Calculus/Differentiation/Applications of Derivatives/Exercises

Approximating Values of Functions Integration/Contents Relative Extrema Find the relative maximum(s) and minimum(s), if any, of the following functions. 1. none 2. Minimum at the point 3. Relative minimum at 4. Relative minimum at Relative maximum at 5. Relative minimum at 6. Relative minimum at Relative maximum at Solutions Range of Function 7. Show that the expression cannot take on any value strictly between 2 and -2. Since is negative, corresponds to a relative maximum. For , is positive, which means that the function is increasing. Coming from very negative -values, increases from a very negative value to reach a relative maximum of at . For , is negative, which means that the function is decreasing. Since is positive, corresponds to a relative minimum. Between the function decreases from to , then jumps to and decreases until it reaches a relative minimum of at . For , is positive, so the function increases from a minimum of . The above analysis shows that there is a gap in the function's range between and . Absolute Extrema Determine the absolute maximum and minimum of the following functions on the given domain 8. on Maximum at ; minimum at 9. on Maximum at ; minimum at Solutions Determine Intervals of Change Find the intervals where the following functions are increasing or decreasing 10. Increasing on ; decreasing on 11. Decreasing on ; increasing elsewhere 12. Increasing on ; decreasing elsewhere 13. Increasing on ; decreasing elsewhere 14. Decreasing on ; increasing elsewhere 15. Decreasing on ; increasing elsewhere Solutions Determine Intervals of Concavity Find the intervals where the following functions are concave up or concave down 16. Concave down everywhere 17. Concave down on ; concave up on 18. Concave up on ; concave down on 19. Concave up on ; concave down on 20. Concave down on ; concave up on 21. Concave down on ; concave up on Solutions Word Problems 22. You peer around a corner. A velociraptor 64 meters away spots you. You run away at a speed of 6 meters per second. The raptor chases, running towards the corner you just left at a speed of meters per second (time measured in seconds after spotting). After you have run 4 seconds the raptor is 32 meters from the corner. At this time, how fast is death approaching your soon to be mangled flesh? That is, what is the rate of change in the distance between you and the raptor? 23. Two bicycles leave an intersection at the same time. One heads north going and the other heads east going . How fast are the bikes getting away from each other after one hour? 24. You're making a can of volume with a gold side and silver top/bottom. Say gold costs 10 dollars per and silver costs 1 dollar per . What's the minimum cost of such a can? $878.76 25. A farmer is investing in of fencing so that he can create an outdoor pen to display three different animals to sell. To make it cost effective, he used one of the walls of the outdoor barn as one of the sides of the fenced in area, which is able to enclose the entire area. He wants the internal areas for the animals to roam in to be congruent (i.e. he wants to segment the total area into three equal areas). What is the maximum internal area that the animals can roam in, given these conditions? thumb|277x277px|Question 27 figure: A sphere with radius and center . The corners of the sphere are labeled and fitted perfectly inside the sphere.|left 26. What is the maximum area of a rectangle inscribed (fitted so that the corners of the rectangle are on the circumference) inside a circle of radius ? . 27. A cylinder is to be fitted inside a glass spherical display case with a radius of . (The sphere will form around the cylinder.) What is the largest volume that a cylinder will have inside such a display case? . 28. A tall man is walking away from a light that is -feet above the ground. The man is walking away from the light at feet per second. How fast (speed not velocity) is the shadow, cast by the man, changing its length with respect to time? . 29. A canoe is being pulled toward a dock (normal to the water) using a taut rope. The canoe is normal to the water while it is being pulled. The rope is hauled in at a constant . The dock is above the water. Answer items (a) through (b). (a) How fast is the boat approaching the dock when of rope are out? . (b) Hence, what is the rate of change of the angle between the rope and the dock? . 30. A very enthusiastic parent is video taping a runner in your class during a race. The parent has the runner center frame and is recording from the straight-line track. The runner in your class is running at a constant . What is the rate of change of the shooting angle if the runner passes the parent half a second after the parent's direct shot (after the point in which the runner's motion and the parent's line of sight are perpendicular)? Solutions Graphing Functions For each of the following, graph a function that abides by the provided characteristics 30. There are many functions that satisfy all the conditions. Here is one example:center|thumb 31. There are many functions that satisfy all the conditions. Here is one example:center|thumb Solutions Approximation Problems By assumption, for these problems, assume and unless stated otherwise. One may use a calculator or design a computer program, but one must indicate the method and reasoning behind every step where necessary. 35. Approximate using whatever method. If you use Newton's or Euler's method, do it in a maximum of THREE (3) iterations. Example: using Euler's method with step size and . See solution page for details 36. Approximate using whatever method. If you use Newton's or Euler's method, do it in a maximum of THREE (3) iterations. Example: using Newton-Rhapson method through iterations. See solution page for details 37. Approximate using whatever method. If you use Newton's or Euler's method, do it in a maximum of THREE (3) iterations. Example: using local-point linearization. See solution page for details Solutions Advanced Understanding 45. Consider the differentiable function for all and continuous function below, where is linear for all and differentiable for all , and and are continuous for all . thumb|center|400x400px a. Approximate . b. Using your answer from (a), find . c. Assume . Find an approximation of the first positive root of shown on the graph. Use only ONE (1) iteration. Let allow . Using only one iteration of the Newton-Rhapson method, . d. A computer program found that there exists only one local maximum and minimum on the function and found no local maximum or minimum for . Based on this finding, what flaw exists in the program and how can it be fixed? Flaw: the program fails to consider the case where the derivative does not exist. Fix: add additional code considering this case. More details in the solutions page. Solutions Approximating Values of Functions Integration/Contents