f(7) = local maimum f(4) = 4, f(6) = ; local minimum f() =, f(5) = (b) Absolute maimum f(7) = 5; absolute minimum f() = local maimum f() =, f() = 4, f(5) = local minimum f() =, f(4) =, f(6) =

3 MA : Prepared b Dr. Archara Pacheenburawana 8. (a) 5, absolute minimum (b), local maimum;, local minimum (c), no local etreme values 9 (d), no local etreme values;, local minimum 4 6 (e), no local etreme values;, local minimum 9 (f), local minimum;, local maimum (g) π, 5π, local maimum; π, 7π, local minimum (h), no local etreme values (i), local minimum;, local maimum (j), local minimum (k),, local minimum (l), local minimum (m), local maimum π (n),, local minimum; π, 5π, local maimum 4. (a) f() = 5,f() = 7 (b) f() = 9,f( ) = (c) f( ) = 47,f(± ) = (d) f() =,f() = (e) f() =,f() = (f) f(π/4) =,f() = (g) f() = /e,f() = (h) f() =,f() = ln (i) f() =,f() = (j) f(7) =,f( ) = Eercise.4. Find the intervals on which f is increasing or decreasing. (a) f() = + (b) f() = (c) f() = (d) f() = (+) / (e) f() = sin, [,π] (g) f() = e (f) f() = sin, [,π] (h) f() = (ln)/. At what values of does f have a local maimum or minimum? Sketch the graph. (a) f() = + (b) f() = + (c) f() = e (d) f() = ln (e) f() = (f) f() = (g) f() = sin+cos (h) f() = + (i) f() = / /. Show that 5 is a critical number of g() = + ( 5) but g has no local etreme values at Find a cubic function f() = a +b +c+d that has a local maimum value of at and a local minimum value of at.

4 MA : Prepared b Dr. Archara Pacheenburawana 9 Answer to Eercise.4. (a) Inc. on (, ) (, ); dec. on (,) (b) Inc. on (,) (, ); dec. on (, ) (,) (c) Inc. on (, ); dec. on (, ) (d) Inc. on (, ); dec. on (, ) (e) Inc. on ( π, 5π ) (7π,π); dec. on (, π ) (5π, 7π ) (f) Inc. on (, π 6 ) (π, 5π); dec. on 6 6 (π, π 6 6 ) (5π,π) 6 (g) Inc. on (, ); dec. on (,) (h) Inc. on (,e ); dec. on (e, ). (a) Loc. ma. at = 7 ; loc. min. at = + 7 (b) None (c) Loc. ma. at =. (d) None

5 MA : Prepared b Dr. Archara Pacheenburawana (e) None (f) Loc. ma. at = ; Loc. min. at = 4 (g) Loc. ma. at = π +nπ; Loc. min. at = 5π +nπ 4 4 π (h) Loc. ma. at = ; Loc. min. at = (i) Loc. min. at = 4. f() = 9 ( + +7)

6 MA : Prepared b Dr. Archara Pacheenburawana Eercise.5. Use the given graph of f to find the intervals of concavit. (a) (b) 4 4 (c) (d) 5 5. Given the following functions. Find the interval of increase and decrease. Find the local maimum and minimum values. Find the intervals of concavit and the inflection points.

7 MA : Prepared b Dr. Archara Pacheenburawana Use the above information to sketch the graph. (a) f() = (b) f() = 4 6 (c) f() = (d) f() = + (e) f() = / (+) / (f) f() = sin Answer to Eercise.5. (a) CU on (, ) (, ); CD on (,) (b) CU on (,); CD on (, ) (c) CU on (, ); CD on (,) (d) CU on (,) (, ); CD on (, ) (,). (a) Inc. on (, ) (, ); dec. on (,); loc. ma. f( ) = 7; loc. min. f() = ; CU on (, ); CD on (, ); IP (, ) 5 (b) Inc. on (,) (, ); dec. on (, ) (, ); loc. min. f(± ) = 9; loc. ma. f() = ; CU on (, ) (, ); CD on (,); IP (±, 5) (c) Inc. on (, ) (, ); dec. on (, ); loc. ma. f( ) = 5; loc. min. f() = ; CU on ( /,) (/, ); CD on (, / ) (/, ); IP (,), (±/, 8 7 )

