Pre‐Calculus, 1 st Semester Exam Study Guide, 2011–12 Directions: Graph the following functions. Be careful to identify the points where the graphs start, stop, and/or change directions. 1) = 2 − 1 2) = −! + 3, < 2 −4, ≥ 2 3. Determine if f and g are inverse functions: ! " () = !"" + # !" () = !" "# Determine if the following functions are even, odd or neither. SHOW ALL WORK. 4. ! " () = ! " ! ! ________________ 5. ! " () = ! ! ! " ________________ 6. Use the remainder theorem and synthetic division to find the indicated value. !" () = " ! + "" " ! !" + #$ " = " ! ! () = ____________________
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1st Semester Pre-calculus Sample Exam Semester Pre...Pre‐Calculus, 1st Semester Exam Study Guide, 2011–12 Directions: Graph the following functions. Be careful to identify the
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Pre‐Calculus, 1st Semester Exam Study Guide, 2011–12 Directions: Graph the following functions. Be careful to identify the points where the graphs start, stop, and/or change directions.
1) 𝑓 𝑥 = 2 𝑥 − 1 2) 𝑓 𝑥 = −𝑥! + 3, 𝑥 < 2−4, 𝑥 ≥ 2
3. Determine if f and g are inverse functions: ! "( )=!""+# ! "( )=!" "# Determine if the following functions are even, odd or neither. SHOW ALL WORK. 4. ! "( )=! " !! ________________
5. ! "( )= !! !" ________________
6. Use the remainder theorem and synthetic division to find the indicated value. ! "( )= "! +""" !!"+#$%%" ="
! !( )= ____________________
7. Use the factor theorem and synthetic division to determine whether the first polynomial is a factor of the second. !+!"###$
%! !
&!'!
'+(!+(
yes or no:_______________ 8. Determine the maximum number of relative extrema for this polynomial function: ! "( )= "! +""# !$"+% _____________ 9. Graph this polynomial function by hand. Identify the zeros.
! "( )= !!!( )!
!+"( ) !!#( )"
10. Determine the zeros of this polynomial: ! "( )= !! +"!" !#!$ !%&!+$' _______________________________ 11. Determine a polynomial function of lowest degree with rational coefficients that has the following zero: !+ "
For 12–14, find the indicated sum using the formulas for the sums of powers of integers.
12. !! !"( )
!=#
#$
" ________________
13. !!
!="
"#
! ________________
14. !!+!( )
!="
"!
! ________________
For 15, use the principle of mathematical induction to prove that each statement is true for all positive integers n. Show all work. I won’t have much imagination as to how or where you got the steps in your proofs.
In 20–22, determine whether each sequence converges or diverges. If it converges, give the limit.
20. !"=!!
"+"
!"
________________
21. !"= !
!
"
"
#$
%
&'
"
________________
22. !"=!!+"
#!#!# ________________
23. Which statement about the function ! "( )= !!
"+!( ) #!$ + !!( ) is true? Circle the
correct answer. A. f(x) is both even and odd. B. f(x) is even but not odd. C. f(x) is odd but not even. D. f(x) is neither even nor odd. 24. Show that ! "( )=!"" !#"! +!" is an odd function. 25. What are the coordinates of the center of this ellipse?
A. B. C.
D.
!+!( )!
!"+!!#( )
!
$=%
!"#( ) !"!#( ) !!"#( ) !!"!#( )
26 An ellipse has its center at the origin, a horizontal major axis of length 100 units, and a focus at a distance of 40 units from the origin. What is the equation of this ellipse?
A. !!
"#!+!!
$#!=$ B.
!!
"#!+!!
$#!="
C.
!!
"#!+!!
$#!=% D.
!!
"#!+!!
$#!=%
In 27, determine whether the sequence converges. If so, give the sum.
27. !!!!
!
"#
$
"
# $ %
& '
"=%
(
)"
A. 2/5 B. 16/5 C. 16 D. diverges 28. Convert this general form of a circle into standard form: !!!!
!
""
+ #"
+#" "$"# +%=&?
A. !!!!
!
" +"( )#
+ # "$( )#
=%
B. !!!!
!
" +"( )#
+ # "$( )#
="%
C. !!!!
!
" +"( )#
+ # "$%( )#
=&#
D. !!!!
!
" +"( )#
+ # "$( )#
=%# 29. Determine the eccentricity of this hyperbola: !!!!
!
"#"$"%# #
$="&&
A. !!
!
"#
$ B.
!!
!
"#
# C. !!
!
"# D. !!
!
"
#
30. Find the center, endpoints of the major and minor axes, locations of foci, and eccentricity of !!!!
!
""
+# #""$"# +#%=&. Then graph the ellipse.
(4 points)
Center:___________________ Major axis endpoints:___________________ Minor axis endpoints:___________________ Foci:______________________ Eccentricity:_____________