NCEA Level 3 Calculus (91578) 2015 — page 1 of 6 Assessment Schedule – 2015 Calculus: Apply differentiation methods in solving problems (91578) Evidence Q1 Expected Coverage (a) 30sec 2 (5x) (b) 2 dy = 3 4x − 3x 2 ( 4 − 6x ) dx dy At x = 1, = 3 × 1× −2 = −6 dx (c) ) 2 x + ( 1)2 f ′(x) > 0 ⇒ 8 > ( x + 1)2 > Merit r A correct expression for the derivative. ( f ′( x) = 8 − Achievement u Correct solution with correct derivative. Correct derivative. Correct solution with correct derivative. Correct derivative. Correct solution with correct derivative. 2 ( x + 1)2 1 4 1 −1 or x + 1 < 2 2 −1 −3 x> or x < 2 2 Either x + 1 > (d) f ′(x) = x(x − 5) − (x + 4)(2x − 5) 2 x2 ( x − 5) f ′(x) = 0 ⇒ x(x − 5) − (x + 4)(2x − 5) = 0 ( ) x 2 − 5x − 2x 2 + 3x − 20 = 0 −x − 8x + 20 = 0 2 x 2 + 8x − 20 = 0 (x + 10)(x − 2) = 0 x = −10 or + 2 Excellence t NCEA Level 3 Calculus (91578) 2015 — page 2 of 6 (e) Let V = volume (m3) S = slant height (m) h = height (m) r = radius (m) cos 30 = Valid statement of the relationship between rates. dS dV or dr dr correct. r S Correct solution with correct derivatives. r cos 30 dS 1 = dr cos 30 h tan 30 = r h = r tan 30 1 V = π r 2h 3 1 3 = π r tan 30 3 dV = π r 2 tan 30 dr dS dS dr dV = × × dt dr dV dt 1 1 = × ×2 cos 30 π r 2 tan 30 When r = 10 m, dS 1 1 = × ×2 dt cos 30 π10 2 × tan 30 = 0.01273 m / minute S= NØ N1 N2 A3 A4 M5 M6 E7 E8 No response; no relevant evidence. ONE answer demonstrating limited knowledge of differentiation techniques. 1u 2u 3u 1r 2r 1t with minor error(s). 1t NCEA Level 3 Calculus (91578) 2015 — page 3 of 6 Q2 Expected Coverage (a) 1 x − 3x 2 5 (b) dy 16 = 1+ 2 dx x dy At x = 4, =2 dx ( (c)(i) 1. x = 1 2. x = –1, 1, 2 3. –1 < x < 1 (ii) 3 (iii) Does not exist. (d) Excellence t Correct solution with correct derivative. −1 2 Two correct answers. dx correct. dL x+L L = 5 1.5 1.5x + 1.5L = 5L 1.5x = 3.5L 7L x= 3 dx 7 = dL 3 dx =2 dt dL dL dx = × dt dx dt 3 = ×2 7 6 = = 0.857 m s –1 7 Merit r A correct expression for the derivative. −4 5 ) ⋅ (1− 6x ) ∴Gradient of normal = Achievement u Four correct answers. Correct solution with correct derivatives. (Units not required.) NCEA Level 3 Calculus (91578) 2015 — page 4 of 6 (e) Depth of water = x h = x + 20 1 1 V = h 3 − 20 3 3 3 1 1 3 = ( x + 20 ) − 20 3 3 3 dV 2 = ( x + 20 ) dx A = ( x + 20 ) Correct dV dx OR Correct dV dx AND dA dx dA dx Correct solution. 2 dA = 2 ( x + 20 ) dx dV = 3000 dt dA dA dx dV = × × dt dx dV dt 1 = 2 ( x + 20 ) × × 3000 ( x + 20 )2 When x = 15 dA 1 = 2 × 35 × 2 × 3000 = 171.4 cm 2 min −1 dt 35 NØ N1 N2 A3 A4 M5 M6 E7 E8 No response; no relevant evidence. ONE answer demonstrating limited knowledge of differentiation techniques. 1u 2u 3u 1r 2r 1t with minor error(s). 1t NCEA Level 3 Calculus (91578) 2015 — page 5 of 6 Q3 (a) Expected Coverage f ′(x) = 5 10 ×2= 2x − 3 2x − 3 10 =4 2x − 3 8x − 12 = 10 Achievement u Merit r Correct solution with correct derivative. x = 2.75 (b) f ′(x) = = e 3x − x.3e 3x (e ) 3x 2 1− 3x e 3x f ′(x) = 0 ⇒ x = (c) Correct solution with correct derivative. 1 3 dx = −3sint dt dy = 3cos 3t dt dy 3cos 3t − cos 3t = = dx −3sint sint ⎛ 3π ⎞ − cos ⎜ ⎟ ⎝ 4⎠ π dy At t = , = =1 4 dx ⎛ π⎞ sin ⎜ ⎟ ⎝ 4⎠ Correct and dx dt Correct solution with correct derivatives. dx dt Parts (i) and (ii) both correct. dy dt ∴Gradient of normal = −1 (d)(i) dx = −Ak sin kt + Bk cos kt dt d2 x = −Ak 2 cos kt − Bk 2 sin kt dt 2 = −k 2 ( A cos kt + Bsin kt ) = −k 2 x (ii) x(0) = 0 ⇒ A cos 0 + Bsin 0 = 0 A=0 v(0) = 2k ⇒ 2k = −Ak sin(0) + Bk cos(0) B= 2 Correct d2 x dt 2 Consistent with dx incorrect dt Or Excellence t NCEA Level 3 Calculus (91578) 2015 — page 6 of 6 (e) Differentiate corretedly related but incorrect expression for w. cos A = Correct dw dA 2 f 2 cos A w sin A = 5− f w= (5 − f )sin A f= 2 ⎞ ⎛ = ⎜5− ⎟ sin A ⎝ cos A ⎠ = 5sin A − 2 tan A dw = 5 cos A − 2sec 2 A dA dw = 0 ⇒ 5 cos A − 2sec 2 A = 0 dA 5 cos 3 A − 2 = 0 2 cos 3 A = 5 A = 42.5° Correct solution. w = 1.55 m NØ N1 N2 A3 A4 M5 M6 E7 E8 No response; no relevant evidence. ONE answer demonstrating limited knowledge of differentiation techniques. 1u 2u 3u 1r 2r 1t with minor error(s). 1t Cut Scores Not Achieved Achievement Achievement with Merit Achievement with Excellence 0–7 8 – 12 13 – 18 19 – 24