8 MA : Prepared b Dr. Archara Pacheenburawana 5 (d) Inc. on (, ); No loc. ma and loc. min; CU on (, ); CD on (,); IP (,) e) Inc. on (, ) (, ); dec. on (, ); loc. ma. f( ) = ; loc. min. f( ) = 4; CU on (, ) (,); CD on (, ); IP (,) (f) Inc. on (,π/) (π,π/); dec. on (π/,π) (π/,π) loc. ma. f(π/) = f(π/) = ; loc. min. f(π) = CU on (,π/4) (π/4,5π/4) (7π/4,π); CD on (π/4,π/4) (5π/4,7π/4) IP (π/4, ),(π/4, ),(5π/4, ),(7π/4, ) π π π π

9 MA : Prepared b Dr. Archara Pacheenburawana 4 Eercise.6. Find two positive numbers whose product is and whose sum is minimum.. Find two numbers whose product is 6 and the sum of whose squares is minimum.. For what number does the principal fourth root eceed twice the number b the largest amount. 4. Find the dimensions of a rectangle with area m whose perimeter is as small as possible. 5. A bo with a square base and open top must have the volume of, cm. Find the dimensions of the bo that minimize the amount of material used. 6. If cm of material is available to make a bo with a square base and an open top, find the largest possible volume of the bo. 7. A farmer wishes to fence off two identical adjoining rectangular pens, each with 9 square feet of area, as shown in the following Figure. What are and so that the least amount of fence is required? 8. Find the point on the line = 4+7 that is closest to the origin. 9. Find the point on the line 6+ = 9 that is closest to the point (,).. Find the point on the parabola = that is closest to the point (,5).. Find the point on the parabola + = that is closest to the point (, ).. Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r. 4. A right circular clinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a clinder. 5. A right circular clinder is inscribed in a sphere of radius r. Find the largest possible surface area of such a clinder. 6. A right circular clinder is inscribed in a sphere of radius r. Find the largest possible volume of such a clinder. 7. Show that the rectangle with maimum perimeter that can be inscribed in a circle is a square.

10 MA : Prepared b Dr. Archara Pacheenburawana 5 8. (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with largest area is a square. 9. At 7 : am. one ship was 6 miles due east from a second ship. If the first ship sailed west at miles per hour and the second ship sailed southeast at miles per hour, when were the closest together?. A piece of wire m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maimum? (b) A minimum? Answer to Eercise.6.,. 4, , cm ( 8, ) ( 45, ) ( ) (, 9,, 9. (, ). r r. Base r, height r/ πr h 5. πr ( + 5 ) 6. 4πr / 9. 8 : 9am.. (a) Use all of the wire for the square (b) 4 / ( 9+4 ) m for the square Eercise.7. Verif that the function satisfies the three hpotheses of Rolle s Theorem on the given interval. Then find all numbers c that satisf the conclusion of Rolle s Theorem. (a) f() = 4+, [,4] (b) f() = ++5, [,] (c) f() = sinπ, [,] (d) f() = +6, [ 6,]. Use the graph of f to estimate the values of c that satisf the conclusion of the Mean Value Theorem for the interval [, 8]. ) = f()

11 MA : Prepared b Dr. Archara Pacheenburawana 6. Verif that the function satisfies the hpotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisf the conclusion of the Mean Value Theorem. (a) f() = ++5, [,] (b) f() =, [,] (c) f() = e, [,] (d) f() = +, [,4] Answer to Eercise.7. (a) (b) ± (c) ± 4,± 4 (d) 4..8,., 4.4, 6.. (a) (b) (c) ln[( e 6 )/6] (d) Eercise 4.. Find the most general antiderivative of the function. (a) f() = 5 (b) f() = +π (c) f() = 5/4 (d) f() = / (e) f() = (f) f() = 4 5 (g) f() = (h) f() = (i) f() = Evaluate the integral and check our answer b differentiating. (a) 4 d (b) ( +)d (c) ( 4 )d (d) (+) d (e) d (f) ( ) d 4 / ( +) (g) d (h) d / (i) (sin cos)d (j) sectand (k) 5sec d (l) (e )d (m) (cos /)d (n) (5 e ) d e + (o) d (p) /4 ( 5/4 4)d e

