Assessment - Curve Sketching Achievement Level Outcome: Curve Sketching WX Name: Date: For the given function you must determine: x & y - intercepts, Critical Points & Points of Inflection. Show Intervals of Increase & Decrease in a table and Concavity in a separate table. Show all of your work in an organized way and use the points found to sketch the graph. For any points that need to be written as a decimal, round to 2 decimal places. f(x) = 2x - 5x + 4x - 1Assessment Optimization and Applications Achievement Level Outcome 10: Applications of Derivatives - Summative W 1. A rectangular piece of land is to be fenced in using two kinds offencing. Two opposite sides will he fenced using standard fencing that costs SS/m, while the other two sides will require heavy-duty fencing that costs S9/m. What are the dimensions of the rectangular lot of greatest area that can be fenced in for a cost of $9000? 2. Calculate the derivative using Implicit Differentiation: xy+y3=x2+x+1 W 3. Two poles, one 5 meters tall and one 15 meters tall, are 20 meters apart. A length of wire is attached to the top of each pole and it is also staked to the ground somewhere between the two poles. Where should the wire be staked so that the minimum amount of wire is used? 4. For the given function you must determine: x at y - intercepts, Critical Points (you do not need to solve for Points of Inflections, meaning you only need to complete the 1St derivative to find local max/min points]. Show Intervals of Increase & Decrease in a table (you do not need to show concavity]. Show all of your work in an organized way and use the points found to sketch the graph. For any points that need to be written as a decimal, round to 2 decimal places. Use this video, if needed, to help with using the derivative to find local max/min points. htt 5: www. outubecom watch?v: a zs67VZ Assessment Optimization Problems Achievement Level Outcome 9: Curve Sketching Outcome 10: Applications of Derivatives Part 1: For the given function you must determine: 1: 8: y - intercepts, Critical Points & Points of Inflection. Show Intervals of Increase & Decrease in a table and Concavity in a separate table. Show all of your work in an organized way and use the points found to sketch the graph. For any points that need to be written as a decimal, round to 2 decimal places. f(x) = 2135x2+4x1 Part 2: Choose TWO Problems to submit a FULL solution (answers are at the bottom so it is essential that your solution shows all steps) 1. Determine the area of the largest rectangle that can be inscribed in a circle of radius 4. 2. Find the dimensions of the rectangle of largest area which can be inscribed in the closed region bounded by the x-axis, y-axis, and graph of y = 8 - x3. 3. A large garden is to be made in the shape shown. The bricks along the straight sections cost $2/m. The bricks along the curved sections cost $3/m. If $1000 is used to pay for the bricks, determine the maximum area that can be enclosed. 4. A rectangular pigpen will be surrounded by a fence and then divided into two sections by a block wall. The area of the pigpen must be 32 me. Fencing cost $20/m and the block wall costs $40/m. What should be the dimensions of the pigpen to minimize costs? 5. Find the area of the largest rectangle that can be inscribed in a right triangle with legs adjacent to the right angle of lengths 4 cm and 12 cm. The two sides of the rectangle lie along the legs. 6. A cylindrical chemical storage tank with a capacity of 1000 m' is to be constructed. The specifications call for the base to be made of steel, which costs $100/m , the top of steel, which costs $50/m , and the wall of steel costing $80/m . Determine the proportions of the tank that meet the conditions and that minimize the cost of construction. 7. A Norman window is made up of a semi-circle and a rectangle. The total perimeter of the window is 16 m. What is the maximum area? 8. A rectangle lies in the first quadrant with one vertex at the origin and two of the sides along the coordinate axes. If the fourth vertex lies on the parabola defined by y = -(x - 6)2 + 63, then determine the dimensions of the rectangle with the greatest area. 9. A window is to be designed in the shape of an isosceles right angled triangle on top of a rectangle as shown. The rectangular piece is divided from the triangular piece by a stile. If all sections of the window are bordered with trim, determine the minimum perimeter required to make the window have an area of 3 m-.10. A new compost bin is going to be made out of a new high tech material that speeds up the composting process! The bin will be in the shape of a cylinder with a half sphere on top. The volume of the compost bin needs to be 30 ft'. The material for the new compost bin is very expensive and so using less material will be beneficial. Determine the minimum amount of material required to build the compost bin. 1 1. A right angled, triangular garden is to be made against the wall of a building as shown in the diagram. If we have 50 metres of edging to border the two required sides of the E garden, what is the maximum area that can be covered and 3 what are the dimensions of the garden? Edging 12. James is standing 90 feet from a nearby river. He sees a fire in the distance and quickly calculates that the fire is a distance of 60 feet from the river. Determine 90 feet the shortest distance that James can travel in order to fill a bucket with water at the river and then rush to put out the blazing fire. 60 feet 13. Smarties are trying to build a new box for a commemorative Smarties package. The new box will be shaped like a regular hexagonal based prism. Determine the minimum surface area required if the volume of the container needs to be 1000 cm'.14. A large aluminum pipe is carried down a hallway that is 10 feet wide. At the end ofthe hallway then turns to the right and it narrows down to 8 feet wide. What is the maximum length for the pipe so that if we always keep in horizontal it can be carried passed the turn in the hallway? 15. A piece of paper has dimensions 6 inches by 12 inches as shown. The bottom corner is folded over so that it touches the left side ofthe page (See diagram). Determine the smallest possible length for the crease in the page. 6 in. 12 in. 12 in. \\creajse 6 in Final Answers: 1. 32 m1 2. 7.6 sq units 3. 9948.9 m2 4. 4m x 8m 5. 12 cm2 6. r=5.53m h = 10.38m 7. 17.9 m2 8. 9 x 54 9. P=9.1m 10. SA = 39ft2 11. 240 m2 12. 17011: 13. SA=5TL9 sq units 14. 25.4 m 15. x = 9/2,y = 9N5, crease 179 in