Download 21 Questions with Answers - Calculus with Analytic Geometry I - Exam 3 | MAT 270 and more Exams Analytical Geometry and Calculus in PDF only on Docsity! MAT 270 Test 3 Review Chapter 3 (NNAI) Sample Problems 1. For the function ( ) 321618 23 ++−= xxxxf use derivatives to find the coordinates of any relative minimum and maximum points and points of inflection correct to 3 decimal places. Label the points 2. For the function ( ) xexf x sin 2 3 1 −= , use derivatives to find the coordinates of any relative minimum and maximum points. Label the points. 3. Suppose the temperature, T, of a yam put into a hot oven maintained at 180°C is given as a function of time by ( ) beaT kt +−= −1 where T is in degrees Celsius and t is in minutes. a) If the initial temperature of the yam is 10°, find a and b. b) If the temperature of the yam is initially increasing at the rate of 2°C per minute, find k. c) How long does it take for the yam to reach a temperature of 100°C? 4. For what values of a and b will the function ( ) cbxaxxf ++= 3 have 2 critical points? For what values will it have no critical points? 5. A canal with vertical sides is to have a perimeter of 10 meters. Find the dimensions of the canal which would maximize the area of a cross–section and thus maximize the amount of water which could flow through it. 6. At 2 A.M., the HMS Maggot is 60 kilometers due north of an oil tanker and sailing south at 15 kilometers per hour. The tanker is sailing west at 10 kilometers per hour. At what time are the two ships closest to each other? Who would give a ship a name like that? 7. A rectangular garden 200 square feet in area is to be fenced off against rabbits. Find the dimensions that will require the least amount of fencing if one side of the garden is already protected by a barn. 8. Squares of equal size are to be cut from each corner of a rectangular piece of cardboard of dimensions 8 inches by 15 inches. The sides will then be folded up to make an open box. Find the size of the square which should be cut from each corner to maximize the volume of the box. 9. A manufacturer wants to design an open box with a square base and a surface area of 108 square inches. What dimensions will produce a box of maximum volume? 10. Which points on the graph of 24 xy −= are closest to the point ( )2,0 ? 11. A right triangle is formed in the first quadrant by the x and y axes and a line through the point ( )3,2 . Find the vertices of the triangle so that its area is a minimum. 12. Find the exact values of the following limits algebraically: a) x e x x 1lim 2 0 − → b) x x x lnlim ∞→ c) xx e x2lim ∞→ d) x x x ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∞→ 11lim e) 30 sinlim x xx x − → f) 3 3lim 3 − − → x x x 13. Use Newton’s Method to find 2x to 4 decimal places with 0 3x = for the function ( ) 3 24 4f x x x= − + .