12 MA : Prepared b Dr. Archara Pacheenburawana 7 Answer to Eercise 4.. (a) 5+C (b) +π+c (c) 4 9 9/4 +C (d) +C (e) +C (f) C (g) C (h) + +C (i) 4 + +C. (a) 5 5 +C (b) + +C (c) 5 5 +C (d) (+) +C (e) / +C (f) + +C (g) / 9 / +C (h) 9 9/ / + / +C (i) cos sin+c (j) sec+c (k) 5tan+C (l) e +C (m) sin ln +C (n) 5 +e +C (o) e +C (p) 5 5/ 6 5 5/4 Eercise 4.. Evaluate the integral b making the given substitution. (a) cosd, u = (b) ( +)d, u = + 4 (c) d, u = + (+) ( +) (d) d, u = +. Evaluate the indefinite integral (a) ( +) 4 d d (c) (e) (g) d 5 +4 d ++ (i) cosd (k) sin( )d (m) cos 4 sind (o) sectan +secd (b) (+)( +) d (d) d (f) (h) + + d d ( +) (j) cos sin+d sin (l) d cos (n) sin(cos+) /4 d (p) cose sin d

13 MA : Prepared b Dr. Archara Pacheenburawana 8 (q) e +e d (r) e + d d 4 (s) (t) ln (ln+) d cot (u) csc d (v) sin(cos ) d e (w) sec e tand () d e +e + + () d (z) + +7 d Answer to Eercise 4.. (a) sin+c (b) ( +) / +C (c) /(+) +C (d) ( +) 4 +C. (a) 5 ( +) 5 +C (b) 4 ( +) 4 +C (c) ( )/ +C (d) +C (e) ln 5 +C (f) ln + +C (g) ++ +C (h) ( +) +C (i) sin+c (j) (sin+)/ +C (k) cos( )+C (l) cos+c (m) 5 cos5 +C (n) 4 7 (cos+)7/4 +C (o) (+sec)/ +C (p) e sin +C (q) (+e ) / +C (r) e + +C (s) ln ln +C (t) 4(ln+) +C (u) (cot)/ +C (v) 4 (cos )4 +C (w) sec +C () ln(e +e )+C () tan + ln(+ )+C (z) (+7) ln +7 +C Eercise 4.. Epress the limit as a definite integral on the given interval. (a) lim n (b) lim n (c) lim n (d) lim n n i sin i, [,π] i= n i= e i + i, [,5] n [( i) 5 i], [,] i= n i, [,4] i=

14 MA : Prepared b Dr. Archara Pacheenburawana 9. Use the form of the definition of the integral given in (4.7) to evaluate the integral. (a) (c) (e) 5 5. Prove that 4. Prove that d (+ )d (+ )d b a b a d = b a. d = b a. (b) (d) (f) 5 (+)d ( )d d 5. Write the given sum or difference as a single integral in the form b f()d. a 6. If 7. If (a) (c) (e). (a) 8 π f()d+ f()d+ f()d+ 6 f()d =.8 and f()d =,, 4 sind (b) f()d f()d f()d (b) (d) f()d f()d =., find f()d = 7, and 4 f()d f()d+ 5 f()d. f()d =, find Answer to Eercise 4. 5 e d (c) +. (a) (b) 4 (c) (d) 4 (e) 7.5 (f) (a) f()d (b) f()d (c) ( 5)d (d) f()d (d) f()d f()d 4 f()d. d f()d (e) f()d

15 MA : Prepared b Dr. Archara Pacheenburawana 4 Eercise 4.4. Evaluate the definite integrals using the First Fundamental Theorem of Calculus. (a) (c) (e) (g) (i) (k) (m) (o) (q) (s) (t) 4 π π/ π/ π/6 5 d (b) ( )d 4 d (d) ( +)d d 4 5 +d sectand d (f) (h) (j) (l) d (n) ( + ) sin d (p) d (r) π/ π π/ π/4 ( + / )d sind (sin cos)d (e e )d (e )d 6 + d cos d {, f()d, where f() =, < { f()d, where f() = 4, < 5,. Use the Second Fundamental Theorem of Calculus to find the derivative. (a) f() = (t t+)dt (b) g() = tdt (c) g() = (e) f() = (g) h() = (i) = / t sintdt (d) F() = (e t +)dt (f) f() = arctantdt (h) = u du (j) = +u cos(t )dt cost t ln(t +)dt dt tsintdt

16 MA : Prepared b Dr. Archara Pacheenburawana 4 Answer to Eercise 4.4. (a) 64 (b) (c) 6 (d) 88 (e) 7 8 (f) 5 (g) (h) (i) Does not eist (j) (k) ln (l) e+e (m) 8 ln (q) 5 (r) (s) 6 (t).7 (n) e (o) π 9 + (p) π. (a) f () = + (b) g () = + (c) g () = sin (d) F () = cos( ) (e) f () = (e 4 +) (f) f () = ln( +) (g) h () = arctan(/)/ (h) = cos (j) = 7/ sin( ) (sin )/( 4 ) Evaluate the definite integral, if it eists ( ) 5 d. (+ ) 5 d 4. Eercise 4.5 π (i) = ( ) +( ) +d ( +) d 4cos (sin+) d cosπd 6. π/ + π/ d 8. cotd d + π/ a. sin d. cos d 4. +a d (a > ) π/4 4 e 4 e d d (+) d ln Answer to Eercise ln 9. ln ( )a

