St. Francis Preparatory School



Name:___________________________________________Date:_________Period:_______Calculus Final Part 2Calculus HonorsCalculus Final Part IIApplications of Derivative: Word ProblemsFor questions that do not tell you what to round to, round to three decimal places. 1.) A box manufacturer desires to create a box with a surface area of 900 inches squared. What is the maximum size volume that can be formed by bending this material into a box? The box is to be closed. It is to have a square base, square top, and rectangular sides.2.) A two-pen corral is to be built. The outline of the corral forms two identical adjoining rectangles. If there is 240 ft of fencing available, what dimensions of the corral will maximize the enclosed area?3.) Sophy has 60 (linear) feet of fencing with which she plans on enclosing a rectangular space for a garden. Find the largest area that can be enclosed with this much fencing and the then determine the dimensions of Emily’s garden.4.) A rectangular page is to contain 30 square inches of print. The margins at the top and bottom of the page are to be 2 inches and the margins on the left and the right are to be 1.5 inches. What should the dimensions of the page be so that the least amount of paper is used?5.) Vinnie wants to enclose his yard with a fence. He has 1000 feet of fencing material and his house is on one side of the field, so he does not need any fencing there. Determine the dimensions of the field that will enclose the largest area?6.) A spherical balloon is inflated with helium at a rate of 50π ft3/minute. How fast is the balloon’s radius increasing at the instant the radius is 5 feet? How fast is the surface area of the balloon increasing?7.) The radius r of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area when (a) r = 12 cm and (b) r = 40 cm.8.) A 30 foot ladder is resting against a wall. The bottom is initially 20 feet away from the wall and is being pushed towards the wall at a rate of 0.35 ft/sec. How fast is the top of the ladder moving up the wall after 24 sec after we start pushing.9.) Air is being pumped into a spherical balloon at a rate of 9 cubic feet per minute. Find the rate of change of the radius when the radius is 3 feet.10.) A person on a pier pulls in a boat using a rope fastened to the bow. The person pulls in on the rope at 5 ft/min and is holding the rope 15 ft above the fastening. How fast is the boat moving when it is 20ft away from the pier? Round your answer to the nearest thousandth.11.) At time t = 0, a diver jumps from a platform diving board that is 48 feet above the water. The position of the diver is given by s(t) = -16t2 + 32t + 48, where s is measured in feet and t is measured in seconds.(a) When does the diver hit the water?(b) What is the diver's velocity at impact?12.) A projectile is shot upward from the surface of the Earth with an initial velocity of 120 meters per second. It’s path is given by the equation y = -4.9t2 +120t. Determine the equation for the instantaneous velocity of the projectile. What is its velocity after 5 seconds? After 10 seconds?13.) A ball is thrown straight down from the top of a 220-foot building with an initial velocity of -22 feet per second. The path of the ball is given by the equation y = -16t2 – 22t + 220. What is its velocity after 3 seconds? What is its velocity after falling 108 feet?14.) Supposed s(t) = 4t3 represents the position of a race car along a straight track measured in feet from the starting line at time, t, seconds. (a) What is the average rate of change s(t) from t = 3 to t = 5 seconds?(b) What is the instantaneous rate of change of the same car at t = 5 seconds?15.) A poster is to contain 250 square inches of picture surrounded by a 4-inch margin at the top and bottom and a 2-inch margin on each side. Find the overall dimensions that will minimize that total area of the poster.16.) What is the radius of a cylindrical soda can with a volume of 128 cubic inches that will use the minimum material? Round to the nearest hundredth.17.) An open-top box with a square bottom and rectangular sides is to have a volume of 636 cubic inches. Find the dimensions that require the minimum amount of material. Round to the nearest thousandth.18.) Thomas became a rancher! He has 630 feet of fencing to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum? Round your answer to the nearest hundredth of a foot, if necessary.19.) Tina decided she wants to become a farmer. She plans to fence a rectangular pasture adjacent to a river. The pasture must contain 270,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river. Round to the nearest thousandth, if necessary.First & Second Derivative TestsIn the examples below, (a) find the critical numbers of f (if any), (b) find the open interval(s) on which the function is increasing or decreasing, and (c) apply the First Derivative Test to identify all relative extrema.20.) f(x) = x3 – 6x2 + 12x21.) f(x)= 2x3 – 3x2 – 12x + 522.) f(x) = 2x2 – 823.) fx=2x+5324.) f(x) = 2x3 + 3x2 – 12x25.) g(x) = 2x3 – 216 26.) fx=x-5x+227.) gx=x4-100xxIn the examples below, find the points of inflection and discuss the concavity of the graph of the function.28.) f(x) = x3 – 6x2 + 12x29.) f(x)= 2x3 – 3x2 – 12x + 530.) fx=x5x-931.) fx=-4x6+6x4+332.) f(x) = x4 – 4x3 + 233.) fx=-2x3+3x2-12x 34.) f(x) = x4 – 500x – 14035.) fx=x5x-36Answer KEY:1.) 12.247 in by 12.247 in by 12.248 in; max volume = 1837.065 in32.) dimensions 60 ft by 40 ft3.) 15 ft by 15 ft; largest area = 225 ft24.) 7.743 in by 10.325 in5.) 250 ft by 500 ft6.) (a) dr/dt = ? ft/min (b) dSA/dt = 20π ft2/min7.) (a) 96π cm2/min; (b) 320π cm2/min8.) .147 ft/sec9.) .080 ft/min10.) -6.25 ft/min11.) (a) 3 sec; (b) -64 ft/sec12.) v(t) = -9.8t + 120; v(5) = 71 m/s; v(10) = 22 m/s13.) v(3) = -118 ft/s; falling 108 after t = 2 sec; v(2) = -86 ft/s14.) (a) 196 ft/s(b) 300 ft/s15.) 30.361 in by 15.180 in16.) 2.73 in17.) 5.418 in by 10.835 in by 10.835 in18.) 105 ft by 157.5 ft19.) 734.848 ft by 367.423 ft20.)x=2; increasing:-∞,2,2,∞; no max/min21.)x=-1,2 decreasing: (-1,2); increasing:-∞,-1,2,∞; max: (-1,12); min: (2,-15) 22.) x=0; decreasing:-∞,0; increasing:0,∞; minimum: (0,-8) 23.) increasing:-∞,∞; no max/min24.) x=-2,1; decreasing: (-2,1); increasing:-∞,-2,1,∞; max: (-2,20); min: (1,-7) 25.) increasing on entire interval; no max/min26.) increasing on entire interval; no max/min; (critical value x = -2)27.) increasing on entire interval; no max/min (critical value x = 0)28.) concave down: (-∞,2); concave up: (2, ∞); POI: (2,8)29.) concave down: (-∞, 0.5); concave up: (0.5, ∞); POI: (0.5, -1.5)30.) concave down: (0, 6); concave up: (-∞,0) & (6, ∞); POI: (0,0), (6, -23328)31.) concave down: (-∞,-0.775), (0.775, ∞); concave up: (-0.775, 0) (0,0.775) POI: (-0.775, 4.298), (0.775, 4.298)32.) concave down: (0,2); concave up: (-∞,0), (2, ∞); POI: (0,2), (2,-14)33.) concave down: (1/2, ∞); concave up: (-∞,1/2); POI: (0.5, -5.5)34.) concave up: (-∞,0), (0, ∞); No POI35.) concave down: (0,24); concave up: (-∞,0) & (24, ∞); POI: (0,0), (24,-95551488) ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download