3.1 Higher Order Derivatives. Velocity and Acceleration 1 . Find f' , f" , f" , f ( 4 ) , f (5 ) , f ( 6 ) for f ( x ) = x5 + x 3 - x 2 + 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 = - x+1 - satisfies xy" +2 yy'=0.3.1 Higher Order Derivatives. Velocity and Acceleration 4. An object launched vertically upward has an altitude (height) described by Mr) = 25 + 20: 5t2 ,where Mr) is the height (in meters) after 3 (seconds) from the launch. 3) 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. d) Show that the acceleration for this motion is constant. 5. A particle motion may be modeled by the position function s0) = r02 9) . 3) Find the interval(s) when the particle is moving away trom the origin. b) Find the intervalts) when the particle is slowing down. c) Find the total distance travelled over [3,4] . 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 . 3.Let y=f(x)=2x28x+3 , 1sxs4. 2. Find the global (absolute) minimum or maximum point(s) for the function represented in the above figure, over the interval [-2.2] - Find the global (absolute) minimum or maximum point(s) for the function y = f (I) over the given interval. 4. Find the global (absolute) minimum or maximum point(s) for the function y = f (x) = I x 2 +1 for 1x1. 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) = x3 -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 0 timization 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 =12that is closest to the origin. 3.3 Optimization 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 7500cm 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