17 MA : Prepared b Dr. Archara Pacheenburawana 4. Find the area of the shaded region. Eercise 5. (a) = + (b) = + = (c) = =. Find the area of the region bounded b the given curves. (a) = +, =, (b) =, =, (c) = +, = 9,

18 MA : Prepared b Dr. Archara Pacheenburawana 4 (d) = e, =, (e) = +, = ( ), (f) =, =, (g) = cos, = sin, π/ (h) =, =,. Find the area of the region enclosed b the given curves. (a) =, = 7 (b) = +, = (c) =, = + (d) =, = (e) =, =, = (f) =, = + (g) =, =, = (h) =, =, = 6, = (i) =, = (j) = cos, = /π (k) =, = /( +) (l) =, = 4 Answer to Eercise 5.. (a) 6 (b) 4 (c) 9. (a) (b) 4 (c) 9.5 (d) e (e) (f) (g) (h) 9 6. (a) 64 (b) 6 (c) 7 4 (d) (e) (f) 6 (g) (h) 8 (i) 8 (j) π (k) π (l) 8 5 Eercise 5.. Findthevolumeofthesolidobtainedbrotatingtheregionboundedb =, =, and = (a) about the -ais (b) about the line =.. Find the volume of the solid obtained b rotating the region bounded b =, =, and = (a) about the -ais (b) about the line = 4.. Find the volume of the solid obtained b rotating the region bounded b = e, =, =, and = (a) about the -ais (b) about the line =.

19 MA : Prepared b Dr. Archara Pacheenburawana Findthevolumeofthesolidobtainedbrotatingtheregionboundedb =, =, and = (a) about the -ais (b) about the -ais. 5. The region R enclosed b the curves =, the -ais, and the -ais. Find the volume of the solid obtained b rotating the region R (a) about the -ais, (b) about the -ais (c) about the line =, (d) about the line =, (e) about the line =, and (f) about the line =. 6. The region R enclosed b the curves =, =, and =. Find the volume of the solid obtained b rotating the region R (a) about the -ais, (b) about the -ais, (c) about the line =, (d) about the line =, (e) about the line =, and (d) about the line =.. (a) 8π 4. (a) π 5 (b) 8π (b) π 7 Answer to Eercise 5.. (a) π 5 (b) 4π 5 ( e 4. (a) πe +π (b) π +4e 9 ) 5. (a) 9π (b) 9π (c) 8π (d) 6π (e) 8π (f) 6π 6. (a) π (b) π 5 (c) π 6 (d) 7π 5 (e) 7π 6 (f) π 5 Eercise 5.. Use the method of clindrical shells to find the volume generated b rotating the region bounded b the given curve about the -ais. (a) =, =, =, = (b) = e, =, =, = (c) =, =. Usethemethodofclindricalshellstofindthevolumeofthesolidobtainedbrotating the region bounded b the given curve about the -ais. (a) = +, =, =, = (b) =, = 9 (c) =, =, + =. Use the appropriate method to find the volume generated b rotating the region bounded b the given curve about the specified ais. (a) =, =, = ; about -ais (b) =, = 5, = ; about = 5 (c) = 4 +, =, = ; about -ais (d) =, =, = ; about -ais (e) =, =, = ; about =

20 MA : Prepared b Dr. Archara Pacheenburawana 45 Answer to Eercise 5.. (a) π (b) π( /e) (c) 64π/5. (a) π/ (b) 944π/5 (c) 5π/6. (a) 6 5 π (b) 4 5 π (c) π (d) π (e) 8π Find the length of the curve. Eercise 5.4. = ( +) /,. = ,. = ln(sec), π/4 4. = ln( ), 5. = cosh, Answer to Eercise ln( +) 4. ln 5. sinh Eercise 6. Evaluate the integral.. e d. lnd. sin4d 4. e d 5. cosd 6. e sin4d 7. (ln) d 8. coscosd 9. sec d. cosln(sin)d. cos(ln ) d. cos d. sin d 4. sind e d 6. ln d 8. ln d 4 (ln) d

