+1(978)310-4246 credencewriters@gmail.com
  

Supplementary Problems
1. Find the derivative of each function.
𝑥 7 −5𝑥
a) 𝑦 = 9𝑥 3 − 2𝑥 2 + 4𝑥 + 4
c) 𝑦 =
b) 𝑦 = (2𝑥 − 3)3
d) 𝑦 = 5𝑒 𝜋 − 5𝑒 𝑥 + 𝑥 𝑒
2. Find 𝑙𝑖𝑚ℎ→0
1
2
8
9
1 8
2
5( +ℎ) −5( )
â„Ž
3. For what value of 𝑥 𝑑𝑜𝑒𝑠 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑜𝑓 𝑓(𝑥) = 𝑒 𝑥 − 2𝑥 ℎ𝑎𝑣𝑒 𝑎 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙
tangent?
4. 𝐿𝑒𝑡 𝑓(𝑥) = 4𝑥 3 − 3𝑥 − 1
a) 𝐹𝑖𝑛𝑑 𝑎𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑜𝑓 𝑓 𝑎𝑡 𝑥 = 2.
b) 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 𝑜𝑓 𝑎𝑛𝑦 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ 𝑜𝑓 𝑓 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒
𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 𝑖𝑠 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 𝑦 = 𝑥 + 12.
5. Find equations of both lines through the point (2, −3) 𝑡ℎ𝑎𝑡 𝑎𝑟𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡
to the parabola 𝑦 = 𝑥 2 + 𝑥.
6. Let N be the normal line to the graph of 𝑦 = 𝑥 2 𝑎𝑡 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 (−2,4).
𝐴𝑡 𝑤ℎ𝑎𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑜𝑖𝑛𝑡 𝑄 𝑑𝑜𝑒𝑠 𝑁 𝑚𝑒𝑒𝑡 𝑡ℎ𝑒 𝑔𝑟𝑎𝑝ℎ?
𝑥 2 𝑖𝑓 𝑥 ≤ 2
𝑚𝑥 + 𝑏 𝑖𝑓 𝑥 > 2
Find the values of 𝑚 𝑎𝑛𝑑 𝑏 𝑡ℎ𝑎𝑡 𝑚𝑎𝑘𝑒 𝑓 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 𝑒𝑣𝑒𝑟𝑦𝑤ℎ𝑒𝑟𝑒.
7. Let 𝑓(𝑥) = {
8. Find a formula for the nth derivative of 𝑓(𝑥) = 1/𝑥.
Supplementary Problems
1a. Use the definition of the derivative 𝒇′ (𝒂) = 𝒍𝒊𝒎𝒙→𝒂
𝒇(𝒙)−𝒇(𝒂)
𝒙−𝒂
to show that the function 𝒇(𝒙) = |𝒙 − 𝟑| 𝒊𝒔 𝒏𝒐𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒃𝒍𝒆 𝒂𝒕 𝟑.
b. Sketch the graph of 𝒇 𝒂𝒏𝒅 𝒊𝒕𝒔 𝒅𝒆𝒓𝒊𝒗𝒂𝒕𝒊𝒗𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒔𝒂𝒎𝒆 𝒑𝒍𝒂𝒏𝒆.
2. Use the definition of the derivative 𝒇′ (𝒂) = 𝒍𝒊𝒎𝒉→𝟎
𝟐
𝒇(𝒂+𝒉)−𝒇(𝒂)
𝒉
𝟏
𝒙 𝒔𝒊𝒏 (𝒙) , 𝒙 ≠ 𝟎
to show that the function 𝒇(𝒙) = {
is differentiable
𝟎, 𝒙 = 𝟎
at 0.
