MATH1013 Calculus I, 2013-14 Spring Tutorial Worksheet 6: Derivatives (T1A) Name: ID No.: Tutorial Section: Complete at least TWO questions from the following questions. (Solution of this worksheet will be available at the course website the week after.) 1. (Demonstration) (page 138, Q. 25) Compute the derivative of √ x + 3 from the definition of a deriva- tive, and then use the result to find the equation of tangent to √ x + 3 at x = 1. 2. (Demonstration) (page 139, Q. 51) Plot a graph of f 0 (x) against the given graph of f (x) in the following figure: 3. (Demonstration) (page 139, Q. 61) Use the points A, B, C, D, E in the following graphs of to answer • At which points is the gradient of the curve negative? • At which points is the gradient of the curve positive? • Using A - E to list the gradients in decreasing order. 4. (Demonstration) Determine if the function f (x)= x sin 1 x , if x 6= 0; 0, if x =0. is continuous and if it has a derivative at x = 0. 5. (Demonstration) (page 188, Q. 77) Assume f and g are differentiable on their domains with h(x)= f (g(x)). Suppose the equation of the tangent line of g at the point (4, 7) is y =3x - 5 and tangent line of f at the point (7, 9) is y = -2x + 23. (i) Calculate h(4) and h 0 (4), and (ii) to determine the tangent line to h at x = 4. 6. (Demonstration) (page 169, Q. 81) Identify which f and x = a so that its derivative f 0 (a) is given by the limit: lim h→0 sin( π 6 + h) - 1 2 h . Also evaluate this f 0 (a). 7. (Demonstration) (p. 187, Q. 44, 49) Given that (e x ) 0 = e x . Differentiate sin(sin(e x )) and tan(e √ 3x ). 8. (Demonstration) (p.168, Q. 75) Prove that (cos x) 0 = - sin x, 9. (Class work) (page 138, Q. 24) Compute the derivative of √ x - 1 from the definition of a derivative, and then use the result to find the equation of tangent to √ x - 1 at x = 2. 1