MATH1013 Calculus I, 2013-14 Spring Tutorial Worksheet 10: Graph sketching and Integrations Name: (T1B) ID No.: Tutorial Section: Complete at least THREE questions from the following questions. (Solution of this worksheet will be available at the course website the week after.) 1. (Demonstration) (p. 289, Q. 11) Sketch the graph of f (x) = x4 −6x2 . Identify local extrema, infection points, and x− and y−intercepts when they exist. 2. (Demonstration) (page 289, Q. 17) Sketch the graph of f (x) = inflection points, and x− and y−intercepts when they exist. 3x . Identify local extrema, −1 x2 3. (Demonstration) (p.296, Q. 19) Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)? 4. (Demonstration) (p.296, Q. 25) A rectangle is constructed with its base on the diameter of a semicircle with radius 5 and with its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area? √ 5. (Demonstration) (page 308, Q. 23) Use linear approximation to estimate 146. 6. (Demonstration)(page 308, Q. 33) Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 cm to h = 11.9 cm (V = πr2 h). Z Z √ 3 3 5 7. (Demonstration)(page 346, Q. 33, 34) Compute r2 dr. + 2 − dx, and x4 x2 8. (Demonstration)(page 347, Q. 89) Let a(t) = −32 denote the acceleration function of an object moving on a line such that its initial velocity and positive are given by v(0) = 20, and s(0) = 0 respectively. Find the position function s(t) as a function of t. 9. (Class work) (p. 289, Q. 12) Sketch the graph of f (x) = 2x6 − 3x4 . Answer 1 10. (Class work) (page 289, Q. 19) Sketch the graph of f (x) = points, and x− and y−intercepts when they exist. 2x − 5 . Identify local extrema, inflection x2 − 9 Answer 11. (Class work) (p.296, Q. 20) Find the point P on the line y = x2 that is closest to the point (18, 0). What is the least distance between P and (18, 0)? Answer 2 12. (Class work) (p.297, Q. 30) (a) A rectangle is constructed with one side on the positive x−axis, one side on the positive y−axis, and the vertex opposite the origin on the line y = 10 − 2x. What dimensions maximize the area of the rectangle, and the corresponding area? (b) Is it possible to construct a rectangle with a larger area than that found in part (a) by placing one side of the rectangle on the line y = 10 − 2x, and the two vertices not on that line on the positive x− and y−axes? Find the dimensions of the rectangle constructed this way. Answer 13. (Class work) (page 308, Q. 34) Approximate the change in the volume of a right circular cone of fixed hight h = 4 m when its radius increases from r = 3 m to h = 3 : 05 m (V = πr2 h/3). Answer Z 14. (Class work) (page 346, Q. 32, 34) Compute 4z 1/3 − z −1/3 dz, and Z 4x4 − 6x2 dx. x Answer 15. (Class work) (page 347, Q. 90) Let a(t) = 4 denote the acceleration function of an object moving on a line such that its initial velocity and positive are given by v(0) = −3, and s(0) = 2 respectively. Find the position function s(t) as a function of t. Answer 3