𝟓 − 𝒙, 𝒙 < 𝟎 3. Find the derivative of the piecewise function 𝒇(𝒙) = { 𝟐 𝒙 − 𝟐𝒙 + 𝟓, 𝒙 ≥ 𝟎 4. In problem number 3 above, is the function 𝒇 𝒄𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔? 𝑰𝒔 𝒇′ continuous? 5. Consider the function 𝒇(𝒙) = { 𝒂𝒙 + 𝒃, 𝒙 ≤ −𝟏 𝒂𝒙 + 𝒙 + 𝟐𝒃, 𝒙 > −𝟏
𝟑
For what values 𝒂 𝒂𝒏𝒅 𝒃, 𝒊𝒔 𝒕𝒉𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒃𝒍𝒆 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒓𝒆𝒂𝒍
numbers?
6. Match the graph with its derivative.
7. Where is the function 𝒇(𝒙) = ⟦𝒙⟧ 𝒏𝒐𝒕 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒕𝒊𝒂𝒃𝒍𝒆, 𝒘𝒉𝒆𝒓𝒆 ⟦𝒙⟧ 𝒊𝒔
the greatest integer function.
8a. Sketch the graph of 𝒇(𝒙) = 𝒙 + |𝒙|
b. For what values of x is f differentiable?
c. Find a formula for 𝒇′
Find
𝑑𝑦
𝑑𝑥
(#1-10)
1. 𝑦 = 𝑒 𝑐𝑜𝑠𝑥
2
2. 𝑦 = 𝑥 20 𝑎𝑟𝑐𝑡𝑎𝑛𝑥
3. 𝑦 = 𝑥 𝑙𝑛𝑥
4. 𝑥𝑒 𝑦 = 𝑦𝑠𝑖𝑛𝑥
5. 𝑠𝑒𝑐(sinh (𝑥))
6. 𝑦 = 𝑒 𝑥𝑠𝑒𝑐𝑥
7. 𝑦 = 𝑙𝑜𝑔5 (1 + 2𝑥)2
8. 𝑦 = 3𝑠𝑖𝑛
−1 𝑥 2
9. 𝑦 = (𝑥 + 2)𝑐𝑜𝑡𝑥
10. ln(𝑥 + 𝑦) = 𝑥𝑦 − 𝑦 3
11. 𝐹𝑖𝑛𝑑 𝑦 ′ 𝑖𝑓 𝑥 𝑦 = 𝑦 𝑥
12. 𝐿𝑒𝑡 𝑓(𝑥) = 𝑙𝑜𝑔𝑏 (3𝑥 2 − 2). 𝐹𝑜𝑟 𝑤ℎ𝑎𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏 𝑖𝑠 𝑓 ′ (1) = 3.
𝑥 𝑛
13. Show that 𝑙𝑖𝑚𝑛→∞ (1 + ) = 𝑒 𝑥 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑥 > 0.
𝑛
14. 𝐹𝑖𝑛𝑑 𝑙𝑖𝑚𝑡→0
𝑡3
𝑡𝑎𝑛3 (2𝑡)
15. 𝐼𝑓 𝑔(𝑥) = 2𝑥 3 + 𝑙𝑛𝑥 𝑖𝑠 ℎ𝑡𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑓(𝑥), 𝑓𝑖𝑛𝑑 𝑙𝑖𝑚𝑥→0
𝑓(1+𝑥)−𝑓(1)
𝑥
Supplementary Problems-Chain Rule & Implicit Differentiation
1. Find the derivative
𝒅𝒚
𝒅𝒙
1
a) 𝑦 = (𝑒 𝑥 − 𝑥 2 )2
b) 𝑦 = 𝑐𝑜𝑠(𝑥 3 − 2𝑥 + 5)
2
2
c) 𝑦 = 𝑒 𝑐𝑠𝑐 𝑥 (𝑐𝑜𝑠 2 𝑥 2 )
d) 𝑦 = tan⁡(𝑥 − 𝑦)
e) 𝑦 = |2𝑥 − 1|⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑓)⁡𝑦 2 + 𝑠𝑖𝑛−1 𝑦 = 2𝑥
𝑥
2. Find an equation of the tangent line to the graph of 𝑓(𝑥) = 𝑠𝑖𝑛(2𝑥) + 𝑐𝑜𝑠 ( )
2
at the point (0,1).
