3 Complete this assignment after you have finished Unit 4, and submit your work to your tutor for grading. Total points: 100 Weight: 10% 1 (9 points) An Earth-observing satellite can see only a portion of the Earth's surface. The satellite has horizon sensors that can direct the angle shown in the accompanying figure. Let be the r radius of the Earth (assumed spherical) and h the distance of the satellite from the Earth's surface. a Show that b Using h r(csc 1) r 6378 km, . find the average rate of change of the distance from the satellite to the surface of the Earth, when c 2 At what rate is the distance changes from h /4 changing when (6 points) Consider the graph of the function g to / 3. What are the units? / 6? shown below. Mathematics 265: Introduction to Calculus I (Rev. C13)Assignment 3 1 a What is the domain of the function g? b Where is the function continuous? c Identify on the graph the local maximum. d Identify on the graph the local minimum. e Does it have an absolute maximum value? Explain. f Does it have an absolute minimum value? Explain. Mathematics 265: Introduction to Calculus I (Rev. C13)Assignment 3 2 3 (8 points) Sketch the graph of the function if 4 x2 if f x 2 if x 3 4 f and its derivative function f where x 2 2 x 3 x3 (16 points) In each of the cases below, give the indicated derivative. You may not need to simplify your answers. a b c d x cos x 3 dx d2 2 x tan x|x x2 d sin x cos x dx x3 d If f (0) 1, f (0) 2, g(0) 0 and g(0) 1, find d f ( x) x2g( x) dx f ( x) g( x) x0 5 (8 points) A softball diamond is a square whose sides are 18 m long. Suppose that a player running from first to second base has a speed of 7.5 m/s at the instant she is 3 m from second base. At what rate is the player's distance from home plate changing at that instant? Hint. Draw a diagram and locate the variables that change with respect to time. 6 (5 points) Let x 1 f ( x) x a Find the equation of the tangent line of the function f at the point Mathematics 265: Introduction to Calculus I (Rev. C13)Assignment 3 (4, f (4)) 3 b Use differentials to estimate the value of 7 f (4.02). (6 points) Use implicit differentiation to prove that the curve x2 y2 (2x2 2y2 x)2 (see exercise 27 of page 187 of the textbook) has a vertical tangent line at the point (1,0). 8 (8 points) One side of a right triangle is known to be 25 cm exactly. The angle opposite to this side is measured to be with a possible error of 60, 0.5. a Use differentials to estimate the errors in the adjacent side and the hypothenuse. b Estimate the percentage errors in the adjacent side and hypothenuse. Mathematics 265: Introduction to Calculus I (Rev. C13)Assignment 3 4 9 (8 points) Sketch the graph of one and only one function x $f$ which satisfies all the conditions listed below. a b f ( x) f ( x) lim f ( x) x 4 c lim f ( x) x 4 d lim f ( x) 2 x e f ( x) 0 on the interval (0,4) (8 points) 10 Sketch the graph of the function f ( x) x2 4 x2 6 Clearly indicate each of the steps as listed on pages 232-234 of the textbook. (8 points) 11 The shoreline of a lake is a circle with diameter 3 km. Peter stands at point E and wants to reach the diametrically opposite point W. He intends to jog along the north shore to a point P and then swim the straight line distance to W. If he swims at a rate of 3 km/h and jogs at a rate of 24 km/h. How far should he jog in order to arrive at point W in the least amount of time? Mathematics 265: Introduction to Calculus I (Rev. C13)Assignment 3 5 (10 points) 12 a. Sketch the graphs of the curves a y sin x and y x2 showing their points of intersection. Use the Intermediate Value Theorem to identify an interval where the equation has a sin x x2 0 non-zero solution. b Use Newton's method to approximate the non-zero solution of the equation Mathematics 265: Introduction to Calculus I (Rev. C13)Assignment 3 sin x x2 0. 6