Name__________________________________     Class__________________  (AB  or  BC)    

   

AP  Calculus  Summer  Review  Packet    

Please  complete  each  problem  in  order,  on  notebook  paper,  and  staple  this  packet  to  the  front.    All  of  your  work  will  be  collected  and  graded  on  the  second  day  of  school.    This  summer  packet  requires  you  to  review  many  areas  of  mathematics.    Expect  to  spend  quite  a  few  hours  on  this  assignment.    You  are  welcome  to  use  old  notes,  books,  or  the  internet  (try  Paul’s  online  notes  or  Khan  Academy)  to  look  up  information  that  you  may  have  forgotten.    If  you  have  difficulties  with  any  of  these  topics,  I  encourage  you  to  be  very  proactive  in  getting  help  at  the  beginning  of  the  year  as  we  will  not  be  spending  much  time  on  these  topics  in  class.        Please  note  that  the  TI-­‐Nspire  CAS  is  the  recommended  calculator  for  this  course.    Instruction  will  be  based  on  this  calculator  only.        

Part  1:    Mixed  Topics                                      

       

   

                           

     

                                 

                   

                                                                             

           

 

                     

                                                   

                 

                         

     

                 

       

Part 2: Additional Trig Review. No Calculators allowed!! 1. Let θ be an acute angle of a right triangle. Find the values of the other five trigonometric functions of θ.

cos θ = 7

10

2. If is an acute angle of a right triangle and sin = 40

41

, what is tan ?

Express your answer as a reduced, improper fraction, if necessary.

3. Solve ABC. Round your answer to nearest hundredth, if necessary.

b = 32 A = 30 . 4. Let (2,−2) be a point on the terminal side of an angle in standard position. Evaluate the trigonometric function of . sin = ?

5. Find the reference angle ' for = 23

12

.

6. Evaluate the expression in radians. Express your answer, in radians, as a fraction, using the word pi for , if necessary.

sin−1 − 12

⎛⎝⎜

⎞⎠⎟

7. Evaluate the expression in radians. Express your answer, in radians, as a fraction, using the word pi for , if necessary.

cos−1

− 32

⎛

⎝⎜⎞

⎠⎟

8. Solve the equation cos θ = −0.5 where 180 θ 270 .

9. What is the general solution of sin3 x − 36 sin x = 0?

A. x = 2 n or x = + 2 n

     

B. x =

2

+ 2 n or x = + 2 n

     

C. x = + 2 n

     

D. x =

2

+ 2 n or x = 3

2

+ 2 n

10. Verify the identity tan x + cot x = sec x csc x . 11. Simplify the expression sec θ cos θ + tan2 θ.

12. Simplify the expression sin

2

− θ

sec θ

+ cos2 θ.

13. Write an equation of the graph of y = 4 sin x translated up 2 units and right units. 14. Graph y = 3 sin(4x) + 10. 15. Graph y = 4 cos(x − 4 ) . 16. Graph y = tan(x) 17. Graph y = arctanx and give the domain and range. 18. Graph y = arcsinx and give the domain and range. Now go back and check your answer to question 6. 19. Graph y = arccosx and give the domain and range. Now go back and check your answer to question 7. 20. Solve for x: Pay attention to given domain. 3sin2 x = cos2 x for 0 ≤ x< 2πcos2x − sin2 x = sin x for -π ≤ x < πtan x + sec x = 2cos x for -∞ < x < ∞

Part 4: Limits Review

1.    Evaluate:        4

5 3lim4x

xx→

+ −

−  

 

  A.  0                                          B.     15           C.   1

6                  D.  DNE     E.  None  of  these  

   

2.    Evaluate:         21

18 3limn

n i

in n→∞

=

⎛ ⎞−⎜ ⎟⎝ ⎠

∑  

 

  A.  0                                        B.112     C. 33

2                    D.  12     E.  None  of  these  

     

3.    Evaluate:    2

5 3lim6 4x

xx→−∞

−

+      

 

  A.   52

−                                B.     54

−           C.   54                    D.   5

2     E.  None  of  these  

       

4.    32

3lim 23 −−−

→ xxx

x  is  

   

A.        0     B.    1     C    41     D         E    None  of  these  

                         

∞

Part  5:    You  must  know  the  following  properties/identities.    They  are  assumed  knowledge  in  an  AP  Calculus  class.    Study  them!      Logs:    

   Also:      

                             

 Trigonometry:    List  of  most  commonly  used  identities  in  AP  Calc          

           It  is  expected  that  you  know  the  domain/range  of  the  following  inverse  functions:    

         

       

   

Of  course,  don’t  forget  about  these  special  triangles.    You  must  know  your  trig  ratios  of  these  angles  in  every  quadrant!    Also  know  your  trig  ratios  of  quadrant  boundary  angles  such  as      

 

Special  Triangles  

Families  of  functions:    Please  know  all  the  key  info  of  the  following  parent  functions: