Calculus Tutorial Manual PDF 2024-25

SESH1110-Calculus AY : 2024-25 Tutorial Manual Name of Student:_____________________________________________________________________________...

SESH1110-Calculus AY : 2024-25 Tutorial Manual Name of Student:___________________________________________________________________________________________________ Enrollment No.: ________________________________________Academic Year: _______________________________________ Department of Science & Humanities  P. P. SAVANI SCHOOL OF ENGINEERING CERTIFICATE This is to certify that Mr./Ms. __________________________ of _________________________ Engineering having Enrollment No. __________________________ has completed his/her Term work in the subject of CALCULUS (SESH1110). Marks Obtained: - _____________out of _____________. Sign of Faculty Date:___________ SESH1110 – Calculus 2|Pa ge  List of Tutorial Sr. No. Name of Tutorial Hours 1. Calculus – 1 4 2. Calculus – 2 4 3. Calculus – 3 2 4. Sequence and Series – 1 4 5. Sequence and Series – 2 2 6. Sequence and Series – 3 2 7. Partial Derivatives – 1 4 8. Partial Derivatives – 2 2 9. Curve tracing – 1 4 10. Curve tracing – 2 2 Total Hours 30 SESH1110 – Calculus 3|Pa ge  Module 1: Calculus (10 Hours) SESH1110 – Calculus 4|Pa ge  Tutorial Session – 1 Calculus – 1 1. 1. lim ( x 2  13) Ans: 4 2. lim 8(t  5)(t  7) Ans: 8 x 3 t6 3. 2x  5 5h  4  2 Ans: 5 lim Ans: 3 4. lim x 2 11  x 3 h 0 h 4 x2  7 x  10 5 y3  8 y 2 1 5. lim Ans: 3 6. lim Ans: y 0 3 y 4  16 y 2 2 x 2 x2 t2  t  2 8. x5 Ans: 1 7. lim Ans: 2 lim t 0 t 2 1 x5 x 2  25 10 x4  16 x3  x2  5 x  3 9. lim Ans: 32 10. lim Ans: 4 x 2 x  2 x 1 ( x  1)2 11. lim sec 2 t tan t Ans: 1 5x3  9x2 Ans: 9 3t 3 12. lim 3 2 t0 x0 2x  2x2 2 x2 Ans: 4 1  1 x  Ans:  13. lim 14. limsin  2  x 0 3  3cos x 3 x 1 1 x  6 1 1  Ans:  1 5  x  25 Ans:  1 15. lim x  2 2 4 16. lim x2 x 10 x2 x 1 1 1 x 1 1 17. lim  Ans:  18. lim Ans: x 0 x 1 x x 2 x 1 x 1 2 4 x2 2. Find lim. Ans: 0 x  x  2x 1 2 4x2  x 3. Find lim. Ans: 0 x 2x3  5 x2  1 Ans: 1 4. Find lim. x 2x2  3 2 x2  1 5. Find lim. Ans: Not exist x 2x  3 4x  5 Ans: Continuous, If the f ( x )  function is continuous or discontinuous at x  1 , 6. 9  3x Continuous and x  0 and x  3 ? discontinuous SESH1110 – Calculus 5|Pa ge   x x  0 7. Prove that f ( x )   is continuous at x  0. Ans: Proof x , x  0 2  1  x sin  2  x  0 Ans: Discuss 8. Discuss the continuity of f at origin, where f ( x)   x . 0,x  0 continuity   1  x ,0  x  2   1 1 9. Prove that f ( x)   1,x  is discontinuous at x . Ans: Proof  2 2  1 1  x , 2  x  1   x 2  5 if x  2 Given the function, f  x   . Compute the following 1  3 x if x  2 Ans: I. -17 10. limits: II. Not exist I. lim f  x  II. lim f  x  x 6 x 2 SESH1110 – Calculus 6|Pa ge  Tutorial Session – 2 Calculus – 2 1. Find y n , for y  x 2 log x. 2  1  n  3! n 3 Ans: yn  n2 x 2. 2x  1 Find y n , for y .  x  2  x  1  1 1  Ans: yn   1 n!  n     x  2   x  1 n 1 n 1  3. Find y n , for y  x 2e x cos x.  2 n2  n  3      Ans: yn  e x 2  2 2 x cos  x    2 2 nx  x   n  1   n  n  1 cos  x   n  2     4   4  4  4. 1 Find y n , for y  2. x  6x  8  1 n!   n 1 1 Ans: yn     2   x  4 n 1  x  2 n 1  5. Find y n , for y  x3 log x. Ans: 6. Find y n , for y  x 2 cos x.  n    n  1    n n  1 cos  x   n  2    Ans: yn  x 2 cos  x    2nx cos  x        2   2   2  7. Find y n , for y  e x log x. Ans: yn  e x log x  nC1 x 1  nC2 x 2 ...   