calculus 1 homework
MAT 1339B (Winter 2018) Assignment 1
Professor: Rachid Bentoumi
Deadline: Friday, February 09, 2018 Before 15:00 (585 King Eduard math-stat department drop boxes)
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Late assignments will NOT be accepted; nor will unstapled assignments. Professors in the math department will not lend you a stapler.
You should complete ALL the questions in the assignments. It is possible, however, that not all the questions will be marked. In that case, the same questions will be marked in all assignments. You will not be informed be- forehand which questions will be marked.
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Question 1. Solve the following inequalities:
(a) |2x− 5| > 2
(b) |3 − 2x| ≤ 7
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Question 2. Find the domain of the functions defined below
(a) f(x) = 1
x2 − 8x
(b) g(x) = x
|x| + 4
(c) h(x) =
√ x− 2 x− 7
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Question 3. Calculate the following limits (show all steps of your calcula- tion).
(a) lim x→0
√ 19 − 3x−x
x
(b) lim x→−∞
√ x2 − 1 2x
(c) lim x→∞
cos (x)
x4 + 1
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Question 4. Let f be a function defined as
f(x) =
x2 − 16 2x− 8
, if x < 4
−4, si x = 4 x− 4 √ x− 2
if x > 4
(a) Determine lim x→4
f(x) ?
(b) Is the function f continuous on R? Justify your answer.
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Question 5. Determine, if it is possible, the value of k that makes the functions below continuous on R.
(a) h(x) =
x + 2 if x < 1
k if x = 1
x2 + 3x− 1 if x > 1
(b) f(t) =
t2 − 25 t− 5
if t < 5
kt if t ≥ 5
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Question 6. Consider the following function
f(x) = √
25 − 2x
(a) Use the definition of the derivative to find f ′ (x)
(b) Determine the tangent line to f at x = 0.
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Question 7. Use the derivative rules to calculate the derivatives of the following functions:
(a) y = √
2x2 + 5x + 7
(b) y = (x2 + 3)4(x3 − 5)3
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Question 8. Consider a closed rectangular box with the following dimensions: length= x cm, width= 2x cm and height= x + 1 cm.
(a) According to x, determine
i. the function A(x) giving the total area of the rectangular box.
ii. the function V (x) giving the volume of the rectangular box.
(b) Calculate average rates of change of the area and the volume when x passes from 3 cm to 6 cm.
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Question 9. A manufacturer estimates that his weekly revenue R(x) can be expressed by
R(x) = 1008x− 12x2 − 8x3
where x is the number of units sold, R(x) is expressed in dollars. Under certain constraints, one cannot produce more than 10 units per week.
(a) Using the variation table (or test points), determine on which interval(s) the revenue is increasing.
(b) Determine the quantity of units that gives a maximum revenue and evaluate this maximum revenue.
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Question 10. Determine the concavity intervals and, if it is possible, the inflection points of the function h(x) = (x− 1)2(x + 1)2.
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