Post on 15-Oct-2020
Precalculus First Semester Review
18) Use change of base to approximate the following to three decimal places 𝑎) log! 216 𝑏) log! 91 𝐥𝐨𝐠𝟐𝟏𝟔 𝐥𝐨𝐠𝟖
= 𝟐.𝟓𝟖𝟓 𝐥𝐨𝐠𝟗𝟏𝐥𝐨𝐠𝟐
= 𝟔.𝟓𝟎𝟖 19) Evaluate the following limits 𝑎) lim!→!
!!!!!!!"!!!
𝑏) lim!→!! !!! !!! !!!!
!
𝑐) lim!→!
!!!! !!
𝑑) lim!→!!!
!!!!!!!
20) A certain population (in thousands) grows according to the equation 𝑃 = 40𝑒!.!"#!
where P represents the population and t represent the number of years since 2000. a) Determine the year when the population 60 thousand will be reached. 𝟔𝟎 = 𝟒𝟎𝒆𝟎.𝟎𝟐𝟓𝒕 𝟏.𝟓 = 𝒆𝟎.𝟎𝟐𝟓𝒕 𝐥𝐧 𝟏.𝟓 = 𝟎.𝟎𝟐𝟓𝒕 𝒕 = 𝐥𝐧 𝟏.𝟓
𝟎.𝟎𝟐𝟓= 𝟏𝟔.𝟐𝟏𝟗 𝒚𝒆𝒂𝒓𝒔
𝟐𝟎𝟏𝟕 b) Determine the population in 2003 𝑷 𝟑 = 𝟒𝟎𝒆𝟎.𝟎𝟐𝟓(𝟑) = 𝟒𝟎.𝟏𝟏𝟓 𝒐𝒓 𝟒𝟏,𝟏𝟏𝟓 c) Determine the initial population. 𝟒𝟎,𝟎𝟎𝟎 21) Given 𝐴 = 1 0
3 4 and 𝐵 =−3 20 3
a) Find 3𝐴 + 𝐵 b) Find AB d) Find 𝐴!! 22) Given the geometric sequence with 𝑎! = 32 and 𝑟 = !
!
a) Write the formula 𝑎! for the sequence
𝒂𝒏 = 𝟑𝟐 𝟏𝟐
𝒏!𝟏
b) Write the first five terms of the sequence 𝟑𝟐,𝟏𝟔,𝟖,𝟒,𝟐 c) Determine the term 𝑎!"
𝒂𝟏𝟐 = 𝟑𝟐 𝟏𝟐
𝟏𝟐!𝟏= 𝟏
𝟑𝟐
Precalculus First Semester Review
23) Given the arithmetic sequence with 𝑎! = 20 and 𝑑 = −3 a) Write the formula 𝑎! for the sequence 𝒂𝒏 = −𝟑𝒏+ 𝟐𝟑 b) Write the first four terms of the sequence 𝟐𝟎,𝟏𝟕,𝟏𝟒,𝟏𝟏 c) Determine the term 𝑎!" 𝒂𝟏𝟎 = −𝟑 𝟏𝟎 + 𝟐𝟑 = −𝟕 24) Determine the value of the account with an initial balance of $3000 that is compounded quarterly with an interest rate of 8% after 5 years.
𝑨 = 𝟑𝟎𝟎𝟎 𝟏+ .𝟎𝟖𝟒
𝟒 𝟓= 𝟒𝟒𝟓𝟕.𝟖𝟒
25) Determine the value of an account with an initial balance of $3000 that is compounded continuously with an interest rate of 8% after 5 years. 𝑨 = 𝟑𝟎𝟎𝟎𝒆.𝟎𝟖(𝟓) = 𝟒𝟒𝟕𝟓.𝟒𝟕 26) Determine the coefficient of the following terms in the expansion of 3𝑥 + 6𝑦 ! a) 𝑥!𝑦! b) 𝑥!𝑦 𝟗𝟕𝟐𝟎 𝟐𝟒𝟑𝟎 27) Evaluate the following a) 𝐶! ! b) 𝐶! ! c) 𝐶! ! 28) Write in a + bi form: !!!!
!!!!
𝟐!𝟒𝒊𝟑!𝟐𝒊
𝟑!𝟐𝒊𝟑!𝟐𝒊
= 𝟔!𝟒𝒊!𝟏𝟐𝒊!𝟖𝒊𝟐
𝟗!𝟒𝒊𝟐= 𝟔!𝟏𝟔𝒊!𝟖
𝟗!𝟒= !𝟐!𝟏𝟔𝒊
𝟏𝟑= − 𝟐
𝟏𝟑+ 𝟏𝟔𝒊
𝟏𝟑
29) Graph the function and tell whether or not it has a point of discontinuity at x = 0. If there is a discontinuity, tell whether it is removable or non-‐removable. a) 𝑓 𝑥 = !
! b) 𝑔 𝑥 = !!!!
!
non-‐removable removable
Precalculus First Semester Review
30) Simplify a) log!! 11! b) ln !
! c) log!
!!"
𝟒 −𝟏 −𝟑 31) Find the zeros of the function algebraically. a) 𝑓 𝑥 = 3𝑥! + 2𝑥 − 5 b) 𝑓 𝑥 = 𝑥! − 36𝑥 𝟑𝒙𝟐 + 𝟐𝒙− 𝟓 = 𝟎 𝒙𝟑 − 𝟑𝟔𝒙 = 𝟎 𝟑𝒙+ 𝟓 𝒙− 𝟏 = 𝟎 𝒙 𝒙𝟐 − 𝟑𝟔 = 𝟎 𝒙 = − 𝟓
𝟑,𝟏 𝒙 𝒙− 𝟔 𝒙+ 𝟔 = 𝟎
𝒙 = 𝟎,−𝟔,𝟔 32) Determine whether the functions are even, odd, or neither a) 𝑓 𝑥 = 3𝑥! − 2𝑥 b) 𝑔 𝑥 = 2𝑥! − 3𝑥! + 5 ODD EVEN 33) Do the following functions have an inverse? If so, find it a) 𝑓 𝑥 = 𝑥! + 1! b) 𝑦 = 2 − 𝑥 𝒙 = 𝒚𝟐 + 𝟏𝟑 𝒙 = 𝟐− 𝒚 𝒙𝟑 = 𝒚𝟐 + 𝟏 𝒙𝟐 = 𝟐− 𝒚 𝒙𝟑 − 𝟏 = 𝒚𝟐 𝒚 = 𝟐− 𝒙𝟐 𝒚 = 𝒙𝟑 − 𝟏 34) Divide !
!!!!!!!!
with long division 35) Divide !!!!!!!!!!!!
with synthetic 𝒙𝟐 + 𝟐𝒙+ 𝟑 𝟓𝒙𝟑 − 𝟓𝒙𝟐 + 𝟑𝒙− 𝟑− 𝟏
𝒙!𝟏
36) Where is the following function discontinuous? a) 𝑦 = !!!
!!! b) 𝑦 = !
!!!!!!!
𝒙 = 𝟑 𝒙 = −𝟏,−𝟑