Question Instructions Read today's Notes and Learning Goals

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1 111 Basic: Optimization I ( ) Question Instructions Read today's Notes and Learning Goals 1. Question Details MinMax Graph 1 [ ] An object is launched straight upward from a platform. Its height is a function of time, h(t), with h in meters and t in seconds. The domain is 0 t 6 seconds. The graph of h is shown below. 1. How many critical points does h have in this domain? 2. When does the object reach its greatest height? 3. What is the maximum height? h max = 2. When does the object reach its smallest height? 3. What is the minimum height? h min =

2 2. Question Details MinMax Pre 1 [ ] An object is launched straight upward. Its height is given by h(t) = t 16.2t 2 with h in feet and t in seconds. Suppose that you want to know how high this object can possibly get. 1. Which function would you want to maximize? h dh d 2 h 2 None of these 2. Which function would you set equal to zero? h dh d 2 h 2 None of these 3. Question Details MinMax 1 [ ] An object is launched straight upward. Its height is given by h(t) = t 16.2t 2 with h in feet and t in seconds. 1. Graph h on the domain 0 t 7 seconds. How many critical points does h have in this domain? 2. Locate the instant in time when the object reaches its greatest height. Be accurate to four decimal places. 3. What is the maximum height? Be accurate to one decimal place. h max =

3 4. Question Details MinMax 2 Pre [ ] An object oscillates up and down. Its height is given by y(t) = cos(1.4t 1) with y in meters and t in seconds. Suppose that you want to know how high (or low) this object can possibly get. 1. Which quantity would you want to maximize (or minimize)? height velocity rate of change of velocity none of these 2. What would you set equal to zero? height velocity rate of change of velocity none of these 5. Question Details MinMax 2 [ ] An object oscillates up and down. Its height is given by y(t) = cos(1.4t 1) with y in meters and t in seconds. 1. Graph y on the domain 0 t 4 seconds. How many critical points does y have in this domain? 2. Locate the instant in time (in the given domain) when the object reaches its greatest height. Be accurate to four decimal places. 3. What is the maximum height? Be accurate to four decimal places. y max = 4. Locate the instant in time (in the given domain) when the object reaches its lowest height. Be accurate to four decimal places. 5. What is the minimum height? Be accurate to four decimal places. y min =

4 6. Question Details MinMax 5 Pre [ ] A box with square base has an open top and a volume of 200 cm 3. The base dimension, x, is a variable measured in cm. The height is also variable, but constrained by the total volume, as shown in the figure below. The surface area of the box (base and four sides) is a function of x: Suppose that you want to know the dimensions of the box that uses the least amount of material. A(x) = x 2 + 4x 200 x 2 1. Which quantity would you want to maximize (or minimize)? The surface area of the box. The volume of the box. The derivative of the surface area of the box. The derivative of the volume of the box. 2. What would you set equal to zero? The derivative of the surface area of the box. The derivative of the volume of the box. The volume of the box. The surface area of the box.

5 7. Question Details MinMax 5 [ ] A box with square base has an open top and a volume of 200 cm 3. The base dimension, x, is a variable measured in cm. The height is also variable, but constrained by the total volume, as shown in the figure below. The surface area of the box (base and four sides) is a function of x: A(x) = x 2 + 4x 200 x 2 1. Graph the function A(x). Assume x > 0. How many critical points does A have? 2. What value of x gives the smallest possible surface area? Be accurate to three decimal places. x = 3. What is the minimum surface area? Be accurate to three decimal places. A min =

6 8. Question Details MinMax 5 Pre Symb [ ] A box with square base has an open top and a fixed total surface area, A. The base dimension, x, is a variable. The height is also variable, but constrained by the fixed area, as shown in the figure below. The volume of the box is a function of x: V(x) = x 2 Suppose that you want to know the dimensions of the box with the greatest volume. 1. Which function would you maximize (or minimize)? dv dx A da dx V A x 2 4x 2. What would you set equal to zero? dv dx V A da dx

7 9. Question Details MinMax 5 Symb [ ] A box with square base has an open top and a fixed total surface area, A. The base dimension, x, is a variable. The height is also variable, but constrained by the fixed area, as shown in the figure below. The volume of the box is a function of x: V(x) = x 2 A x 2 4x 1. What value of x gives the maximum volume? Give a symbolic answer that involves A. x = 2. What is the maximum volume? Give a symbolic answer that involves A. V max = 10. Question Details MinMax 4 Pre [ ] A population of insects grows according to a logistic model: p(t) = with p measured in insects and t in days. Suppose that you want to know the fastest possible rate of change of population. 1. Which function would you maximize (or minimize)? p dp d 2 p e 0.3t None of these 2. What would you set equal to zero? p dp d 2 p 2 None of these

8 11. Question Details MinMax 4 [ ] A population of insects grows according to a logistic model: p(t) = e 0.3t with p measured in insects and t in days. dp dp 1. Graph the function. How many critical points does have? 2. Locate the instant in time when the rate of change of population is greatest. Be accurate to three decimal places and include units. 3. What is the maximum rate of change of population? Round to the nearest whole number. Units are not required. insects/day 4. What is the population at the instant when it is changing fastest? Round to the nearest whole number. Units are not required. insects 12. Question Details MinMax 1 Symb [ ] An object is launched straight upward. Its height is given by 1 h(t) = kt at 2 2 with h in feet, t in seconds. k and a are constants. 1. Locate the instant in time when the object reaches its greatest height. Give a symbolic answer that involves k and a. 2. What is the maximum height? Give a symbolic answer that involves k and a. h max =

9 13. Question Details MinMax 3 [ ] An object oscillates up and down, with damping. Its height is given by y(t) = 3 + e 0.35t cos(1.4t 1) with y in meters and t in seconds. 1. Graph y on the domain 0 t 8 seconds. How many critical points does y have in this domain? 2. Locate the instant in time (in the given domain) when the object reaches its greatest height. Be accurate to four decimal places. 3. What is the maximum height? Be accurate to four decimal places. y max = 4. Locate the instant in time (in the given domain) when the object reaches its lowest height. Be accurate to four decimal places. 5. What is the minimum height? Be accurate to four decimal places. y min = Assignment Details