CALCULUS EXAM II Spring 2003 Name: Instructions: WRITE ALL WORK AND ALL ANSWERS ON THIS EXAM PAPER. For all questions, I reserve the right to apply the 'no work means no credit' policy, so make sure you always document your understanding and knowledge by showing your work carefully and clearly. There are twenty two questions, each worth 5 (five) points, of which you must do exactly twenty, and bonus questions collectively worth as much as 20 (twenty) points. You must clearly mark e.g., "DO NOT GRADE" the two questions (from #1 22) you want to skip, or they may be counted against you. In addition, it would be greatly appreciated if you could fit your work and your answers into the space provided. Best of luck... For question 1, consider the graph of f x, shown at below right. 1) Use the graph to estimate the instantaneous rate of change of f x at x = 1. 4 3 2 1-1 1 2 3 For question 2, consider the graph of gx, shown at below right. 2) Using the same set of axes where gx is graphed below, sketch the graph of g x. In the space below either show some supporting work or indicate briefly why you answered as you did. 3 2 1-1 1 2 3 4 5 6-2 -3 Page 1 of 13/Math 190 Calculus Exam 2/Spring 2003
3) Use the definition of the derivative, f x = lim h 0 f x = x 2 3x +2. f x+ h f x h, to find f x if 4) Find the equation of the line tangent to the curve of f x = 1 x 1+x at 2, 3. Give the equation in y = mx + b format. Page 2 of 13/Math 190 Calculus Exam 2/Spring 2003
For questions 5 9, take the derivative of the function. Simplify your answers reasonably no negative exponents allowed in final answers but don't go overboard or get crazy. 5) f x = x x 2 1 6) g x = sin 1 1 x 7) hx = xln x 3 Page 3 of 13/Math 190 Calculus Exam 2/Spring 2003
For questions 8 and 9, take the derivative of the function. Simplify your answers reasonably no negative exponents allowed in final answers but don't go overboard or get crazy. 8) f x = x 2 e x 9) gx = tan π x cos π x 10) You throw a stone from the top of a cliff. The height of the stone, t seconds after being thrown, is h t = 16t 2 +16t + 320 feet. What is the average velocity of the stone between t =1 and t =3 seconds? Page 4 of 13/Math 190 Calculus Exam 2/Spring 2003
11) A bomb is dropped from an airplane in level flight. The height of the bomb at time = t seconds after being dropped is h t = 16t 2 + 14400 feet. a) [2 points] When does the bomb reach the ground? b) [3 points] What is the velocity of the bomb upon impact (ignoring friction)? 12) If f x = sin 1 xe x, find f x. Page 5 of 13/Math 190 Calculus Exam 2/Spring 2003
13) If y x 4 y =6, find dy dx by implicit differentiation and use your answer to evaluate dy dx x, y = 4,9. 14) Where, if anywhere, is the tangent to f x = x2 +16 x horizontal? Use calculus or I won't read your answer. Page 6 of 13/Math 190 Calculus Exam 2/Spring 2003
15) A kite is flying 200 feet above the ground, at the end of a 250 foot string. If the girl flying the kite is letting out string at the constant rate of 2 feet/second, and the kite remains at an altitude of 200 feet, at how many feet per second is the horizontal distance (of the kite) from the girl increasing? 16) [Refer to problem 15 for the necessary information.] How fast is the angle θ (made by the kite string and the horizon) increasing when the kite is 150 feet from the girl (as measured along the ground)? Page 7 of 13/Math 190 Calculus Exam 2/Spring 2003
17) Two straight roads intersect at right angles in Newtonville. Car A is on one road moving toward the intersection at a speed of 50 mph. Car B is on the other road moving away from the intersection at 30 mph. How fast is the distance between the cars changing when Car A is 2 miles from the intersection and Car B is 4 miles from the intersection? [Hint: draw a picture before attempting analysis.] 18) Find the differential: d sin 2 πx Page 8 of 13/Math 190 Calculus Exam 2/Spring 2003
5 19) Estimate, using Newton's method, 30. CLEARLY show what equation you fed into your Newton's Method software. 20) Use logarithmic differentiation to find dy dx for y = x 1 x. 21) Find the second derivative of f x = xe 1 x. Page 9 of 13/Math 190 Calculus Exam 2/Spring 2003
22) Consider the following information: Company X sells research papers to students. If the price of a paper is p, then f p = q gives the quantity of papers sold at that price. Given that f 100 = 3000 and f 100 = 75, and that the revenue generated by the sale of these papers is Rp = pfp, use this information to estimate R 105. Page 10 of 13/Math 190 Calculus Exam 2/Spring 2003
BONUS SECTION: CHOOSE NO MORE THAN TWO! Otherwise, indicate clearly which ones you want graded, or I'll just grade the first two problems! 23) BONUS [10 points] Recall: Newton's method is based on the iterative formula x n +1 = x n f x n. f x n Use Newton's method (computing each iteration individually) to find an approximation to the solution of the equation e x = x +4. Start with x 0 =1, and fill in the table below with your results. 1 i 0 1 2 3 4 x i 1 These problems are much easier to work if, as shown in class, you first find a general formula which gives x n +1 in terms of x n, etc., before you start plugging in values. Page 11 of 13/Math 190 Calculus Exam 2/Spring 2003
24) BONUS [10 points] Use differentials to approximate the value of 32. Clearly show what you're using for f x, x, and so forth. 25) BONUS [10 points] If 3+e xy = x ln y 2, find dy dx at the point 0, e. Page 12 of 13/Math 190 Calculus Exam 2/Spring 2003
26) BONUS [10 points] Consider the graphs of the functions below. If it is known that each collection of four functions pictured below contains a function and its derivative, which of the graphs is the derivative of the indicated function? In each case, support your answer/explain briefly, or NO CREDIT will be awarded. a) 5 4 3 2 1 1 2 3 4 5 4 3 C 2 B 1-1 -2-3 -4 fhxl -1-2 -3 A -4 1 2 3 4 5-5 b) 5-5-4-3-2-1 1 2 3 4 5 4 ghxl 4 3 A 3 2 2 1 1-1 -2-3 -4-5-4-3-2-1 1 2 3 4 5 C B -1-2 -3-4 -5 Page 13 of 13/Math 190 Calculus Exam 2/Spring 2003