1  n  1! nCn x  n  n 1  8. Verify Rolle’s theorem   5  I. f ( x)  e x (sin x  cos x), x   , . Ans: Verified and c   4 4   x 2  ab  II. f ( x)  log   , x   a, b  ,a  0,b  0. Ans: Verified and c   ab  x(a  b)  2 III. f ( x)  2 x 3  x 2  4 x  2,x    2, 2 . Ans: Verified and c  1,   3     IV. f ( x )  cos 2 x, x   , . Ans: Verified and c  0  4 4 9. Lagrange’s mean value theorem I. f  x   log x, x  1,e Ans: Verified and c  e  1 ba II. f  x   2 x 2  3 x  4 in  a, b  Ans: Verified and c    a, b  2 SESH1110 – Calculus 7|Pa ge  62 3 III. f  x    x  1 x  2 x  3 , x   0,4 Ans: Verified and c  3 1 IV. f  x   , x   1,1 Ans: Not verified x 10. Verify Cauchy’s mean value theorem ab I. f ( x)  x 2 ,g ( x)  x,x  [a, b] Ans: Verified and c  2 14 II. f ( x)  x 2  2, g ( x)  x 3  1,x  [1,2] Ans: Verified and c  9 13 III. f ( x)  x 2  3, g ( x)  x3  1,x  [1,3] Ans: Verified and c  6 1 e IV. f  x   log x, g  x   , x  1, e Ans: Verified and c  x e 1 SESH1110 – Calculus 8|Pa ge  Tutorial Session – 3 Calculus – 3 Q.1 Write the definitions of global maxima, global minima, local maxima, local minima, critical point, local extremum and global extremum. Q.2 Find the maxima and minima of the function 10 x 6  24 x 5  15 x 4  40 x 3  108. Ans: For x  2 , minimum and x  0 neither maximum nor minimum Q.3 Find the maximum and minimum values of x 5  5 x 4  5 x 3  1. Ans: Max. f 1  0 , Min. f  3  28 , No extremum f  0   0 Q.4 Find the maximum and minimum values of a 2 sin 2 x  b 2 cos 2 x.     a , Min. value f    b 2 2 Ans: Max. value f  2 Q.5 Find the local and global extremum values of f ( x )  sin x ,0  x  2.   Ans: Local and global max. value f   1 2  3  Local and global min. value f    1  2  Q.6 Find the local extremum values f  x   2 x 2  9 x  1, 2  x  5. 9 89 Ans: f     4 8 x2  x  1 Q.7 Find the maximum and minimum values of. x2  x  1 1 Ans: Max. f (1)  3, Min. f (  1) . 3 SESH1110 – Calculus 9|Pa ge  Module 2: Sequence and Series – I (4 Hours) SESH1110 – Calculus 10 | P a g e  Tutorial Session – 4 Sequence and Series – 1 Q-1 Test the convergence of the following sequences:  n2  n  a)  2  Ans: Convergent;  2n  n  b) 2   1  n Ans: Oscillating Sequence; 1 and 3 3n c) Ans: Convergent; 0 n  7n2  n   2 d)    Ans: Convergent; 1  n  1    2n3  7n  e)  3 2  Ans: Convergent;  5n  3n  2n  1 f) Ans: Convergent 1  3n 1 g) 1  Ans: Convergent n h)  n 1  n  Ans: Convergent  1 Q-2 Is the sequence sin    convergent? Ans: Yes  6 n Q-3 Test the convergence of the series: 1 3 5 a)    Ans: Convergent 1 2  3 2  3  4 3  4  5 2 13  5 2  23  5 2  n3  5 b)    Ans: Convergent 4 15 1 4  25 1 4  n5 1 1 1 1 c) p  p  p  Ans: Convergent 3 5 7    n  1 n  2   d)    Ans: Convergent n 1  n2 n     n  1 n  2   e)    Ans: Divergent n 1  n2 n  Q-4 Test the convergence of the series: 1 1 1 1 a)     Ans: Convergent 2 5 10 1  n2 12 22 32 42 b)     Ans: Convergent 1! 2! 3! 4!  