Sample Test-2|||| Calculus I I am listing some problems and these are taken directly from the book and have also noted the section from which the problems are taken. 1. Find the derivative using the denition (only one function) (from page 127): f(x) = px g(t) = 1 t2 f(x) = p3x + 1 2 Find the derivative using rules of dierentiation (from section 3.2): f(x) = e3 f(x) = x100 f(x) = px 4 f(x) = 10×4 ?? 32x + e2 f(x) = 6px ?? 4×2 + 9 f(x) = x2??x+1 x 3 [15] Find derivatives using product and quotient rules (from section 3.3) (A) f(x) = x2ex (B) g(x) = x2+1 x2??4 (C)h(x) = x+1 x2ex (D)h(x) = (5×7 + 5x + 1)(6×3 + 3×2 + 5×1) (E)h(x) = tan x 4 Prove that (a) d dx(cos x) = ??sin x (b) d dx (sin x) = cos x Use limx!0 sin x x = 1 and limx!0 1??cos x x = 0 from section 3.4 5. Use chain rule to nd derivatives (from section 3.6) (a) y = (5×2 + 11x)20 (b) y = px2 + 1 (c) y = cos(5x + 1) 1
(d) y = esin t (e) y = px + px (f) y = sin3(cos(3x)) 6. Use implicit dierentiation to nd dy dx(from section 3.7) (a) x2 + xy ?? y3 = 7 also nd the equation of the tangent line at (3; 2) (b) 2(x + y) 13 = y also nd the equation (c)(xy + 1)3 = x ?? y2 + 8 (d) px4 + y2 = 5x + 2y3 7(fron section 3.8)Use logarithmic dierentiation to nd dy dx where y = (x+1)3=2(x??4)5=2 (5x+3)2=3 2