SUPER SECRET NUMBER PUZZLE IMPLICIT DIFFERENTIATION Find the answer to each question. Write your answer in the answer blank to the left. Add up all of your answers. Check to see if your number matches the super secret number! 1. Find the value of the derivative of x3 + xy - 2 = 0 at the point (1, 1). 2. Find the value of the derivative of x3 + y3 = 6xy at (3, 3). 3. What is the y-intercept of the tangent line to x3 + y3 = 8 at the point (0, 2)? 4. Find the value of the second derivative of x = y? at the point where y= 5. Find the slope of the normal line to the curve x2 - xy + y? =7 at the point where x = 1. (There are two possible answers - write the smaller answer on the line.) 6. The slope of the tangent to the curve x3 - 6xy - ky3 = a is -1 at the point (0, 1). Find the sum of k and a. 7. Consider the curve x2 + 4y2 = 7+ 3xy. There is a horizontal tangent line to this curve at a point where x = 3. What is the y-coordinate of this point? 8. Consider the curve x3+ 2x + y+ + 4y = 5. Find the sum of the x-coordinates of the two points on the curve where the line tangent to the curve is vertical. THE SUPER SECRET NUMBER ISCalculus AB 2.7 B Worksheet Name: 1 . A kite is flying at a height of 40 ft. A child is flying the kite so that the kite is moving horizontally at a rate of 3 ft/sec. If the string is taut, at what rate is the string being let out when the length of the string is 50 ft? 2. Every day a flight from San Diego to Dallas flies directly over my home at a constant altitude of 4 miles. If I assume that the plane is flying at a constant speed of 400 miles/hour, at what rate is the angle of elevation of my line of sight changing with respect to time when the horizontal distance between the approaching plane and my location is exactly 3 miles? 3 . An automobile traveling at a rate of 30 ft/sec is approaching an intersection from the west. When the automobile is 60 feet from the intersection, a truck traveling south at the rate of 40 fu/sec is 80 feet south of the intersection. How fast are the automobile and truck separating? 4. A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 fu/sec. a) How fast is the top of the ladder moving down the wall when the base of the ladder is 7 feet from the wall? b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. c) Find the rate at which the angle between the ladder and the wall of the house is changing when th base of the ladder is 7 feet from the wall