3.1 Higher Order Derivatives. Velocity and Acceleration 1 . Find f' , f" , f " , f ( 4) , f (5) , f () for f ( x ) = x5 + x3 - x2 + x+ 2. 2. Find f' and f" for f(x) =V1-x2 . Write the answers in a simplified and factored form. 3. Show that y = - satisfies xy"+2yy'=0. x+13.1 Higher Order Derivatives. Velocity and Acceleration 4. An object launched vertically upward has an altitude (height) described by 11(1) = 25+ 20! 5:2 , where h(t) is the height (in meters) after t (seconds) from the launch. a) Find the moment of time when the object is at rest. b) Find the maximum height reached by the object. c) Find the velocity of the object when the object hits the ground. cl) Show that the acceleration for this motion is constant. 5. A particle motion may be modeled by the position function 5(1) = t(t2 9) . a) Find the interval(s) when the particle is moving away from the origin. b) Find the interval(s) when the particle is slowing down. c) Find the total distance travelled over [14] . 3.2 Maximum and Minimum on an Interval. Extreme Values 1. Find the extremum (local or global, minimum or maximum) point(s) for the function represented, by its graph, in the figure to the right . 2. Find the global (absolute) minimum or maximum point(s) for the function represented in the above figure, over the interval [2,2] . 3. Let y=f(x)=2x2 8x+3 , ISxS4. Find the global (absolute) minimum or maximum point(s) for the function y = f (x) over the given interval. x x2+1 4. Find the global (absolute) minimum or maximum point(s) for the function y = f(x) = for 1 s x 51. 3.2 Maximum and Minimum on an Interval. Extreme Values 5. Find the global (absolute) minimum or maximum point(s) for the function y = f (x) = x -3x2 +5 for -15x53. 6. Find the extremum (local or global, minimum or maximum) point(s) for the function y = f (x) = - 1 x2 + 13.3 Optimization Problems 1. A rectangle has a perimeter of 400m. What length and width should it have so that its area is a maximum. What is the maximum value of its area? 2. A rectangle is inscribed in a semicircle of radius 10m. Find the dimensions of the rectangle that will maximize the area of the rectangle and the maximum value of the area. 3.3 Optimization Problems 3. Find a positive number such that the sum of its square and its reciprocal is minimum. 4. Find the point on the line 3y + 2x = 12 that is closest to the origin.3.3 O timization Problems 5. A farmer with 800m of fencing wants to enclose a rectangular area and then divide it into five pens (of equal size) with fencing parallel to one side of the rectangle (see the figure to the right). What are the dimensions (length and width) of a pen that will produce the largest possible area of each pen. 6. If 75004::2 of material is available to make a box with a square base and open top, find the dimensions of the box that give the largest volume of the box. What is the maximum value of the volume? 3.3 Optimization Problems 7. A box with an open top is to be constructed from a rectangular piece of cardboard, 2m by 3m , by cutting out a square from each of the four corners and bending up the sides. Find the dimensions of the box corresponding to a maximum volume. In: 8. Find the dimensions of the largest right-cylinder that can be inscribed in a cone of radius R = 6m and height H = 9m. 3.3 Optimization Problems 9. A north-south road intersects an east-west road at a point 0 . A motorcycle crosses 0, at noon, travelling West at a constant speed of 80 km/h . At the same time, a car is 50 km South of 0 , travelling North at 60km] h. Find the time at which they are closest to each other, and approximate the minimum distance between them. 10. The sum of two non-negative numbers is 16 . Find the maximum possible value and the minimum possible value of the sum of their cube roots. 3.4 Ogtimization Problems in Economics and Science 1. A farmer wants to fence an area of 9600m2 in a rectangular field and divide it in half with a fence parallel to one of the sides of the rectangle. Find the dimensions (length and width) of the rectangular field that minimize the cost of the fence? 2. The selling price (in $) of an item is p(x) = 600.02x , where x is the number of items sold per day. lithe cost (in :3) of manufacturing 1: items is C(x) = 1000+ 10x , find the number of items to be manufactured per day in order to maximize the proiit. 3.4 OEtimization Problems in Economics and Science 3. The cost (in $5) of manufacturing 1 thousands units of USB memory sticks is given by C(x) = 2x2 50x + 800 . How many items must be produced in order to minimize the unit cost u(x) = c(x)lx. 4. A farmer wishes to fence in a rectangular field of 60,000 m2 . The North and the South fences cost $2/m while the East and the West fences cost $31r m . Find the dimensions of the field that will minimize the cost. 3.4 Optimization Problems in Economics and Science 5. A closed box with a square base is to contain 252cm3 . The bottom costs $5] cm2 , the top costs $2142:2 , and the sides cost $3 / cm2 . Find the dimensions that will minimize the cost. 6. A soda cracker package (the top is closed) is to be constructed in the shape of a rectangular prism with a square base. The total capacity is 512 cm3 . Find dimensions (length, width, and height) that will minimize the cost. 3.4 OBtimization Problems in Economics and Science 7. A cylindrical can (the top is closed) is to be made to hold 1000cm3 of oil. Find the dimensions (radius and height) of the can that will minimize the cost of the metal to make the can. 8. Corn silos are usually in the shape of a cylinder (with a closed base) surmounted by a hemisphere. If the volume of a silo is 1000m3, what dimensions (radius and height) of the silo would use the minimum amount of materials? 3.4 Optimization Problems in Economics and Science 9. A tank has hemispherical ends and a cylinder center. The cost (per square meter) of manufacturing the hemispherical ends is double in comparison with the cost (per square meter) of manufacturing the cylindrical part. Find the proportions (the ratio between height and radius) of the cylinder that will maximize the volume for a given total cost. 10. A lifeguard can run on the beach at 8 ml 3 and swim at 4m/s. If an incident happens at 40 m from the shore and the lifeguard is on the shore at IOOmfrom the incident place. find the minimum amount of time it takes for the lifeguard to reach the place where the incident has occurred