Partial derivative

Partial derivative


Page 1 of 4 FACULTY OF HUMAN SCIENCES DEPARTMENT OF EDUCATION SUBJECT : MATHEMATICS 1B (FET) CODE : HEMAYIB LECTURER : EPT TEBO MODERATOR : W BARROS ASSESSMENT : TEST 1 SUBMISSION : ON VUTELA ONLY DURATION : 120 MIN (INCLUSIVE OF SUBMISSION TIME) START & END : 8:00 AM – 10:00 AM TOTAL MARKS : 40 ISSUE DATE : 18TH SEPTEMBER 2021 INSTRUCTIONS: 1. Answer all questions. 2. Question can be answered in any order, but Subsections must be kept together. 3. Non-Programmable calculators may be used. 4. When doing calculations, give the final answer to at least three decimal places. 5. NO COPYWORK FROM ANY SOURSE IS ALLOW (NOT AN OPEN BOOK ASSESSMENT) This question paper consists of 4 - typed pages, including the Front page. ***DO NOT TURN THIS PAGE BEFORE PERMISSION IS GRANTED*** Page 2 of 4 Question 1 [marks 12] 1.1. Find the equation of the tangent plane to ? = ?√?2 + ?2 + ?3 ?? (−4, 3) (6) 1.2. Determine ?? ?? of ?? = 1+tanh ? 1−tanh ? (6) Question 2 [marks 8] 2. If ? = sin ? and ? = sin 2? , prove that (1 − ?2) ( ?? ?? ) 2 = 4(1 − ?2) (8) Question 3 [marks 10] 3. A box with an open top is to be constructed from a square piece of cardboard, 3m wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (10) Question 4 [marks 10] 4. The Louvre Museum in Paris features an inverted square pyramid with a height of 10 m. The side of the square is 13 m. Supervillains decide to fill the pyramid with water and do so at a rate of 10 m3 per second. How quickly is the water level rising when it reaches the top? (10) Page 3 of 4 FORMULA SHEET FOR HEMAY1A 1.) ( ) ( )  ( )  1; 1 1 1 −+= + + ncxfdxxfxf n n n 2.) ( ) ( ) ( ) cxfdx xf xf +=   ln (3.) ( ) ( ) ( ) c a a dxaxf xf xf += ln 4.) ( ) ( ) ( ) cedxexf xf xf += (5.) ( ) ( ) ( ) cxfdxxfxf +−= cossin 6.) ( ) ( ) ( ) cxfdxxfxf += sincos (7.) ( ) ( ) ( )  cxfdxxfxf += seclntan 8.) ( ) ( ) ( )  cxfdxxfxf += sinlncot 9.) ( ) ( ) ( ) ( )  cxfxfdxxfxf ++= tanseclnsec 10.) ( ) ( ) ( ) ( )  cxfxfdxxfxf +−= cotcoseclncosec 11.) ( ) ( ) ( ) cxfdxxfxf += tansec 2 12.) ( ) ( ) ( ) cxfdxxfxf +−= cotcosec 2 13.) ( ) ( ) ( ) ( ) cxfdxxfxfxf += sectansec 14.) ( ) ( ) ( ) ( ) cxfdxxfxfxf +−= coseccotcosec 15.) ( ) ( ) ( ) cxfdxxfxf += coshsinh 16.) ( ) ( ) ( ) cxfdxxfxf += sinhcosh 17.) ( ) ( ) ( )  cxfdxxfxf += coshlntanh 18.) ( ) ( ) ( )  cxfdxxfxf += sinhlncoth 19.) ( ) ( ) ( ) cxfdxxfxf += tanhsech 2 20.) ( ) ( ) ( ) cxfdxxfxf +−= cothcosech 2 21.) ( ) ( ) ( ) ( ) cxfdxxfxfxf +−= sechtanhhsec 22.) ( ) ( ) ( ) ( ) cxfdxxfxfxf +−= cosechcothcosech 23.) ( ) ( )  ( ) c a xf a dx axf xf += +  −  1 22 tan 1 24.) ( ) ( )  ( ) ( ) ( ) c axf axf a c a xf a dx axf xf + + − +−= −  −  ln2 1 coth 1 1 22 or 25.) ( ) ( )  ( ) ( ) ( ) c xfa xfa a c a xf a dx xfa xf + − + += −  −  ln2 1 tanh 1 1 22 or Page 4 of 4 26.) ( ) ( )  ( ) ( ) ( ) c a xf a xf c a xf dx axf xf +         +      ++= +  −  1lnsinh 2 1 22 or 27.) ( ) ( )  ( ) ( ) ( ) c a xf a xf c a xf dx axf xf +         −      ++= −  −  1lncosh 2 1 22 or 28.) ( ) ( )  ( ) ( ) c a xf c a xf dx xfa xf +−+= −  −−  11 22 cossin or 29.) ( ) ( )  ( ) ( )  ( ) c a xf aaxfxfdxaxfxf +++=+ − 12 2 122 2 122 sinh 30.) ( ) ( )  ( ) ( )  ( ) c a xf aaxfxfdxaxfxf +−−=− − 12 2 122 2 122 cosh 31.) ( ) ( )  ( ) ( )  ( ) c a xf axfaxfdxxfaxf ++−=− − 12 2 122 2 122 sin 32.) ∫ ? ?? = ?? − ∫ ? ?? USEFUL IDENTITIES 33.) 1cossin 22 =+ AA 38) ( )AA 2cos1sin 21 2 −= 34.) AA 22 sectan1 =+ 39) ( )AA 2cos1cos 21 2 += 35.) AA 22 coseccot1 =+ 40) ( ) ( ) BABABA ++−= sinsincossin 2 1 36.) AAA cossin22sin = 41) ( ) ( ) BABABA +−−= coscossinsin 2 1 AAA 22 sincos2cos −= 42) ( ) ( ) BABABA ++−= coscoscoscos 2 1