math 1a, exam #1. - berkeley...
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Math 1A, Exam #1. 1. Find
(a)
𝑑𝑥𝑥(1 + 𝑙𝑛𝑥 !)
(b). (tan!!(𝑥))!
1 + 𝑥!!
!!𝑑𝑥
2. Prove that
(sin(𝑥!!
!!) + cos(𝑥!))𝑑𝑥 ≤ 2 .
3. Find the equation for the line through the point (2,4) ∈ 𝑹! that cut off the least area from the first quadrant. 4. Rotate 𝑦 = 2 − 𝑎𝑥 around the x-axis and determine the volume for 0 ≤ 𝑥 ≤ 1. (b). Find a such that the volume is as small as possible. 5. Write True or False for each of the following. (a). If 𝑓 𝑥 = cos 𝑡! 𝑑𝑡,!"
! then 𝑓! 𝜋 = 1. (b). If a function f satisfies 𝑑𝑓/𝑑𝑥 ≥ 0 on the real line, then 𝑓 𝑎 < 𝑓 𝑏 , whenever 𝑎 < 𝑏. (c). If 𝑓(𝑥) > 0, then 𝑓 𝑥 𝑑𝑥 ≥ 0!
! , for any real numbers a and b. (d). If a continuous function 𝑓 𝑥 defined on the real line satisfies 𝑓 0 = 5 and 𝑓 1 = 2, then there is a real number t so that 𝑓 𝑡 = 𝜋.
6. (a). Find. !"!"
if, 𝑦 = 𝑥!. (b). Find.
𝑑𝑑𝑥 𝑡! + 1𝑑𝑡.
!
!!
7. Find.
(a).
lim!→!
1𝑥! cos! 𝑡 𝑑𝑡
!
!
(b).
lim!→!
2𝑛
2𝑖𝑛
!!
!!!
8. Let 𝑎 > 0, 𝑏 > 0 and prove using 𝜀 − 𝛿 that,
lim!→!
1𝑎 + 𝑥
=1
𝑎 + 𝑏.