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THEME 3.1 - 3.2: DIFFERENTIABILITY AT A POINT DUE DATE: MARCH 24, 2026 This worksheet. will be based upon the Lecture videos 1Theme 3-Part 1 to 4_Theme 3_Part 4. There is a total of 6 questions. Total marks: 64. 1, DERIVATIVE AS SLOPE OF TANGENT LINE Watch the video 1 Theme 3 Part 1 before attempting. Question 1.[2, 5 ; 2; 2,5 marks] Find ihe equation of the tangent and nor- mal line to the cur at (8,2) as follows (a) Show that the slope of tangent line is 4; at x = 8 as follows: (i) Prove that the gradient of the secant line h away from 2 — 8 is given by YBTh-2 A (ii) Prove that the limit as A — 0 of the gradient of the secant. line A away [rom x= lis: 1 12° (b) Using (a) find the equation of the normal and tangent line. (c) Use the definition of the derivative to calculate the derivative of function y — y/a as follows: (i) Prove that the gradient of the nt linc h away from a fixed x is given by Veth— h (ii) Prove that the limit as h — 0 of the gradient of the secant line h away from x is: L 3L Question 2.[3, 3 marks] Show (hat if f(#) = lanaa: then the derivative f’(i) = asec? ax ax follows: (i) Prove that the gradient of the secant line h away from a fixed a is given by tana(e+h)—tan(aw) — tan(ah)(1 + tan? (ax) h h(L tan(aa)tan(ah))” (ii) Prove that the limit as h > 0) of the gradiont of the secant line h away from x is: ase#(az). 1 TUEMIS 3.1 - 3.2: DIFFERENTIABILITY AY A POINT DUR DATE: MARCH 24, 2026 2. DIFFERENTIATION RULES Watch the video 2-Theme 3Part 2 before attempting. Question 3. [2, 4; 2, 4;2 , 2 marks] Use the appropriate differentiation rule to cvaluate the first, f’ and second f” derivative of . 1 (@) fl) = 9 (ii) g(a) = Va? +0 si (iii) h(a) = Question 4. [1, 1, 4; 1, 1, 5 marks] Use mathematical induction to prove for any positive integer n: (a) Prove that 7" — 1 is divisible by 6 whore n = 1,2, as follows .. is a positive integer (i) Prove thal. the statement is true for the case n= 1. (ii) State the assumption that the statement is truc for the n = k-th odd num- ber. (ii) Only assuming that is true, show that the statement is true for the n — k+1- th odd number. (b) Consider the function f(z) = «2 where n is a positive integer. Use mathe- matical induction to prove that (1) for n = 1,2,3......... as follows: (i) Prove that (1) is true or is valid for the n = L-th case. (ii) State the assumption that the statement (1) is true for the A’th case that isn=h. (iii) Onl k+1-the assuming that (ii) is truc, show that the statement (1) is true for the thal is, n = (k +1). 3. NOn-DIPFERENTIABLLIVY: VERVICAL TANGENT Watch the video 3'Theme 3Part 3 before attempting. Question 5. [3, 2, 2, 1 marks] Consider the the function y= f(t) = 08 — er. at the point z = 0 (i) Prove that f(x) is continuous at x = 0. 2 rhe) C) | ‘2 aocaat y- ~ 7 op _ +) — el ,2\ : / -_ w~ O (@W-k ; Toe ugh oft Tk (gly + 2 thk 8 O fie Toad “ os (x ys 2-*fan “4 ~ se © = ee - ie 4a hy he iY, le) , te tang nt Oni Cove onyro dn. P yt ] Ss i no eZ neko ae ia) Q) Newwh De ok Blt 2) feo whey 12° qct = -\% 2S “2 (a8) +2, (1) n-¥ We POn, SRY Qe Cob, ir Cd) mee) ne ce ‘ya = ) () Q2 _ J an (a, ae alata) Lo A A(t (4, on am) Olu, be ler tah)) a lft / ¥ (x, tan an) outs a tonstank ™ cetnk = ra ~ baa (ax + al ) — taner — ben tr eben oh ¥ Sans _~ — bun an \ mm bee nt bow ef h kaa oa + ken ad —_ tan aM (\- ben ae ben ah) k (\- ban gm ban ob) CQ) aa deo kD 4 ten | “ is | — bean en 7 |e. ay. (we GD Sey - he | = baneca len ak § 4 ; ( hewn yin ahs re Gn a = at _ 7" (aL) (\) aT = & = ee + \ = sec” n | Noe | bbe Tan = ase aH o eS ee VN ~ 6 wm how ns 1,2,3,. ot (3) \oal wlan leer me 4 = 4) 126+6-4- oe co Nai. Looe Vaan : Aseuen Vx dw ark: Seokave, BAe bm Gomme ty oa Teel Nad de ate: Panye (<0 ne beat = = Aw Py. qtye-s biqaaiv ——- ) ~ GET OQ) fri = 6 ~ die - = E(w) OD Wh %) =) (de ero ie w = a ” Aw ne L,% be \ J vd Lev NM= A Lom lye (y)-) - da * - ah > 5% Q) Wada rhe By poten » Avenue Wee Qe me & bod y i 2” s “L, Cal | C1) Now yt J hy Cher) Ue ) -| * nan. aa 2 o Pal a a \ feo gy Wy CO C) r 2 @ ~ - l Aa An very “, ath = % 4% + dn h-) \, v Ware \ = \latrtra (‘x ) 2 Voy bgp ar eo “at * a “eo ; .) Park a wf fat (2 | — () wv 7 | eee: Dye) (2) ABS? = L of t\ * a ” a la y\) 2 _@ 4 Ws (2} ( ie 7 2.) ; pe 
