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Supplementary Problems
1. Find the derivative of each function.
ð‘¥ 7 âˆ’5ð‘¥
a) ð‘¦ = 9ð‘¥ 3 âˆ’ 2ð‘¥ 2 + 4ð‘¥ + 4
c) ð‘¦ =
b) ð‘¦ = (2ð‘¥ âˆ’ 3)3
d) ð‘¦ = 5ð‘’ ðœ‹ âˆ’ 5ð‘’ ð‘¥ + ð‘¥ ð‘’
2. Find ð‘™ð‘–ð‘šâ„Žâ†’0
1
2
8
9
1 8
2
5( +â„Ž) âˆ’5( )
â„Ž
3. For what value of ð‘¥ ð‘‘ð‘œð‘’ð‘  ð‘¡â„Žð‘’ ð‘”ð‘Ÿð‘Žð‘â„Ž ð‘œð‘“ ð‘“(ð‘¥) = ð‘’ ð‘¥ âˆ’ 2ð‘¥ â„Žð‘Žð‘£ð‘’ ð‘Ž â„Žð‘œð‘Ÿð‘–ð‘§ð‘œð‘›ð‘¡ð‘Žð‘™
tangent?
4. ð¿ð‘’ð‘¡ ð‘“(ð‘¥) = 4ð‘¥ 3 âˆ’ 3ð‘¥ âˆ’ 1
a) ð¹ð‘–ð‘›ð‘‘ ð‘Žð‘› ð‘’ð‘žð‘¢ð‘Žð‘¡ð‘–ð‘œð‘› ð‘œð‘“ ð‘¡â„Žð‘’ ð‘¡ð‘Žð‘›ð‘”ð‘’ð‘›ð‘¡ ð‘™ð‘–ð‘›ð‘’ ð‘¡ð‘œ ð‘¡â„Žð‘’ ð‘”ð‘Ÿð‘Žð‘â„Ž ð‘œð‘“ ð‘“ ð‘Žð‘¡ ð‘¥ = 2.
b) ð¹ð‘–ð‘›ð‘‘ ð‘¡â„Žð‘’ ð‘ð‘œð‘œð‘Ÿð‘‘ð‘–ð‘›ð‘Žð‘¡ð‘’ð‘  ð‘œð‘“ ð‘Žð‘›ð‘¦ ð‘ð‘œð‘–ð‘›ð‘¡ð‘  ð‘œð‘› ð‘¡â„Žð‘’ ð‘”ð‘Ÿð‘Žð‘â„Ž ð‘œð‘“ ð‘“ ð‘¤â„Žð‘’ð‘Ÿð‘’ ð‘¡â„Žð‘’
ð‘¡ð‘Žð‘›ð‘”ð‘’ð‘›ð‘¡ ð‘™ð‘–ð‘›ð‘’ ð‘–ð‘  ð‘ð‘Žð‘Ÿð‘Žð‘™ð‘™ð‘’ð‘™ ð‘¡ð‘œ ð‘¦ = ð‘¥ + 12.
5. Find equations of both lines through the point (2, âˆ’3) ð‘¡â„Žð‘Žð‘¡ ð‘Žð‘Ÿð‘’ ð‘¡ð‘Žð‘›ð‘”ð‘’ð‘›ð‘¡
to the parabola ð‘¦ = ð‘¥ 2 + ð‘¥.
6. Let N be the normal line to the graph of ð‘¦ = ð‘¥ 2 ð‘Žð‘¡ ð‘¡â„Žð‘’ ð‘ð‘œð‘–ð‘›ð‘¡ (âˆ’2,4).
ð´ð‘¡ ð‘¤â„Žð‘Žð‘¡ ð‘œð‘¡â„Žð‘’ð‘Ÿ ð‘ð‘œð‘–ð‘›ð‘¡ ð‘„ ð‘‘ð‘œð‘’ð‘  ð‘ ð‘šð‘’ð‘’ð‘¡ ð‘¡â„Žð‘’ ð‘”ð‘Ÿð‘Žð‘â„Ž?
