Unit 5 Progress Check FRQ Part B

Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B 1. NO CALCULATOR IS ALLOWED FOR THIS QUESTION. Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit. Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point. Unless otherwise specified, the domain of a function which is a real number. The graph of the continuous function differentiable, except at . Let be the function with is assumed to be the set of all real numbers is shown above for and derivative given by . The function for is twice . Page 1 of 9 Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B (a) Find the -coordinate of each critical point of . Classify each critical point as the location of a relative minimum, a relative maximum, or neither. Justify your answers. Please respond on separate paper, following directions from your teacher. (b) Find all values of answers. at which the graph of has a point of inflection. Give reasons for your Please respond on separate paper, following directions from your teacher. (c) Fill in the missing entries in the table below to describe the behavior of . Indicate Positive or Negative. Give reasons for your answers. and on the interval Please respond on separate paper, following directions from your teacher. (d) Let be the function defined by a reason for your answer. . Is increasing or decreasing at ? Give Please respond on separate paper, following directions from your teacher. Part A Note: Sign charts are a useful tool to investigate and summarize the behavior of a function. By itself a sign chart for or is not a sufficient response for a justification. The first point requires reference to A maximum of 1 out of 3 points is earned for only one correct critical point with correct identification and justification, and no incorrect critical points are included. Page 2 of 9 Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B Select a point value to view scoring criteria, solutions, and/or examples and to score the response. 0 1 2 3 The student response accurately includes all three of the criteria below. critical points relative maximum at relative minimum at with justification with justification Solution: has a relative maximum at has a relative minimum at because changes from positive to negative there. because changes from negative to positive there. Part B Note: Sign charts are a useful tool to investigate and summarize the behavior of a function. By itself a sign chart for or is not a sufficient response for a justification. A maximum of 1 out of 2 points is earned if only one point of inflection with reason and no incorrect points of inflection. Select a point value to view scoring criteria, solutions, and/or examples and to score the response. 0 1 2 The student response accurately includes both of the criteria below. Page 3 of 9 Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B answers with reasons Solution: Because changes sign at and the graph of has points of inflection there. Part C A maximum of 1 out of 2 points is earned if only behavior for reason. with reason OR behavior for Select a point value to view scoring criteria, solutions, and/or examples to score the response. 0 1 2 The student response accurately includes both of the criteria below. behavior for and reasons Solution: Page 4 of 9 with Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B is positive for is positive for because the graph of and because because the graph of is concave down there. is increasing there. is concave up there, and is negative for Part D The response is not eligible for the second point with an incorrect product rule. Select a point value to view scoring criteria, solutions, and/or examples to score the response. 0 1 2 The student response accurately includes both of the criteria below. product rule answer with reason Solution: Because 2. and it follows that and is decreasing at NO CALCULATOR IS ALLOWED FOR THIS QUESTION. Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit. Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point. Page 5 of 9 Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B Unless otherwise specified, the domain of a function which is a real number. is assumed to be the set of all real numbers The number of mosquitoes in a field after a major rainfall is modeled by the function , where is the number of days after the rainfall ended and (a) Using correct units, interpret the meaning of for defined by . in the context of the problem. Please respond on separate paper, following directions from your teacher. (b) Based on the model, what is the absolute maximum number of mosquitoes in the field over the time interval ? Justify your answer. Please respond on separate paper, following directions from your teacher. (c) For what values of is the rate of change of the number of mosquitoes in the field increasing? Please respond on separate paper, following directions from your teacher. (d) For , the number of bats in the field is modeled by the differentiable function , where is a function of the number of mosquitoes in the field. Based on the models, write an expression for the rate of change of the number of bats in the field at time . Please respond on separate paper, following directions from your teacher. Part A The point requires a time reference, rate, and units. Select a point value to view scoring criteria, solutions, and/or examples and to score the response. Page 6 of 9 Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B 0 1 The student response accurately includes a correct interpretation of Solution: At time days, the number of mosquitoes in the field is decreasing at a rate of 171 mosquitoes per day. Part B Note: Sign charts are a useful tool to investigate and summarize the behavior of a function. By itself a sign chart for or is not a sufficient response for a justification. The response is eligible for additional points based on consistent answers using an incorrect maximum of one computational error. with a The fourth point does not require a simplified answer. Select a point value to view scoring criteria, solutions, and/or examples and to score the response. 0 1 2 3 4 The student response accurately includes all four of the criteria below. sets identifies as a candidate answer with justification Solution: Because changes from positive to negative at has a relative maximum at Page 7 of 9 Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B Because is the only critical point of on the interval that is the location of a relative maximum. It is also the location of the absolute maximum of on the interval The absolute maximum number of mosquitoes in the field over the time interval is Part C The third point may be earned based on a consistent answer using an incorrect computational error. with a maximum of one Select a point value to view scoring criteria, solutions, and/or examples to score the response. 0 1 2 3 The student response accurately includes all three of the criteria below. considers expression for answer Solution: The rate of change of the number of mosquitoes in the field is increasing for Part D The point is earned with Select a point value to view scoring criteria, solutions, and/or examples and to score the response. Page 8 of 9 Calculus AB Scoring Guide Unit 5 Progress Check: FRQ Part B 0 1 The student response accurately includes a correct expression. Solution: The number of bats in the field at time is given by Therefore, the rate of change of the number of bats in the field at time is given by Page 9 of 9