3. If 𝑦 = 𝐴𝑒 −2𝑥 + 𝐵𝑒 −𝑥 , 𝑤ℎ𝑒𝑟𝑒⁡𝐴⁡𝑎𝑛𝑑⁡𝐵⁡𝑎𝑟𝑒⁡𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠, 𝑠ℎ𝑜𝑤⁡𝑡ℎ𝑎𝑡
𝑦 ′′ + 3𝑦 ′ + 2𝑦 = 0.
4. 𝐹𝑖𝑛𝑑⁡𝑎𝑙𝑙⁡𝑝𝑜𝑖𝑛𝑡𝑠⁡𝑜𝑛⁡𝑡ℎ𝑒⁡𝑐𝑢𝑟𝑣𝑒⁡𝑥 2 𝑦 2 + 𝑥𝑦 = 2⁡𝑤ℎ𝑒𝑟𝑒⁡𝑡ℎ𝑒⁡𝑠𝑙𝑜𝑝e⁡𝑜𝑓⁡𝑡ℎ𝑒⁡
tangent line is -1.
5a) Suppose 𝑓⁡𝑖𝑠⁡𝑎⁡𝑜𝑛𝑒 − 𝑡𝑜 − 𝑜𝑛𝑒⁡𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒⁡𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛⁡𝑎𝑛𝑑⁡𝑖𝑡𝑠⁡𝑖𝑛𝑣𝑒𝑟𝑠𝑒
⁡⁡⁡𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛⁡⁡𝑓 −1 ⁡𝑖𝑠⁡𝑎𝑙𝑠𝑜⁡𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒. 𝑈𝑠𝑒⁡𝑖𝑚𝑝𝑙𝑖𝑐𝑖𝑡⁡𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑡𝑖𝑜𝑛⁡𝑡𝑜⁡
𝑠ℎ𝑜𝑤⁡𝑡ℎ𝑎𝑡⁡
(𝑓 −1 )′ (𝑥) =
1
𝑓′ (𝑓−1 (𝑥))
⁡𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑⁡𝑡ℎ𝑎𝑡⁡𝑡ℎ𝑒⁡𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
is not zero.
2
b) 𝐼𝑓⁡𝑓(4) = 5⁡𝑎𝑛𝑑⁡𝑓 ′ (4) = ⁡, 𝑓𝑖𝑛𝑑⁡(𝑓 −1 )′ ⁡(5).
3
6a) 𝑆ℎ𝑜𝑤⁡𝑡ℎ𝑎𝑡⁡𝑓(𝑥) = 𝑥 + 𝑒 𝑥 ⁡𝑖𝑠⁡𝑜𝑛𝑒 − 𝑡𝑜 − 𝑜𝑛𝑒.
b) 𝑊ℎ𝑎𝑡⁡𝑖𝑠⁡𝑡ℎ𝑒⁡𝑣𝑎𝑙𝑢𝑒⁡𝑜𝑓⁡𝑓 −1 (1)?
c) 𝑈𝑠𝑒⁡𝑡ℎ𝑒⁡𝑓𝑜𝑟𝑚𝑢𝑙𝑎⁡𝑓𝑟𝑜𝑚⁡6𝑎)⁡𝑡𝑜⁡𝑓𝑖𝑛𝑑⁡(𝑓 −1 )′ ⁡(1).
7. 𝐹𝑖𝑛𝑑⁡𝑦 ′′ ⁡𝑖𝑛⁡𝑡𝑒𝑟𝑚𝑠⁡𝑜𝑓⁡𝑥⁡𝑎𝑛𝑑⁡𝑦⁡𝑜𝑛𝑙𝑦.