ð‘¥ 2 ð‘–ð‘“ ð‘¥ â‰¤ 2
ð‘šð‘¥ + ð‘ ð‘–ð‘“ ð‘¥ > 2
Find the values of ð‘š ð‘Žð‘›ð‘‘ ð‘ ð‘¡â„Žð‘Žð‘¡ ð‘šð‘Žð‘˜ð‘’ ð‘“ ð‘‘ð‘–ð‘“ð‘“ð‘’ð‘Ÿð‘’ð‘›ð‘¡ð‘–ð‘Žð‘ð‘™ð‘’ ð‘’ð‘£ð‘’ð‘Ÿð‘¦ð‘¤â„Žð‘’ð‘Ÿð‘’.
7. Let ð‘“(ð‘¥) = {
8. Find a formula for the nth derivative of ð‘“(ð‘¥) = 1/ð‘¥.
Supplementary Problems
1a. Use the definition of the derivative ð’‡â€² (ð’‚) = ð’ð’Šð’Žð’™â†’ð’‚
ð’‡(ð’™)âˆ’ð’‡(ð’‚)
ð’™âˆ’ð’‚
to show that the function ð’‡(ð’™) = |ð’™ âˆ’ ðŸ‘| ð’Šð’” ð’ð’ð’• ð’…ð’Šð’‡ð’‡ð’†ð’“ð’†ð’ð’•ð’Šð’‚ð’ƒð’ð’† ð’‚ð’• ðŸ‘.
b. Sketch the graph of ð’‡ ð’‚ð’ð’… ð’Šð’•ð’” ð’…ð’†ð’“ð’Šð’—ð’‚ð’•ð’Šð’—ð’† ð’Šð’ ð’•ð’‰ð’† ð’”ð’‚ð’Žð’† ð’‘ð’ð’‚ð’ð’†.
2. Use the definition of the derivative ð’‡â€² (ð’‚) = ð’ð’Šð’Žð’‰â†’ðŸŽ
ðŸ
ð’‡(ð’‚+ð’‰)âˆ’ð’‡(ð’‚)
ð’‰
ðŸ
ð’™ ð’”ð’Šð’ (ð’™) , ð’™ â‰  ðŸŽ
to show that the function ð’‡(ð’™) = {
is differentiable
ðŸŽ, ð’™ = ðŸŽ
at 0.
ðŸ“ âˆ’ ð’™, ð’™ < ðŸŽ 3. Find the derivative of the piecewise function ð’‡(ð’™) = { ðŸ ð’™ âˆ’ ðŸð’™ + ðŸ“, ð’™ â‰¥ ðŸŽ 4. In problem number 3 above, is the function ð’‡ ð’„ð’ð’ð’•ð’Šð’ð’–ð’ð’–ð’”? ð‘°ð’” ð’‡â€² continuous? 5. Consider the function ð’‡(ð’™) = { ð’‚ð’™ + ð’ƒ, ð’™ â‰¤ âˆ’ðŸ ð’‚ð’™ + ð’™ + ðŸð’ƒ, ð’™ > âˆ’ðŸ
ðŸ‘
For what values ð’‚ ð’‚ð’ð’… ð’ƒ, ð’Šð’” ð’•ð’‰ð’† ð’‡ð’–ð’ð’„ð’•ð’Šð’ð’ ð’…ð’Šð’‡ð’‡ð’†ð’“ð’†ð’ð’•ð’Šð’‚ð’ƒð’ð’† ð’‡ð’ð’“ ð’‚ð’ð’ ð’“ð’†ð’‚ð’
numbers?
6. Match the graph with its derivative.
7. Where is the function ð’‡(ð’™) = âŸ¦ð’™âŸ§ ð’ð’ð’• ð’…ð’Šð’‡ð’‡ð’†ð’“ð’†ð’ð’•ð’Šð’‚ð’ƒð’ð’†, ð’˜ð’‰ð’†ð’“ð’† âŸ¦ð’™âŸ§ ð’Šð’”
the greatest integer function.