21 MA : Prepared b Dr. Archara Pacheenburawana 46 Answer to Eercise 6.. e 4 e +C. ln 9 +C. 4 cos4+ 6 sin4+c 4. e 9 e 7 e +C 5. cos+ 9 cos 7 sin+c 6. 7 e sin4 4 7 e cos4+c 7. (ln) ln++c 8. sincos cossin+c 9. tan+ln cos +C. sinln(sin) sin+c. [sin(ln)+cos(ln)]+c. cos +C. cos +sin +C 4. 4 sin cos ln 7. ln4 8. e (ln) 64 6 ln+ 5 5 Eercise 6. Evaluate the integral.. cossin 4 d. cos sind π/4. sin 5 cos d 4. cos 5 sin 4 d π/ π/ 5. sin d 6. cos sin d π/4 7. ( sin) d 8. sin 4 cos d 9. sin cosd. cos tan d sin. d. tan d cos. sec 4 d 4. tan 5 d π/4 5. tansec d 6. tan 4 sec d 7. tan secd 8. π/ π/ tan secd sec 9. d. cot d cot π/6. cot csc d. cot ωcsc 4 ωdω

22 MA : Prepared b Dr. Archara Pacheenburawana cscd 4. tan d sec 6. cos7θcos5θdθ 8. sin tan d sin5sind sin4cos5d Answer to Eercise sin5 +C. cos +C sin5 7 sin7 + 9 sin9 +C 5. π sin4+c 7. +cos 8 sin4+c 8. 9 (π 4) 9. [ 7 cos cos] cos+c. cos ln cos +C. ln(+sin)+c. tan +C. tan +tan+c 4. 4 sec4 tan +ln sec +C 5. sec +C sec sec+c tan +C. π. 5 csc5 + csc +C. cot ω 5 cot5 ω +C. ln csc cot +C 4. sin+c 5. sin+c 6. 6 sin 4 sin7+c 7. 4 sinθ+ 4 sinθ+c 8. cos 8 cos9+c Eercise 6. Evaluate the integral.. t dt. t 4 d d. d d d 6. d d 8. d 9. d. (a ) / 7 d.. 5. / d d d 6. +8d d d ( ++)

23 MA : Prepared b Dr. Archara Pacheenburawana e t 9 e t dt 8. d +a d Answer to Eercise 6. π C C 4. ( )ln 5. ( +4) / 4 +4+C 6. 4 sin ()+ 4 +C ln C sec ( )+C ( + ) +C a sin ( a) +C. 7+C. ln(+ ). ( +4) / +C [sin ( )+( ) ]+C ln C 6. [ tan ( )+ ] (+) +C ( ++) [ 7. e t 9 e t +9sin ( et)] +C 8. ln(+ +a )+C Eercise 6.4 Evaluate the integral d. + d d 4. 6 d d d + 7. d 8. (+) d d d 4 6. d. 4 d d 4. d d d

24 MA : Prepared b Dr. Archara Pacheenburawana 49 Answer to Eercise 6.4. ln + ln +C. ln + +4ln +C. 5 ln ln +C 4. ln + + ln 5 ln +C 5. ln +4 +ln + +C 6. ln + ln +ln +C 7. ln + (+) +C 8. ln + 4 (+) ln +C 9. ln + ln + +ln +C 4. ln( +)+tan +ln +C. ln + + ln ln( +)+tan +C. +ln + +ln +C. ln + (+) +C 4. tan +ln +C 5. ln + ln ++ 7 tan (+ )+C 6. +ln + ln( +) tan +C Eercise 6.5 Determine whether each integral is convergent or divergent. Evaluate those that are convergent d. (+) e d 4. e d 6. cosd 8. ln d. ln d. d 4. d 6. 4 d 8. π/4 π w dw d (+)(+) d e d + d d csc tdt secd z lnzdz

