1. Use the Intermediate Value Theorem to show that: () vx - 2 = -vi + 1 has a solution. (ii) f(t) = 14 - 3t takes on the value 27 for some t E (1, 2). 2. Determine the derivatives of the following functions using the definition of the derivative. () f(x) =14 (ii) g(x) = VI 3. Determine the derivatives of the following functions: () f(x) = csc(x) (ii) g(x) = V2r cos(x) (iii) h(x) = tan(3x) sin(r) (iv) p(x) = cos(x2) + cos? (x) (v) y(x) = - 3.r2 + 4 5-23 (vi) w(x) = (x3 + tan' (iz) ) 4. Use the definition of the derivative of a function at a point to establish the existence/ non-existence of the derivative of the following function, at r = 0 and r = 1. f (x) = 1 x2 +1 , ifx 21 |2x| , if x