The rate at which gasoline flows out of a tank, in gallons per hour, is given...

 

 

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The rate at which gasoline flows out of a tank, in gallons per hour, is given by a differentiable function G of time t. The table below shows the rate as measured every 4 hours for a 24-hour period: t (hours) 0 4 812 16 20 24 G(t) (gallons 12.2 12.1 11.8 10.4 10.8 11.6 12.2 per hour) Part A: Estimate G'(10). Show the work that leads to your answer. Part B: Is there some time t, for 0 < t < 24, such that G'(t) 0? Justify your answer. 5. A gardener wants to design a rectangular garden and has 100 feet of fencing available. A house is on one side of the garden, so no fencing will be needed there. What dimensions will give the maximum area? Honors Project 5 101 2 3 4 5 6 7 8 ! 1. The figure above shows the graph of f', the derivative of the function f on the closed interval 0x8. The graph of f' has horizontal tangent lines at x = 2, x = 4, and x = 6. The function f is twice differentiable. Part A: Find the x-coordinate of each of the points of inflection on the graph of f. Give a reason for your answer. Part B: At what value of x does fattain its absolute maximum value on the closed interval 0 x 8? Show the analysis that leads to your answer. Part C: For what values of x is the graph of f concave upward? Justify your answer. 2. Let g be the function given by g(x) = x 12x. Part A: Identify the relative extrema of g. Justify if your value is a local minimum, local maximum, or neither. Part B: On what interval(s) is the graph of g concave up? Justify your answer. 3. A twice-differentiable function f is defined for all real numbers x. The function f and its derivative have the properties and various values of x indicated in the table below. 00 Part A: Find all values of x at which f has a relative extrema on the interval [0, 8]. Determine whether f has a relative maximum or a relative minimum at each of these values. Justify your answer. Part B: Explain why there must be a value c, for 2 The rate at which gasoline flows out of a tank, in gallons per hour, is given by a differentiable function G of time t. The table below shows the rate as measured every 4 hours for a 24-hour period: t (hours) 0 4 812 16 20 24 G(t) (gallons 12.2 12.1 11.8 10.4 10.8 11.6 12.2 per hour) Part A: Estimate G'(10). Show the work that leads to your answer. Part B: Is there some time t, for 0 < t < 24, such that G'(t) 0? Justify your answer. 5. A gardener wants to design a rectangular garden and has 100 feet of fencing available. A house is on one side of the garden, so no fencing will be needed there. What dimensions will give the maximum area? Honors Project 5 101 2 3 4 5 6 7 8 ! 1. The figure above shows the graph of f', the derivative of the function f on the closed interval 0x8. The graph of f' has horizontal tangent lines at x = 2, x = 4, and x = 6. The function f is twice differentiable. Part A: Find the x-coordinate of each of the points of inflection on the graph of f. Give a reason for your answer. Part B: At what value of x does fattain its absolute maximum value on the closed interval 0 x 8? Show the analysis that leads to your answer. Part C: For what values of x is the graph of f concave upward? Justify your answer. 2. Let g be the function given by g(x) = x 12x. Part A: Identify the relative extrema of g. Justify if your value is a local minimum, local maximum, or neither. Part B: On what interval(s) is the graph of g concave up? Justify your answer. 3. A twice-differentiable function f is defined for all real numbers x. The function f and its derivative have the properties and various values of x indicated in the table below. 00 Part A: Find all values of x at which f has a relative extrema on the interval [0, 8]. Determine whether f has a relative maximum or a relative minimum at each of these values. Justify your answer. Part B: Explain why there must be a value c, for 2