8a. Sketch the graph of ð’‡(ð’™) = ð’™ + |ð’™|
b. For what values of x is f differentiable?
c. Find a formula for ð’‡â€²
Find
ð‘‘ð‘¦
ð‘‘ð‘¥
(#1-10)
1. ð‘¦ = ð‘’ ð‘ð‘œð‘ ð‘¥
2
2. ð‘¦ = ð‘¥ 20 ð‘Žð‘Ÿð‘ð‘¡ð‘Žð‘›ð‘¥
3. ð‘¦ = ð‘¥ ð‘™ð‘›ð‘¥
4. ð‘¥ð‘’ ð‘¦ = ð‘¦ð‘ ð‘–ð‘›ð‘¥
5. ð‘ ð‘’ð‘(sinh (ð‘¥))
6. ð‘¦ = ð‘’ ð‘¥ð‘ ð‘’ð‘ð‘¥
7. ð‘¦ = ð‘™ð‘œð‘”5 (1 + 2ð‘¥)2
8. ð‘¦ = 3ð‘ ð‘–ð‘›
âˆ’1 ð‘¥ 2
9. ð‘¦ = (ð‘¥ + 2)ð‘ð‘œð‘¡ð‘¥
10. ln(ð‘¥ + ð‘¦) = ð‘¥ð‘¦ âˆ’ ð‘¦ 3
11. ð¹ð‘–ð‘›ð‘‘ ð‘¦ â€² ð‘–ð‘“ ð‘¥ ð‘¦ = ð‘¦ ð‘¥
12. ð¿ð‘’ð‘¡ ð‘“(ð‘¥) = ð‘™ð‘œð‘”ð‘ (3ð‘¥ 2 âˆ’ 2). ð¹ð‘œð‘Ÿ ð‘¤â„Žð‘Žð‘¡ ð‘£ð‘Žð‘™ð‘¢ð‘’ ð‘œð‘“ ð‘ ð‘–ð‘  ð‘“ â€² (1) = 3.
ð‘¥ ð‘›
13. Show that ð‘™ð‘–ð‘šð‘›â†’âˆž (1 + ) = ð‘’ ð‘¥ ð‘“ð‘œð‘Ÿ ð‘Žð‘›ð‘¦ ð‘¥ > 0.
ð‘›
14. ð¹ð‘–ð‘›ð‘‘ ð‘™ð‘–ð‘šð‘¡â†’0
ð‘¡3
ð‘¡ð‘Žð‘›3 (2ð‘¡)
15. ð¼ð‘“ ð‘”(ð‘¥) = 2ð‘¥ 3 + ð‘™ð‘›ð‘¥ ð‘–ð‘  â„Žð‘¡ð‘’ ð‘‘ð‘’ð‘Ÿð‘–ð‘£ð‘Žð‘¡ð‘–ð‘£ð‘’ ð‘œð‘“ ð‘“(ð‘¥), ð‘“ð‘–ð‘›ð‘‘ ð‘™ð‘–ð‘šð‘¥â†’0
ð‘“(1+ð‘¥)âˆ’ð‘“(1)
ð‘¥
Supplementary Problems-Chain Rule & Implicit Differentiation
1. Find the derivative
ð’…ð’š
ð’…ð’™
1
a) ð‘¦ = (ð‘’ ð‘¥ âˆ’ ð‘¥ 2 )2
b) ð‘¦ = ð‘ð‘œð‘ (ð‘¥ 3 âˆ’ 2ð‘¥ + 5)
2
2
c) ð‘¦ = ð‘’ ð‘ð‘ ð‘ ð‘¥ (ð‘ð‘œð‘  2 ð‘¥ 2 )
d) ð‘¦ = tanâ¡(ð‘¥ âˆ’ ð‘¦)
e) ð‘¦ = |2ð‘¥ âˆ’ 1|â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð‘“)â¡ð‘¦ 2 + ð‘ ð‘–ð‘›âˆ’1 ð‘¦ = 2ð‘¥
ð‘¥
2. Find an equation of the tangent line to the graph of ð‘“(ð‘¥) = ð‘ ð‘–ð‘›(2ð‘¥) + ð‘ð‘œð‘  ( )
2
at the point (0,1).