25 MA : Prepared b Dr. Archara Pacheenburawana 5 Answer to Eercise D. 4. D ln 7. D 8. e 4 9. D. D... D 4. D 5. D 6. D 7. D 8. 8 ln 8 9 Eercise 7.. List the first five terms of the sequence. (a) a n = (.) n (b) a n = n+ n (c) a n = ( )n (e) { sin nπ n! } (d) { 4 6 (n)} (f) a =, a n = +a n. Findaformulaforthegeneraltermterma n ofthesequence, assuming thatthepattern of the first few terms continuous. (a) {, 4, 8, 6,...} (b) {, 4, 6, 8,...} (c) {,7,,7,...} (d) { 4, 9, 6, 4 5,...} (e) {,, 4 9, 8 7,...} (f) {,,,,,,...}. Determine whether the sequence converges of diverges. If it converges, find the limit. (a) a n = n(n ) (b) a n = n+ n (c) a n = +5n n (d) a n+n n = + n (e) a n = n n+ (f) a n = (g) a n = ( )n n n + (i) a n = +cosnπ { } +( ) n (k) n { } ln(n ) (m) n (o) { n+ n} n + n (h) a n = sin(nπ/) (j) {arctann} { } n! (l) (n+)! (n) {( ) n sin(/n)} { } ln(+e n ) (p) n (q) a n = n n (r) a n = ln(n+) lnn

26 MA : Prepared b Dr. Archara Pacheenburawana 5 (s) a n = cos n n (t) a n = (+n) /n (u) a n = n + n + + n (v) a n n = ncosn n + (w) a n = n! () a n n = ( )n n! Answer to Selected Eercise 7.. (a).8,.96,.99,.9984, (c),,,, (e) {,,,,,...} 8 4. (a) a n = / n (c) 5n (e) ( ) n. (a) D (c) 5 (e) (g) (i) D (k) (m) (o) (q) (s) (u) (w) D Eercise 7. Determine whether the series is convergent or divergent. If it is convergent, find its sum ( n ( ) ) n n= n 8 n+ 6. n= n= n= n(n+) n +n 8.. arctan n. n= [ ( ) n ( ) 4 + n ] 5 n= 4. ( n ) n 6. ( e ) n+ π 8. n= n= n= n= 4 n n n+5 [(.) n +(.) n ] n= n + n n= 6 n n ln n+ n= n 5 n+ n= n! n n= ( (n ) ) n n=

27 MA : Prepared b Dr. Archara Pacheenburawana 5 Answer to Eercise 7... D D 6. D D.. D. D D D 7. e π(π e) 8. n= Eercise 7. Determine whether the series converges or diverges.... ne n n 4 n+ n= n= n= n n n n=5 n= n 7. ne n n + nlnn n= n= n= arctan n... +n n +n+ n n= n= n= n= n= n= n= n= n= n= 7 4n+ n +n n n. +cosn n. + n 6. n+ n n 9. n n +n+. n! n 5. n+ n n (n)! n= n= n= n= n= n= n= n= n= n= n= (4+n) 7/ n 8. n(n+)(n+). n n5 +4 n + n 4 + n! n n+ n (n)! n n! sinn n 4. n= n= n= n= n= n= ne n n+ n n + n n + n +n+n +n +n 6 sin n= n= n= n= n= 8 n n! ( ) n n n+ 4n +n n 5 4n + n( ) n 4 n

28 MA : Prepared b Dr. Archara Pacheenburawana n= n= n= n (n+)4 n+ 44. n n +n 47. n n (n)! 5. n! ( ) n 45. ( ) n n + n n +n n! 5. n= n= n= cos(nπ/) n! +sin n ( n n n= Answer to Eercise 7. n= n= n= 4 6 (n) n!. C. D. C 4. C 5. C 6. C 7. C 8. D 9. D. C. C. D. D 4. C 5. C 6. C 7. C 8. D 9. C. C. D. C. C 4. C 5. D 6. C 7. D 8. C 9. C. D. D. C. C 4. D 5. C 6. D 7. C 8. C 9. C 4. C 4. C 4. C 4. C 44. D 45. C 46. D 47. C 48. C 49. C 5. C 5. C 5. C 5. C 54. C 55. D 56. C Eercise 7.4. Find the Maclaurin series for the function. (a) e (b) e a (c) cosπ (d) sinπ (e) ln(+) (f) + (g) cosh (h) sinh (i) sin (j) e. Find the Talor series of the function f() centered at the given value of a. (a) e ; a = (b) e ; a = ln (c) ; a = (d) + ; a = (e) sinπ; a = (f) cos; a = π (g) ln; a = (h) ln; a = e ) n

29 MA : Prepared b Dr. Archara Pacheenburawana 54. (a) (i). (a) (g) ( ) n n (c) n! n= n= n= ( ) n (n+)! n+ e n! ( )n (c) Answer to Selected Eercise 7.4 ( ) n π k n (e) (n)! n= ( ) n ( ) n n n= ( )(+) n (e) n= ( ) n+ n (g) n n= ( ) n π n n= (n)! n= ( ) n (n)! n

Questions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.

Questions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt. Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,