3n n c)  n 1 4 n SESH1110 – Calculus 11 | P a g e  Tutorial Session – 5 Sequence and Series – 2 Q-1 Test the convergence of the following series:  an1 a)  n Ans: l  1 ; Convergent n1 n 2 3 n 1 2 3  n  b)           Ans: l  1 ; Convergent 3 5 7  2n  1   nnxn Ans: The series is convergent c)  if x  1 and is divergent if  n  1 n n 1 x 1 1 1 1 1 d) 1  2  3  4    n   Ans: l  1 ; Convergent 2 3 4 n n   n 1 e)   n 1  3n   Ans: l  1 ; Convergent Q-2 Test the convergence of the following series:  2 tan 1 n 3 2 a)  n 1 1  n 2 Ans: I  16 ; Convergent  1 b)  Ans: I   ; Convergent n1 1 n 2 4  1 c)  n log n n1 Ans: Convergent  1  d)  n log n n 3 log n  1 2 Ans: I  2  sec  1  log 3  ; Convergent Q-3 Test the convergence of the following series: 1 1 1 1 a) 1      Ans: Convergent 2 4 8 16 1 2 3 4 b)     Ans: Convergent 2 5 10 17   1 n 1  c)  log( n  1) n 1 Ans: Convergent  1 n 1  d) n n 1 2 ( n  1) Ans: Convergent   n 1   1 n1 e) log   Ans: Convergent n1  n  SESH1110 – Calculus 12 | P a g e  Module 3: Sequence and Series – II (4 Hours) SESH1110 – Calculus 13 | P a g e  Tutorial Session – 6 Sequence and Series – 3 Q-1. Expand log x in powers of ( x  1) and hence evaluate log 1.1 correct up to four decimal places. 1 1 1 Ans: log x   x  1    x  1   x  1   x  1   ; log 1.1  0.09 53 08 2 3 4 2 3 4 Q-2. Expand 3 x 3  8 x 2  x  2 in powers of  x3. Ans: f ( x)  154  130  x  3  35  x  3  3  x  3  2 3 Q-3. Expand x 4  11x 3  43 x 2  60 x  14 in powers of  x3. Ans: f ( x)  5  9  x  3  2  x  3   x  3  2 3 Q-4.   Expand sin   x  in powers of x. Hence find the value of sin 44  and sin 46 . 4   Ans: sin   x   1  x 2 x3  1  x      ; sin 44   0.6947 & sin 46   0.7193 4  2 2! 3!  Q-5. Expand x 4  3 x 3  2 x 2  x  1 in powers of  x3. Ans: 16  38  x  3  29  x  3  9  x  3   x  3 2 3 4 Q-6. Express ( x 1)4  2(x 1)3  5(x 1)  2 in ascending power of x. Ans: f (x)  4  7x  2x3  x4 Q-7. Determine the Maclaurin’s series for cosh3x up to x6. 9 27 4 81 6 Ans: 1  x 2  x  x  2 8 80 Q-8. Determine the Maclaurin’s series for cos4t as far as the term in t 6. 32 4 256 6 Ans: 1  8t 2  t  t  3 45 3 x Q-9. Expand e 2 using Maclaurin’s series up to x3. 3 9 9 3 Ans: 1  x  x 2  x  2 8 16 Q-10. Use Maclaurin’s series to determine the expansion of  3  2t . 4 Ans: 81  216t  216t 2  96 t 3  16 t 4   SESH1110 – Calculus 14 | P a g e  Tutorial Session – 7 Sequence and Series – 4 1  x  n 1 Q-1 Prove that lim  n. x 0 x xex  log(1 x) Ans: 3 Q-2 Evaluate lim. 2 x0 x2 x log x   x  1 1 Q-3 Evaluate lim. Ans: x 1  x  1 log x 2 ex  ex  2log(1 x) Q-4 Evaluate lim. Ans: 1 x0 x sin x ex  ex  x2  2 Ans:  1 Q-5 Evaluate lim. x0 sin2 x  x2 4 log 1  kx 2  Q-6 Evaluate lim. Ans: 2k x 0 1  cos x x  sin x 1 Q-7 Evaluate lim. Ans: x0 x3 6 tan x  sin x 1 Q-8 Evaluate lim. Ans: x0 sin 3 x 2 log x Q-9 Prove that lim  0,  n  0 . x  xn   log  x   Q-10  2 Prove that lim  0.  tan x x 2 Q-11 Prove that lim log x tan x  1. x 0 x  x 1 2x 1 1 Q-12 Prove that lim 3 . x 6x 3 Q-13 Prove that lim x log x  0. x0  x 1 Q-14 Prove that limlog  2   cot  x  a   . xa  a a  1 log 1  x   1 Q-15 Evaluate lim   . Ans: x0 x  x2  2  x 1  1 Q-16 Prove that lim   . x 1  x  1 log x  2  1 1  1 Q-17 Prove that lim   . x2 x  2  log  x 1  2 SESH1110 – Calculus 15 | P a g e  1 1  1 Q-18 Evaluate lim 2  2 . Ans:  x0  x sin x  3  3  Q-19 Prove that lim  cos 2 x  x 2   e  6. x0 1 1 Q-20  tan x  x2 Prove that lim   e. 3 x0  x  Evaluate lim  cot x  sin x Q-21. Ans: 1 x0 tan x 1 Q-22 Evaluate lim  . Ans: 1 x 0 x 1 Q-23 Prove that lim 1  x 2  log 1 x   e. x 1 1 Prove that lim  cos ecx  tan 2 x Q-24  e2.  x 2 SESH1110 – Calculus 16 | P a g e  Module 4: Partial Derivatives (6 Hours) SESH1110 – Calculus 17 | P a g e