𝑥3 − 𝑦3 = 7
8. Show
𝑑
𝑑𝑥
⁡(𝑠𝑒𝑐 −1 𝑥) =
1
𝑥√𝑥 2 −1
Supplementary Problems-Product and Quotient Rules
1. Find the derivative
1
1
1
𝑥
𝑥
𝑥
b) 𝑔(𝑡) = (2𝑡 5 − 𝑡)(𝑡 3 − 2𝑡 + 1)
a) 𝑓(𝑥) = + 2 + 3
c) 𝑓(𝑥) =
𝑥𝑒 𝑥
1
𝑑) 𝑦 = (𝑠𝑒𝑐𝑥 − 2𝑐𝑜𝑠𝑥𝑡𝑎𝑛𝑥)
𝑥 2 −𝑥
𝑒
2. Given the function 𝑓(𝑥) =
𝑥3
𝑥+1
1
𝑎𝑡 (1, )
2
a) Find the slope of the tangent line for the given function at the given point.
b) Find an equation of the tangent line to the graph of f at the given point.
c) Find the points, if any where the graph of the function has a horizontal
tangent line.
3. Find 𝑦 ′′ 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑦 = 2𝑠𝑖𝑛𝑥 − 3𝑐𝑜𝑠𝑥
4. Find all points on the graph of f where the tangent line is horizontal.
a) 𝑓(𝑥) = 2𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥
𝑏) 𝑓(𝑥) = 𝑠𝑒𝑐𝑥
5. Find the limits
a) 𝑙𝑖𝑚𝑥→0
6. If 𝑔(𝑥) =
𝑠𝑖𝑛𝑥
𝑠𝑖𝑛𝜋𝑥
𝑥
𝑒𝑥
, 𝑓𝑖𝑛𝑑 𝑔(𝑛) (𝑥).
7. 𝐼𝑓 𝑓(𝑥) = 𝑙𝑖𝑚𝑡→𝑥
8. Prove
𝑑
𝑑𝑥
𝑏) 𝑙𝑖𝑚𝑥→1
𝑠𝑒𝑐𝑡−𝑠𝑒𝑐𝑥
𝑡−𝑥
𝜋
, find 𝑓 ′ ( )
(𝑠𝑒𝑐𝑥) = 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥
4
sin (𝑥−1)
𝑥 2 +𝑥−2
Exam 2
Math 190 Summer 2022
1. Exam 2 will cover sections 3.1-3.9, lecture notes, handouts.
2. Open book and notes are allowed. A calculator will not be needed on this exam and should not be used.
3. If you use notation that is different from the book and/or notes then a zero will be assigned for that
problem.
4. You must show all work for credit.
5. Do not use pen. Erase anything you do not want graded. Sloppy writing will incur penalties.
6. I prefer the exam to be printed out. If you cannot print out the exam, then write your solutions
on a blank piece of paper. If writing on blank paper, keep to the structure the way the exam
is written. Example: If there are three questions on the first page of the exam, then the
first page of your paper should have only three questions.
7. Your exam must be uploaded in pdf form under one file.
8. When you upload your exam save it with your last name. After you upload your exam,
double check it was uploaded correctly. If you fail to upload correctly, then your exam
may not be graded and you will not be allowed to resubmit after the deadline.
9. You must submit the cover page with your honor code signature or it will not be accepted and it will
incur a late penalty.
Failure to follow any of the conditions will result in points taken off of your exam.
I will not address any questions during the exam. Do your very best to answer each problem.
Honor Code Pledge:
I pledge that on my honor, as an El Camino College student,
I have neither given nor received unauthorized assistance on this
exam.
