MATH1013 Calculus I, 2013-14 Fall Week 11 Tenth Tutorial Worksheet: Graphing Functions, Optimization Problems, Linear Approximation and Differentials (L10, T11A) 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 263, Q. 11) Sketch the graph of f (x) = x4 − 6x2 . Identify local extrema, inflection points, and x- and y-intercepts when they exist. 2. (Demonstration) (page 263, Q. 15) Sketch the graph of f (x) = points, and x- and y-intercepts when they exist. x2 . Identify local extrema, inflection x−2 3. (Demonstration) (page 263, Q. 47) Use the following f 0 (x) to determine the local maximum/minimum x+2 of f and intervals of increase/decrease: f 0 (x) = 2 . x (x − 6) 4. (Demonstration) (p.270, 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)? 5. (Demonstration) (p.270, 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? √ 6. (Demonstration) (page 282, Q. 23) Use linear approximation to estimate 146. 7. (Demonstration) (page 282, 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). 8. (Class work) (page 263, Q. 12) Sketch the graph of f (x) = 2x6 − 3x4 . Answer 9. (Class work) (p. 238, Q. 47) (page 263, Q. 16) Sketch the graph of f (x) = Answer 10. (Class work) (page 263, Q. 30) Sketch the graph of f (x) = x2 ln x. 1 x2 . x2 − 4 Answer 11. (Class work) page 263, Q. 46) Use the following f 0 (x) to determine the local maximum/minimum of x−1 f and intervals of increase/decrease: f 0 (x) = . (x − 2)2 (x − 3) Answer 12. (Class work) (p.270, 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 13. (Class work)(p.271, 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 (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 14. (Class work) (page 282, 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 2