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MATH 2413 Calculus I
WRITTEN HOMEWORK ASSIGNMENT 8
REVIEW PACKAGE FOR TEST 3
Selected problems from Chapters 4 and 5.
Name: ________________________________
PSID: _________________________________________
Instructions:
Print out this file (or download and work directly on your tablet / laptop) and complete the problems. If the problem is
from the text, the section number and problem number are in parentheses. You must submit your work in this format
to get credit.
If you choose not to print the homework file and wish to use your own paper, you must copy the problem and set up
your pages to follow the same structure of the homework file. Your submission should be an exact replica of the
online file (in terms of pages and questions on each page).
Use a blue or black pen or a pencil (dark) and write neatly. Do not use any red pen(s).
Write your work in the space provided and your solutions in the boxes provided. You must show all your work in
order to receive credit for a problem. The graders are looking for proper notation, good logic, and correct answers.
Neat and organized work is important for full credit.
Your work is to be uploaded into CANVAS before the posted date and time. It should be submitted as a
SINGLE PDF file, not pictures, not separate files, not a word doc. Follow the instructions given on CANVAS to
upload the file and click on the submit button. After submitting, check that you have submitted a single PDF file
successfully, and all pages are readable. If the grader can’t read your work, the problem will not be graded. It is your
responsibility to check that the file is saved in the system and is complete. No exceptions.
Late work cannot be submitted in lab or via email. Only uploaded ON TIME homework will be graded. Please plan
ahead.
Students are expected to adhere UH Academic Honesty Policy. Submitting someone else’s work is considered a
violation of this policy. Unless otherwise stated, calculators or other software are not allowed.
Sign Below to Promise that you will follow UH Academic Honesty Policy:
Signature:__________________________________________________________________
Solve the problems in this review sheet; if you can’t print, you can use notebook paper but you MUST follow
the same order and create an exact replica. You must show work unless otherwise stated. For free response
problems on the test: you must show work to receive credit.
This review sheet should not be your only source while studying for the exam. Take and retake the practice test.
We recommend that your first attempt on PT3 is at least one week prior to the testing window. You will receive
valuable feedback from this attempt. Use the feedback, study and keep taking the PT (before the deadline) to
strengthen your skills. PT is designed to help you in your studies; consider it to be a tool to use in your
preparations.
- Given the following information: f (2) 5, f '(2) 4, f (5) 6, f'(5) 7 , find
1
f '(5)?
(^)
- The slope of the tangent line to the curve f ( )x at (2,8) is 10. Given f ( )x is invertible, find the
equation of the tangent line to
1 f ( )x
at x^ ^8.
- Are the following functions monotonic and hence invertible? Verify your answer using calculus. You
don’t need to include the work.
(a) (^)
4 2 f x 2 x 5 x 2 Yes or no: _______
(b) (^)
5 f x 4 x 10 x 2 Yes or no: _______
(c) (^)
5 f x 2 x 6 x 3 Yes or no: _______
(d) (^)
2 x 1
f x e
Yes or no: _______
(e) f (^) x (^) 5cos(4 )x Yes or no: _______
(f) (^) 2
f x
x
Yes or no: _______
(g) (^)
2 f x x 1 Yes or no: _______
- Find the indicated derivatives:
(a)
2 f ( )x 3arcsin 2x ; find f ' x (^) ?
(b)
4 f ( )x 10arctan(5 x ); find f ' x (^) ?
(c) f ( )x arcsec 4 x; find f ' (^) x (^) ?
(d) (^)
10
( ) 2arcsin
x
f x e ; find f ' x (^) ?
(e) (^)
7
( ) 5arctan
x f x e ; find f ' (^) x (^) ?
(f) f^ ( )x^ ^ arcsin 2 x; find
f
- Find the slope of the tangent line to the following curve at x 1.
4 4
3
( ) ln
x x
f x
x
- Use logarithmic differentiation to find the derivatives:
(a) (^)
12 2
x
y x
(b)
sin(2 ) 2 3
x
y x
- Let m and n be to positive real numbers. Given 8 m 5 n 20 , find the maximum value for the
product mn.
- A rectangle is drawn in the first quadrant so that the base lies on the x-axis, bottom left corner on the
origin, top right corner is on the line 2 y 4 x 30. Draw a diagram. Find the maximum area of such
a rectangle.
- A rectangle is drawn in the first and second quadrants so that the base lies on the x-axis, upper left
corner on the line y 4 x 10 , top right corner is on the line y 4 x 10. Draw a diagram. Find the
maximum area of such a rectangle.
- A rectangle is drawn so that the base lies on the x-axis, and the top corners are on the curve
2
y 48 x. Draw a diagram. Find the maximum area of such a rectangle.
NOTE: Solve all optimization problems on the corresponding WHW, correspond quiz; and in class notes.
Section 5.3:
- Compute the following limits – if possible. State the method(s) you are using, and use proper notation.
(a) 2 0
lim
x x
x
e e x
x
(b)
2
0
2sin( )
lim
x
x x
x
- Compute the following limits – if possible. State the method(s) you are using, and use proper notation.
(a)
3
lim 1
x
x
e x
(b)
3
ln( ) lim 4 5 x
x
x
- Compute the following limits – if possible. State the method(s) you are using, and use proper notation.
(a)
lim 12 sin
x
x
x
^
(b)
5
2 ln( ) lim 6 2
x
x
x
Exercise: Solve all limit (L’Hospital’s rule) questions from the corresponding WHW and from class notes.
Exercise: Use logarithmic differentiation to find the derivative of f(x). Also, find the slope of the line that is
tangent to the curve at x=0. Find the equation of this tangent line.
2
3
x x
f x
x
Exercises (5.1): Solve but do not have to turn in as part of the review package. Make sure you can solve
them!
(i) An open top box will be made using a rectangular tin of 24 in by 36 in by cutting congruent squares
from all 4 corners and then folding. If the goal is to get the largest possible area, what are the dimensions
of the squares cut from the corners?
(ii) An open top box with a square base will be made to hold 64 cubic feet. What are the dimensions that
will minimize the surface area?
(iii) A closed top box with a square base will be made to hold 128 cubic feet. What are the dimensions
that will minimize the surface area?
Exercise: A rectangle is inscribed on the unit circle; each corner of the rectangle is on the unit circle. Draw a
diagram; label it. Find the largest possible area for such a rectangle.
Hint: The equation of the unit circle that resides in quadrant I is 𝑦 = √1 − 𝑥
ଶ
Take and retake the practice test. Good luck on your test!
