Exam Title: Practice Test - OSH7
First Name (All CAPS):
Last Name (All CAPS):
CCID (All CAPS):
Student ID:
Bubble in your CCID below
TUYBFHLN-1
Page 1 of 14
Assignmetn 7 - Practice Midterm
Practice Midterm Exam
Duration: 1 hour
This test has 14 questions on 13 pages, for a total of 50 points.
Disclaimer: the format of your actual exam will be the same, however the content and
difficulty may vary.
• Read all the questions carefully before starting to work.
• Q1-Q10 are short-answer questions[3pts each]; put your answer in the boxes provided.
• Q11 - Q14 are long-answer; you should give complete arguments and explanations for all
your calculations.
• Continue on pages 9 and 10 AND leave a note “continued on page 9” (otherwise it won’t
get looked at) if you need more space.
• Students are permitted a single letter sized, two-sided, reference sheet. There are no
restriction on this page other than you must make your own.
• This is a closed-book examination. None of the following are allowed: documents
(other than the reference sheet) or electronic devices of any kind (including calculators,
computers, cell phones, etc.)
• Do not write over the QR code at the bottom the page. This will result in a grade of
0 for that problem because there will be around 10,000 pages that are sent to scanners.
That QR code is what links the page to you! Tampering with is equivalent to removing a
page from the examination and not submitting.
TUYBFHLN-2
Page 2 of 14
Assignmetn 7 - Practice Midterm
Short-Answer Questions. Put your answer in the box provided. Full marks will be given
for a correct answer placed in the box, while part marks may be given for work shown. Unless
otherwise stated, calculator ready answers are acceptable.
3 marks
1. Determine the domain of f (x) = ln (ln(x + 154))
Answer:
3 marks
2. Find the second derivative of f (x) = sin(3x2 )
Answer:
TUYBFHLN-3
Page 3 of 14
Assignmetn 7 - Practice Midterm
3 marks
3. Determine when the graph of y = xex is concave up.
Answer:
3 marks
4. A company’s monthly profit realized from renting x apartments is
P (x) = −10x2 + 1, 760x − 50, 000.
Compute the marginal profit when x = 50.
Answer:
TUYBFHLN-4
Page 4 of 14
Assignmetn 7 - Practice Midterm
3 marks
5. Determine when the function y =
ln(x)
is decreasing.
x
Answer:
3 marks
6. Find the values of a and b so that the function is differentiable at x = 1:
(
ax3 + b, if x ≤ 1
f (x) =
e2x ,
if x > 1
Answer:
TUYBFHLN-5
Page 5 of 14
Assignmetn 7 - Practice Midterm
3 marks
7. Let
√
2, x ∈ Q
f (x) =
π,
x∈
6 Q
Then lim f ◦ f (x) is equal to:
x→e
Answer:
3 marks
8. Find
dy
if xy − y 3 = 4.
dx
Answer:
TUYBFHLN-6
Page 6 of 14
Assignmetn 7 - Practice Midterm
3 marks
9. The price p (in dollars) and the demand q for a product are related by the following demand
equation:
p3 + q + q 3 = 38.
Find the price elasticity of demand in terms of p and q for this product.
Answer:
3 marks 10. An investment grows for three years at an annual rate of 10% compounded continuously.
If its matrurity value (after the three years) is $20,000, determine how much interest was
earned?
Answer:
TUYBFHLN-7
Page 7 of 14
Assignmetn 7 - Practice Midterm
Full-solution problems: Justify your answers and show all your work. Place a box around
your final answer. Unless otherwise indicated, simplification of answers is required in
these questions.
5 marks 11. Suppose that a company’s Cost function, C(x), is a differentiable function. Use a calculus
based argument to show that average cost will have a horizontal tangent line at the level of
production when the average cost is equal to the marginal cost.
TUYBFHLN-8
Page 8 of 14
Assignmetn 7 - Practice Midterm
5 marks 12. Find an equation of the tangent line to the curve given by x2 + y 2 = (2x2 + 2y 2 − x)2 at the
point (0, 1/2).
TUYBFHLN-9
Page 9 of 14
Assignmetn 7 - Practice Midterm
√
5 marks 13. Let f (x) = 3x + 4 − 3. Use the definition of the derivative to find f 0 (4). No marks will be
awarded for using derivative rules.
TUYBFHLN-10 Page 10 of 14
Assignmetn 7 - Practice Midterm
5 marks 14. Prove that the functions f (x) = x4 + x − 3 and g(x) =
intersection.
TUYBFHLN-11 Page 11 of 14
√
x − x have at least one point of
Assignmetn 7 - Practice Midterm
This page has been left blank for your workings and solutions. Work here will not be graded
unless there is clear indication in answer box of the problem. Do not remove it!
TUYBFHLN-12 Page 12 of 14
Assignmetn 7 - Practice Midterm
This page has been left blank for your workings and solutions. Work here will not be graded
unless there is clear indication in answer box of the problem. Do not remove it!
TUYBFHLN-13 Page 13 of 14
TUYBFHLN-14 Page 14 of 14