3. If ð‘¦ = ð´ð‘’ âˆ’2ð‘¥ + ðµð‘’ âˆ’ð‘¥ , ð‘¤â„Žð‘’ð‘Ÿð‘’â¡ð´â¡ð‘Žð‘›ð‘‘â¡ðµâ¡ð‘Žð‘Ÿð‘’â¡ð‘ð‘œð‘›ð‘ ð‘¡ð‘Žð‘›ð‘¡ð‘ , ð‘ â„Žð‘œð‘¤â¡ð‘¡â„Žð‘Žð‘¡
ð‘¦ â€²â€² + 3ð‘¦ â€² + 2ð‘¦ = 0.
4. ð¹ð‘–ð‘›ð‘‘â¡ð‘Žð‘™ð‘™â¡ð‘ð‘œð‘–ð‘›ð‘¡ð‘ â¡ð‘œð‘›â¡ð‘¡â„Žð‘’â¡ð‘ð‘¢ð‘Ÿð‘£ð‘’â¡ð‘¥ 2 ð‘¦ 2 + ð‘¥ð‘¦ = 2â¡ð‘¤â„Žð‘’ð‘Ÿð‘’â¡ð‘¡â„Žð‘’â¡ð‘ ð‘™ð‘œð‘eâ¡ð‘œð‘“â¡ð‘¡â„Žð‘’â¡
tangent line is -1.
5a) Suppose ð‘“â¡ð‘–ð‘ â¡ð‘Žâ¡ð‘œð‘›ð‘’ âˆ’ ð‘¡ð‘œ âˆ’ ð‘œð‘›ð‘’â¡ð‘‘ð‘–ð‘“ð‘“ð‘’ð‘Ÿð‘’ð‘›ð‘¡ð‘–ð‘Žð‘ð‘™ð‘’â¡ð‘“ð‘¢ð‘›ð‘ð‘¡ð‘–ð‘œð‘›â¡ð‘Žð‘›ð‘‘â¡ð‘–ð‘¡ð‘ â¡ð‘–ð‘›ð‘£ð‘’ð‘Ÿð‘ ð‘’
â¡â¡â¡ð‘“ð‘¢ð‘›ð‘ð‘¡ð‘–ð‘œð‘›â¡â¡ð‘“ âˆ’1 â¡ð‘–ð‘ â¡ð‘Žð‘™ð‘ ð‘œâ¡ð‘‘ð‘–ð‘“ð‘“ð‘’ð‘Ÿð‘’ð‘›ð‘¡ð‘–ð‘Žð‘ð‘™ð‘’. ð‘ˆð‘ ð‘’â¡ð‘–ð‘šð‘ð‘™ð‘–ð‘ð‘–ð‘¡â¡ð‘‘ð‘–ð‘“ð‘“ð‘’ð‘Ÿð‘’ð‘›ð‘¡ð‘–ð‘Žð‘¡ð‘–ð‘œð‘›â¡ð‘¡ð‘œâ¡
ð‘ â„Žð‘œð‘¤â¡ð‘¡â„Žð‘Žð‘¡â¡
(ð‘“ âˆ’1 )â€² (ð‘¥) =
1
ð‘“â€² (ð‘“âˆ’1 (ð‘¥))
â¡ð‘ð‘Ÿð‘œð‘£ð‘–ð‘‘ð‘’ð‘‘â¡ð‘¡â„Žð‘Žð‘¡â¡ð‘¡â„Žð‘’â¡ð‘‘ð‘’ð‘›ð‘œð‘šð‘–ð‘›ð‘Žð‘¡ð‘œð‘Ÿ
is not zero.
