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Problem set 2
Due date: February 29th Thursday 14:00 (before the class)
You can write your answers by hand or in a word (or Latex) file. You can submit your
assignment in person (written or printed), before the beginning of class on Thursday. Or, you
can upload your submission on Canvas. When you upload your assignment, please check
resolution of your file.
You can cooperate with others or rely on some materials on the internet. But you have to
submit your own work individually. You need to clarify process of your work, especially for
calculation.
1. Find difference quotient of below functions.
(a) π¦ = 2π₯3 β 3 (b) π¦ = π₯ β 9 c) π¦ = βπ₯2 β π₯ + 1
2. Given π =
[(π£+2)3β8]
π£
(π£ β 0), find
a. lim
π£β0
π b. lim
π£β2
π
3. Check the statements below by each and verify whether they are TRUE or FALSE, and
shortly explain why.
a. When a function π¦ = π(π₯) has the same left-side and right-side limit at π₯ = π, this
function has a limit value at π₯ = π
b. π¦ = |π₯ β 3| has a right-side limit value at π₯ = 3
c. π¦ = |π₯ β 3| has a limit value at π₯ = 3
d. If π¦ = π(π₯) is continuous everywhere, then it is differentiable at any value of π₯.
e. If π¦ = π(π₯) is differentiable everywhere, then it is continuous at any value of π₯.
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4. Solve the following inequalities
a. |π₯ + 1| < 6 b. |4 β 3π₯| < 2
5. Find the limits of the function π = 7 β 9π£ + π£2
a. As π£ β 0 b. As π£ β 3 c. As π£ β β1
6. For a function π¦ = π(π₯) =
3π₯2
(π₯+1)
, its derivative is πβ²(π₯) =
3π₯2+6π₯
(π₯+1)2 . Prove this result.
(You can utilize proof in the textbook and chapter 7 slide)
7. Find πβ²(1) and πβ²(2) from the following functions
a. π¦ = π(π₯) = ππ₯3 b. π(π₯) = β5π₯β2 c. π(π₯) =
3
4
π₯
4
3 d. π(π€) = β3π€β
1
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8. For a cost function π3 β 3π2 + 10, check the statements below by each and verify
whether they are TRUE or FALSE, and shortly explain why
a. Marginal cost function is
ππΆ
ππ
= 3π2 β 6π
b. AC is decreasing when 0 < π < 1
c. When π = 5, average cost 10
d. When π = 10, average cost is greater than marginal cost
e. When π = 8, the slope of average cost curve is positive
9. Given the average cost function π΄πΆ = π2 β 4π + 174, find 1) total cost and 2) marginal
cost functions.
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10. Differentiate the following by using the product rule
a. (9π₯2 β 2)(3π₯ + 1) b. (π₯2 + 3)π₯β1 c. (ππ₯ β π)(ππ₯2)
11. Find the derivatives of
a.
6π₯
π₯+5
b.
ππ₯2+π
ππ₯+π
12. Find an inverse of π¦ = βπ₯ + 1 (π₯ β₯ 0). And check the domain of the inverse function.
13. Check the statements below by each and verify whether they are TRUE or FALSE, and
shortly explain why.
(a) π₯2 + 2π₯ + 1 has an inverse function when its domain is π₯ β₯ 1
(b) π¦ = βπ₯4 + 5 is strictly monotonic when its domain is π₯ > 0
(c) Given π¦ = π(π₯) = π₯3 + 2,
ππ₯
ππ¦
=
1
β3π₯2
(d) If π¦ = π(π₯) is a strictly increasing function, then πβ1(π₯) is strictly decreasing function
(e) If π¦ = π(π₯) is not a strictly increasing function, then it is a strictly decreasing function
14. Use the chain rule to find
ππ¦
ππ₯
for the following
a. π¦ = (3π₯2 β 13)3 b. π¦ = (7π₯3 β 5)9
15. Find
ππ¦
ππ₯1
and
ππ¦
ππ₯2
for each of the following functions
a. π¦ = 2π₯1
3 β 11π₯1
2π₯2 + 3π₯2
2 b. π¦ = 7π₯1 + 6π₯1π₯2
2 β 9π₯2
3 c. π¦ =
5π₯1+3
π₯2β2
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16. Find the differential ππ¦, for given functions
a. π¦ = βπ₯(π₯2 + 3) b. π¦ =
π₯
π₯2+1
17. Find the total differential for each of following functions
a. π = 7π₯2π¦3 b. π =
9π¦3
π₯βπ¦
c. π = β5π₯3 β 12π₯π¦ β 6π¦5
