AP Calculus  
                                                                                             
                                   Writing Justifications and Avoiding Common Errors 
                                                                                             
                                                                                             
                                                                                             
                                                                                             
                                                                         Student Handout 
                                                                                             
                                                                                             
                                                                                             
                                                                                             
                                                                                             
                                                                                             
                                                                            2017-2018 EDITION 
                                                                                             
                                                                                             
                        
                      Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org           
                       
                       
                      Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org           
                       
                                                                        Writing Justifications  
                                                                                            
                      Students are expected to demonstrate their knowledge of calculus concepts in 4 ways: 
                       
                      1. Numerically (Tables/Data) 
                      2. Graphically  
                      3. Analytically (Algebraic equations) 
                      4. Verbally 
                      The verbal component occurs often on the free response portion of the exam and requires 
                      students to explain and/or justify their answers and work.  It is important that students understand 
                      what responses are valid for their explanations and justifications. 
                       
                       
                      1.  On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a 
                                                                                                               2
                                                                                                            
                           gravel processing plant is modeled by Gt( ) 90                        45cos t         , where t is measured in hours 
                                                                                                            
                                                                                                             18
                                                                                                            
                                  08t
                           and                . At the beginning of the workday                         , the plant has 500 tons of unprocessed 
                                                                                               (0t   )
                                                                                  08t
                           gravel. During the hours of operation,                             , the plant processes gravel at a constant rate 
                           of 100 tons per hour.  
                           (a) Find              . Using correct units, interpret your answer in the context of the problem. 
                                        G'(5)
                            
                            
                            
                            
                           (b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of 
                                operation on this workday.  
                            
                            
                            
                            
                            
                            
                           (c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time t 5
                                hours? Show the work that leads to your answer. 
                            
                            
                            
                            
                            
                            
                           (d) What is the maximum amount of unprocessed gravel at the plant during the hours of 
                                operation on this workday? Justify your answer. 
                            
                        
                      Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org             1 
                       
                      2.   
                                                                                         05t
                           A particle moves along a straight line. For                               , the velocity of the particle is given by
                                              26/53
                            vt()2(t 3t) t, and the position of the particle is given by s(t). It is known that 
                            s(0) 10.  
                            
                                                                             05t
                           (c) Find all times t in the interval                          at which the particle changes direction. Justify 
                                 your answer.  
                            
                            
                            
                            
                            
                            
                            
                           (d) Is the speed of the particle increasing or decreasing at time t  4? Give a reason for your 
                                 answer. 
                       
                       
                       
                                                                                             
                                                                  t                                                          
                                                            (minutes)             0        12         20        24        40 
                                                                                                                   
                                                                                                                             
                                                                v (t)             0       200        240       -220       150 
                                                          (meters per                                   
                                                             minute) 
                      3.  
                                                                                    04t       0
                           Johanna jogs along a straight path. For                                , Johanna’s velocity is given by a 
                           differentiable function v. Selected values of v(t), where t is measured in minutes and v(t) is 
                           measured in meters per minute, are given in the table above. 
                                                                                                                        40 vt()
                           (b)  Using correct units, explain the meaning of the definite integral                               dt in the context of the 
                                                                                                                       0
                                                                                  40 vt()
                                 problem. Approximate the value of                         dt using a right Riemann sum with the four 
                                                                                  0
                                 subintervals indicated in the table. 
                       
                        
                      Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org         2