2
b) ð¼ð‘“â¡ð‘“(4) = 5â¡ð‘Žð‘›ð‘‘â¡ð‘“ â€² (4) = â¡, ð‘“ð‘–ð‘›ð‘‘â¡(ð‘“ âˆ’1 )â€² â¡(5).
3
6a) ð‘†â„Žð‘œð‘¤â¡ð‘¡â„Žð‘Žð‘¡â¡ð‘“(ð‘¥) = ð‘¥ + ð‘’ ð‘¥ â¡ð‘–ð‘ â¡ð‘œð‘›ð‘’ âˆ’ ð‘¡ð‘œ âˆ’ ð‘œð‘›ð‘’.
b) ð‘Šâ„Žð‘Žð‘¡â¡ð‘–ð‘ â¡ð‘¡â„Žð‘’â¡ð‘£ð‘Žð‘™ð‘¢ð‘’â¡ð‘œð‘“â¡ð‘“ âˆ’1 (1)?
c) ð‘ˆð‘ ð‘’â¡ð‘¡â„Žð‘’â¡ð‘“ð‘œð‘Ÿð‘šð‘¢ð‘™ð‘Žâ¡ð‘“ð‘Ÿð‘œð‘šâ¡6ð‘Ž)â¡ð‘¡ð‘œâ¡ð‘“ð‘–ð‘›ð‘‘â¡(ð‘“ âˆ’1 )â€² â¡(1).
7. ð¹ð‘–ð‘›ð‘‘â¡ð‘¦ â€²â€² â¡ð‘–ð‘›â¡ð‘¡ð‘’ð‘Ÿð‘šð‘ â¡ð‘œð‘“â¡ð‘¥â¡ð‘Žð‘›ð‘‘â¡ð‘¦â¡ð‘œð‘›ð‘™ð‘¦.
ð‘¥3 âˆ’ ð‘¦3 = 7
8. Show
ð‘‘
ð‘‘ð‘¥
â¡(ð‘ ð‘’ð‘ âˆ’1 ð‘¥) =
1
ð‘¥âˆšð‘¥ 2 âˆ’1
Supplementary Problems-Product and Quotient Rules
1. Find the derivative
1
1
1
ð‘¥
ð‘¥
ð‘¥
b) ð‘”(ð‘¡) = (2ð‘¡ 5 âˆ’ ð‘¡)(ð‘¡ 3 âˆ’ 2ð‘¡ + 1)
a) ð‘“(ð‘¥) = + 2 + 3
c) ð‘“(ð‘¥) =
ð‘¥ð‘’ ð‘¥
1
ð‘‘) ð‘¦ = (ð‘ ð‘’ð‘ð‘¥ âˆ’ 2ð‘ð‘œð‘ ð‘¥ð‘¡ð‘Žð‘›ð‘¥)
ð‘¥ 2 âˆ’ð‘¥
ð‘’
2. Given the function ð‘“(ð‘¥) =
ð‘¥3
ð‘¥+1
1
ð‘Žð‘¡ (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 ð‘¦ â€²â€² ð‘“ð‘œð‘Ÿ ð‘¡â„Žð‘’ ð‘“ð‘¢ð‘›ð‘ð‘¡ð‘–ð‘œð‘› ð‘¦ = 2ð‘ ð‘–ð‘›ð‘¥ âˆ’ 3ð‘ð‘œð‘ ð‘¥
4. Find all points on the graph of f where the tangent line is horizontal.
a) ð‘“(ð‘¥) = 2ð‘ ð‘–ð‘›ð‘¥ + ð‘ð‘œð‘ ð‘¥
ð‘) ð‘“(ð‘¥) = ð‘ ð‘’ð‘ð‘¥
5. Find the limits
a) ð‘™ð‘–ð‘šð‘¥â†’0
6. If ð‘”(ð‘¥) =
ð‘ ð‘–ð‘›ð‘¥
ð‘ ð‘–ð‘›ðœ‹ð‘¥
ð‘¥
ð‘’ð‘¥
, ð‘“ð‘–ð‘›ð‘‘ ð‘”(ð‘›) (ð‘¥).
