1. Determine the equations of both tangent and normal to the curve: y(3x - y') = 16 at (3,2) 2. Given Ax We=(1+ x , - k ). What is the value of k such that y is perpendicular to the curve ya = 3x + 1 atx = 1? 3. A spherical balloon is filled with air at a rate of 1 m /s. Compute the time rate of change of the surface area of the balloon at the instant the balloon's volume is 113.1 m. 4. Water is pouring at a varying rate into a swimming pool that is initially empty. After t hours, there are + 12 liters in the pool. At what rate is the water pouring into the pool when t = 9h? 5. The radius of a snowball rolling down a hill is increasing at a rate of 20 cm/min. How fast is its volume increasing when its diameter is Im? 6. Find the two positive numbers whose sum is 8 if the product of one number by the cube of the other number is to be maximum. 7. The cost 'C' of an item is a function of the quantity 'x' of the item: C(x] = x2 - 4000% + 50 Find the quantity for which the cost is minimum. A peach grower estimates that if he picks his crop now, he will obtain 1000 lugs of peaches, which he can sell at $1.00 per lug. However, he estimates that his crop will increase by an addition 60 lugs of peaches for each week that he delays picking but the price will drop at a rate of $0.025 per lug per week. In addition, he will experience a spoilage rate of approximately 10 lugs for each week he delays. To maximize his revenue, how many weeks should he wait before picking the peaches? What is the maximum value of the following function on the interval x $ 07 Y E :283+ 12x2- 30x + 109. What is the maximum value of the following function on the interval x s 07 Y = 2x3+ 12x2 - 30x + 10 10. Find the two positive numbers whose sum is A if the product of one number by the cube of the other number is to be maximum. 11. The cost 'C' of an item is a function of the quantity 'x' of the item: C(x] = x2 - 4000x + 50. Find the quantity for which the cost is minimum. 12. A peach grower estimates that if he picks his crop now, he will obtain 1000 lugs of peaches, which he can sell at $1.00 per lug. However, he estimates that his crop will increase by an additional 60 lugs of peaches for each week that he delays picking, but the price will drop at a rate of $0.025 per lug per week. In addition, he will experience a spoilage rate of approximately 10 lugs for each week he delays. To maximize his revenue, how many weeks should he wait before picking the peaches? 13. What are the minimum and maximum values respectively of the equation f(x] = 5x3- 2x3 + 1 on the interval [-3,3]7 14. What is the standard form of the equation of the line tangent to a circle centered at the origin with a radius of 5 at the point (4,3)? 15. At which value of y does the relative minimum occur for the curve y = 2x3 - x2 - 24x + 25? 16. At what values of y does the inflection point occur for the curve y=3x3 - 3x2 - 24x + 25