Signature__________________________________________
1. Find the following limits. If the limit does not exist, state why.
L’Hopital’s Rule (chapter 4) cannot be used. No tables or graphs. Justify all work.
a) 𝑙𝑖𝑚h→0
b) 𝑙𝑖𝑚𝑥→0
𝜋𝑠𝑖𝑛2 (ℎ)
2ℎ𝑠𝑖𝑛(2ℎ)
1−cos (𝑏𝑥)
𝑎𝑥
𝑤ℎ𝑒𝑟𝑒 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠 𝑎𝑛𝑑 𝑎 ≠ 0, 𝑏 ≠ 0
Continued…
c) 𝑙𝑖𝑚𝑥→𝜋
3
𝑠𝑒𝑐𝑥−2
𝜋
𝑥− 3
d) 𝑙𝑖𝑚𝑛→∞ (1 +
1
3𝑛
𝑛
)
𝑠𝑖𝑛𝑥
2a. If g(x) = 𝑐𝑜𝑠 −1 (𝑐𝑜𝑠𝑥), 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑔′ (𝑥) = |𝑠𝑖𝑛𝑥|
2b) Where is 𝑔(𝑥) 𝑛𝑜𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒?
1
𝑥, 𝑥 < 1 3. Find the derivative of the piecewise function 𝑓(𝑥) = { 2 √𝑥 − 1, 𝑥 ≥ 1 Write your derivative as a piecewise function. No graphs 4. At what point(s), if any, is the line y = x − 1 parallel to the tangent line to the graph of 𝑦 = √25 − 𝑥 2 ? No graphs! 5. 𝐺𝑖𝑣𝑒𝑛 𝑥𝑒 𝑦 = 4𝑥 2 a) Find 𝑦 ′ b) 𝐹𝑖𝑛𝑑 𝑦 ′′ and make sure your answer 𝑦 ′′ 𝑖𝑠 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑥 𝑎𝑛𝑑 𝑦 𝑜𝑛𝑙𝑦 𝑎𝑛𝑑 𝑖𝑛 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑑 𝑓𝑎𝑐𝑡𝑜𝑟𝑒𝑑 𝑓𝑜𝑟𝑚. 6. Find the derivative 𝑦 ′ . a) 𝑦 = ln(𝑠𝑖𝑛−1 (3𝑥 3 )) b) 𝑦 = 2𝜋 𝑒 + 𝜋 𝑒 − 𝑥 𝑒 + 𝑥 𝑒 𝑥 continued… c) y 2 + tan−1 y = 2x (Answer should be in simplified form, no complex fractions) d) 𝑦 = 𝑏 𝑡𝑎𝑛 −1 (𝜋𝜃) 𝑤ℎ𝑒𝑟𝑒 𝑏 > 0 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
7. Given the function 𝑔(𝑥) = 𝑥 2 + 2𝑥 − 1, 𝑥 ≥ −1
𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑎𝑡 𝑥 = 2 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑒𝑥𝑝𝑙𝑖𝑐𝑖𝑡𝑙𝑦
finding the inverse function.
8. Given the equation 𝑥 2 − 𝑥𝑦 + 𝑦 2 = 1.
a) 𝐹𝑖𝑛𝑑
𝑑𝑦
𝑑𝑥
b) Find 𝑡ℎ𝑒 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 (𝑥, 𝑦) 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑜𝑖𝑛𝑡𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑡𝑎𝑛𝑔𝑒𝑛𝑡
to the curve is horizontal.
9. Find the equation of the line which passes thru the point (1,2) and is normal
to the curve 𝑦 =
𝑥2
4
.
10. Suppose an object is traveling along the right side of the parabola 𝑦 = 𝑥 2
at a rate such that its distance from the origin is increasing at 9 cm/min.
a) At what rate is the x-coordinate of the bug increasing at the point (2,4)?
1. Diagram
2. Given:
3. Want:
4. Formula
5. Differentiate/Solve
𝑡
11. A particle moves according to the position function 𝑠(𝑡) = (1+𝑡)2 on the interval
𝑡 ≥ 0, 𝑤ℎ𝑒𝑟𝑒 𝑡 𝑖𝑠 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑖𝑛 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝑎𝑛𝑑 𝑠 𝑖𝑛 𝑓𝑒𝑒𝑡.
a) When is the particle at rest?
b) When is the particle moving in the positive direction?

Purchase answer to see full
attachment

  
error: Content is protected !!