7. ð¼ð‘“ ð‘“(ð‘¥) = ð‘™ð‘–ð‘šð‘¡â†’ð‘¥
8. Prove
ð‘‘
ð‘‘ð‘¥
ð‘) ð‘™ð‘–ð‘šð‘¥â†’1
ð‘ ð‘’ð‘ð‘¡âˆ’ð‘ ð‘’ð‘ð‘¥
ð‘¡âˆ’ð‘¥
ðœ‹
, find ð‘“ â€² ( )
(ð‘ ð‘’ð‘ð‘¥) = ð‘ ð‘’ð‘ð‘¥ð‘¡ð‘Žð‘›ð‘¥
4
sin (ð‘¥âˆ’1)
ð‘¥ 2 +ð‘¥âˆ’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.
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.
Lâ€™Hopitalâ€™s Rule (chapter 4) cannot be used. No tables or graphs. Justify all work.
a) ð‘™ð‘–ð‘šhâ†’0
b) ð‘™ð‘–ð‘šð‘¥â†’0
ðœ‹ð‘ ð‘–ð‘›2 (â„Ž)
2â„Žð‘ ð‘–ð‘›(2â„Ž)
1âˆ’cos (ð‘ð‘¥)
ð‘Žð‘¥
ð‘¤â„Žð‘’ð‘Ÿð‘’ ð‘Ž ð‘Žð‘›ð‘‘ ð‘ ð‘Žð‘Ÿð‘’ ð‘ð‘œð‘›ð‘ ð‘¡ð‘Žð‘›ð‘¡ð‘  ð‘Žð‘›ð‘‘ ð‘Ž â‰  0, ð‘ â‰  0
Continuedâ€¦
c) ð‘™ð‘–ð‘šð‘¥â†’ðœ‹
3
ð‘ ð‘’ð‘ð‘¥âˆ’2
ðœ‹
ð‘¥âˆ’ 3
d) ð‘™ð‘–ð‘šð‘›â†’âˆž (1 +
1
3ð‘›
ð‘›
)
ð‘ ð‘–ð‘›ð‘¥
2a. If g(x) = ð‘ð‘œð‘  âˆ’1 (ð‘ð‘œð‘ ð‘¥), ð‘ â„Žð‘œð‘¤ ð‘¡â„Žð‘Žð‘¡ ð‘”â€² (ð‘¥) = |ð‘ ð‘–ð‘›ð‘¥|
2b) Where is ð‘”(ð‘¥) ð‘›ð‘œð‘¡ ð‘‘ð‘–ð‘“ð‘“ð‘’ð‘Ÿð‘’ð‘›ð‘¡ð‘–ð‘Žð‘ð‘™ð‘’?
