Mathematics 53 Second Long Exam
22 January 2007
This exam is good for 1 hour and 30 min only. Use only black or blue ballpen – no pencils. Use of calculators is not allowed. Do as indicated. Show all necessary solutions and BOX all final answers. God bless! 1. Give the equation of the normal line (slope-intercept form) to the curve x2 − 2xy − 2y 2 = 1 at the point (−3, −1). (4 pts) 2. Given f (x) =
ax2 + bx
,x ≤ 3
|9 − x2 |
,x > 3
Determine the values of a and b such that f is differentiable at x = 3.
(4 pts)
dy . Do not simplify. (4 pts @) dx h q 2 i4 x3 3 3 2 2 3 3 (a) y = cos sin(2x + 1) + [cot x + (x − 1) ] (c) y = sin tan 1 − 4x2 + 3x 3 x csc tan 2 √ (b) y = p (d) xy 2 = tan y + 3x2 + 3 108 sec(4x − 1) − 3x3
3. Find
4. Vryan threw a basketball vertically upward from the edge of the top of a building and its position is s(t) = 120 + 8t − 16t2 feet above the ground at the end of t seconds. (a) Give the height of the building and the initial velocity of the ball.
(3 pts)
(b) After how many seconds will the ball reach its maximum height?
(2 pts)
(c) Determine the speed of the ball when it reaches the ground.
(4 pts)
5. If Vryan’s basketball was measured and its radius was found to be 5 inches with a possible error of no more that 0.01 inch, what is the maximum possible error in its the volume? (4 pts) 6. Vryan decided to √ take his bike for a ride. At noon, he is 3 km west of another cyclist Maria. Maria is heading north at 2 kph. If Vryan is heading south with twice Maria’s rate, how fast is the distance between them changing at 2:00 PM? (5 pts) 7. When Vryan arrived home, he found out that oil is leaking from their inverted conical tank at the rate of 2π m3 /min. If the tank has a radius of 5 meters at the top and a depth of 20 meters, how fast is the oil level changing when it is 16 meters deep? How fast is the surface area of the oil changing at the same moment? (8 pts) TOTAL: 50 POINTS “If you would be a real seeker after truth, you must at least once in your life doubt, as far as possible, all things.” – Ren´e Descartes rtrelator0607