1
ð‘¥, ð‘¥ < 1 3. Find the derivative of the piecewise function ð‘“(ð‘¥) = { 2 âˆšð‘¥ âˆ’ 1, ð‘¥ â‰¥ 1 Write your derivative as a piecewise function. No graphs 4. At what point(s), if any, is the line y = x âˆ’ 1 parallel to the tangent line to the graph of ð‘¦ = âˆš25 âˆ’ ð‘¥ 2 ? No graphs! 5. ðºð‘–ð‘£ð‘’ð‘› ð‘¥ð‘’ ð‘¦ = 4ð‘¥ 2 a) Find ð‘¦ â€² b) ð¹ð‘–ð‘›ð‘‘ ð‘¦ â€²â€² and make sure your answer ð‘¦ â€²â€² ð‘–ð‘  ð‘–ð‘› ð‘¡ð‘’ð‘Ÿð‘šð‘  ð‘œð‘“ ð‘¥ ð‘Žð‘›ð‘‘ ð‘¦ ð‘œð‘›ð‘™ð‘¦ ð‘Žð‘›ð‘‘ ð‘–ð‘› ð‘ ð‘–ð‘šð‘ð‘™ð‘–ð‘“ð‘–ð‘’ð‘‘ ð‘“ð‘Žð‘ð‘¡ð‘œð‘Ÿð‘’ð‘‘ ð‘“ð‘œð‘Ÿð‘š. 6. Find the derivative ð‘¦ â€² . a) ð‘¦ = ln(ð‘ ð‘–ð‘›âˆ’1 (3ð‘¥ 3 )) b) ð‘¦ = 2ðœ‹ ð‘’ + ðœ‹ ð‘’ âˆ’ ð‘¥ ð‘’ + ð‘¥ ð‘’ ð‘¥ continuedâ€¦ c) y 2 + tanâˆ’1 y = 2x (Answer should be in simplified form, no complex fractions) d) ð‘¦ = ð‘ ð‘¡ð‘Žð‘› âˆ’1 (ðœ‹ðœƒ) ð‘¤â„Žð‘’ð‘Ÿð‘’ ð‘ > 0 ð‘–ð‘  ð‘Ž ð‘ð‘œð‘›ð‘ ð‘¡ð‘Žð‘›ð‘¡.
7. Given the function ð‘”(ð‘¥) = ð‘¥ 2 + 2ð‘¥ âˆ’ 1, ð‘¥ â‰¥ âˆ’1
ð¹ð‘–ð‘›ð‘‘ ð‘¡â„Žð‘’ ð‘‘ð‘’ð‘Ÿð‘–ð‘£ð‘Žð‘¡ð‘–ð‘£ð‘’ ð‘œð‘“ ð‘¡â„Žð‘’ ð‘–ð‘›ð‘£ð‘’ð‘Ÿð‘ ð‘’ ð‘“ð‘¢ð‘›ð‘ð‘¡ð‘–ð‘œð‘› ð‘Žð‘¡ ð‘¥ = 2 ð‘¤ð‘–ð‘¡â„Žð‘œð‘¢ð‘¡ ð‘’ð‘¥ð‘ð‘™ð‘–ð‘ð‘–ð‘¡ð‘™ð‘¦
finding the inverse function.
8. Given the equation ð‘¥ 2 âˆ’ ð‘¥ð‘¦ + ð‘¦ 2 = 1.
a) ð¹ð‘–ð‘›ð‘‘
ð‘‘ð‘¦
ð‘‘ð‘¥
b) Find ð‘¡â„Žð‘’ ð‘ð‘œð‘œð‘Ÿð‘‘ð‘–ð‘›ð‘Žð‘¡ð‘’ð‘  (ð‘¥, ð‘¦) ð‘œð‘“ ð‘Žð‘™ð‘™ ð‘ð‘œð‘–ð‘›ð‘¡ð‘  ð‘œð‘› ð‘¡â„Žð‘’ ð‘ð‘¢ð‘Ÿð‘£ð‘’ ð‘¤â„Žð‘’ð‘Ÿð‘’ ð‘¡â„Žð‘’ ð‘¡ð‘Žð‘›ð‘”ð‘’ð‘›ð‘¡
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 ð‘¦ =
ð‘¥2
4
.
10. Suppose an object is traveling along the right side of the parabola ð‘¦ = ð‘¥ 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
ð‘¡
11. A particle moves according to the position function ð‘ (ð‘¡) = (1+ð‘¡)2 on the interval
ð‘¡ â‰¥ 0, ð‘¤â„Žð‘’ð‘Ÿð‘’ ð‘¡ ð‘–ð‘  ð‘šð‘’ð‘Žð‘ ð‘¢ð‘Ÿð‘’ð‘‘ ð‘–ð‘› ð‘ ð‘’ð‘ð‘œð‘›ð‘‘ð‘  ð‘Žð‘›ð‘‘ ð‘  ð‘–ð‘› ð‘“ð‘’ð‘’ð‘¡.
a) When is the particle at rest?
b) When is the particle moving in the positive direction?