question
chengxihe
a year ago
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files (2)Problemset2.pdf
Problemset1.pdf
Problemset2.pdf
1
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) 𝑦 = 2𝑥3 − 3 (b) 𝑦 = 𝑥 − 9 c) 𝑦 = −𝑥2 − 𝑥 + 1
2. Given 𝑞 = [(𝑣+2)3−8]
𝑣 (𝑣 ≠ 0), find
a. lim 𝑣→0
𝑞 b. lim 𝑣→2
𝑞
3. Check the statements below by each and verify whether they are TRUE or FALSE, and
shortly explain why.
a. When a function 𝑦 = 𝑓(𝑥) has the same left-side and right-side limit at 𝑥 = 𝑁, this
function has a limit value at 𝑥 = 𝑁
b. 𝑦 = |𝑥 − 3| has a right-side limit value at 𝑥 = 3
c. 𝑦 = |𝑥 − 3| has a limit value at 𝑥 = 3
d. If 𝑦 = 𝑓(𝑥) is continuous everywhere, then it is differentiable at any value of 𝑥.
e. If 𝑦 = 𝑓(𝑥) is differentiable everywhere, then it is continuous at any value of 𝑥.
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4. Solve the following inequalities
a. |𝑥 + 1| < 6 b. |4 − 3𝑥| < 2
5. Find the limits of the function 𝑞 = 7 − 9𝑣 + 𝑣2
a. As 𝑣 → 0 b. As 𝑣 → 3 c. As 𝑣 → −1
6. For a function 𝑦 = 𝑓(𝑥) = 3𝑥2
(𝑥+1) , its derivative is 𝑓′(𝑥) =
3𝑥2+6𝑥
(𝑥+1)2 . Prove this result.
(You can utilize proof in the textbook and chapter 7 slide)
7. Find 𝑓′(1) and 𝑓′(2) from the following functions
a. 𝑦 = 𝑓(𝑥) = 𝑐𝑥3 b. 𝑓(𝑥) = −5𝑥−2 c. 𝑓(𝑥) = 3
4 𝑥
4
3 d. 𝑓(𝑤) = −3𝑤− 1
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8. For a cost function 𝑄3 − 3𝑄2 + 10, check the statements below by each and verify
whether they are TRUE or FALSE, and shortly explain why
a. Marginal cost function is 𝑑𝐶
𝑑𝑄 = 3𝑄2 − 6𝑄
b. AC is decreasing when 0 < 𝑄 < 1
c. When 𝑄 = 5, average cost 10
d. When 𝑄 = 10, average cost is greater than marginal cost
e. When 𝑄 = 8, the slope of average cost curve is positive
9. Given the average cost function 𝐴𝐶 = 𝑄2 − 4𝑄 + 174, find 1) total cost and 2) marginal
cost functions.
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10. Differentiate the following by using the product rule
a. (9𝑥2 − 2)(3𝑥 + 1) b. (𝑥2 + 3)𝑥−1 c. (𝑎𝑥 − 𝑏)(𝑐𝑥2)
11. Find the derivatives of
a. 6𝑥
𝑥+5 b.
𝑎𝑥2+𝑏
𝑐𝑥+𝑑
12. Find an inverse of 𝑦 = √𝑥 + 1 (𝑥 ≥ 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) 𝑥2 + 2𝑥 + 1 has an inverse function when its domain is 𝑥 ≥ 1
(b) 𝑦 = −𝑥4 + 5 is strictly monotonic when its domain is 𝑥 > 0
(c) Given 𝑦 = 𝑓(𝑥) = 𝑥3 + 2, 𝑑𝑥
𝑑𝑦 =
1
−3𝑥2
(d) If 𝑦 = 𝑓(𝑥) is a strictly increasing function, then 𝑓−1(𝑥) is strictly decreasing function
(e) If 𝑦 = 𝑓(𝑥) is not a strictly increasing function, then it is a strictly decreasing function
14. Use the chain rule to find 𝑑𝑦
𝑑𝑥 for the following
a. 𝑦 = (3𝑥2 − 13)3 b. 𝑦 = (7𝑥3 − 5)9
15. Find 𝜕𝑦
𝜕𝑥1 and
𝜕𝑦
𝜕𝑥2 for each of the following functions
a. 𝑦 = 2𝑥1 3 − 11𝑥1
2𝑥2 + 3𝑥2 2 b. 𝑦 = 7𝑥1 + 6𝑥1𝑥2
2 − 9𝑥2 3 c. 𝑦 =
5𝑥1+3
𝑥2−2
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16. Find the differential 𝑑𝑦, for given functions
a. 𝑦 = −𝑥(𝑥2 + 3) b. 𝑦 = 𝑥
𝑥2+1
17. Find the total differential for each of following functions
a. 𝑈 = 7𝑥2𝑦3 b. 𝑈 = 9𝑦3
𝑥−𝑦 c. 𝑈 = −5𝑥3 − 12𝑥𝑦 − 6𝑦5
Problemset1.pdf
1
Problem set 1
Due date: February 13th Tuesday 14:00 (before beginning of 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 beginning of class on the Tuesday. Or, you
can upload your submission on Canvas. When you upload your assignment, please check
resolution of your file. And I recommend you to upload it as .pdf file.
You can cooperate with others or can utilize 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. A set 𝐶 is defined to be set difference between two set 𝐴 and 𝐵: 𝐶 = 𝐴 − 𝐵. And set 𝐴 and
𝐵 are defined as like below.
𝐴: 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝐵: 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
𝑐𝑖 is an element of the set 𝐶. Check the statements below by each and verify whether they are
TRUE or FALSE, and shortly explain why.
a. For any element 𝑐𝑖 ∈ 𝐶, it holds 𝑐𝑖 2 > 0
b. There is an element 𝑐𝑗 ∈ 𝐶, satisfying this condition; 𝑐𝑗 𝑖𝑠 𝑎 𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔 𝑑𝑒𝑐𝑖𝑚𝑎𝑙
c. 𝜋 ∈ 𝐶
d. One of the two solutions of equation (𝑥 − 2) (𝑥 + 7
9 ) = 0 is an element of the set 𝐶
e. 𝐶 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡
f. 𝐵⋂𝐶 = ∅
g. For any two different elements 𝑐𝑖 and 𝑐𝑗 in the set 𝐶, their product is also an element of 𝐶:
𝑐𝑖𝑐𝑗 ∈ 𝐶 (𝑐𝑖 ≠ 𝑐𝑗)
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2. Textbook exercise 2.3.4. from (a) to (g)
3. Textbook exercise 2.4.6.
4. Check the statements below by each and verify whether they are TRUE or FALSE, and
shortly explain why.
a. A relation 𝑥2 + 𝑦2 = 9 (𝑥| − 3 ≤ 𝑥 ≤ 3) is a function.
b. There is a function 𝑦 = 3𝑥 + 1 and domain of 𝑥 is all integers. Then, image of 𝑦 includes
some irrational numbers.
c. 𝑥3 ÷ 𝑥7 = 𝑥10
d. 𝑥√2(𝑥2√2 + 𝑥√3) = 𝑥4+√6
e. 𝑥𝑎 × 𝑥𝑏 × 𝑥𝑐 = 𝑥𝑎𝑏𝑐
f. 𝑎𝑥 × 𝑏𝑥 × 𝑐𝑥 = (𝑎𝑏𝑐)𝑥
5. Find equilibrium market price and quantity the market given below
𝑄𝑑 = 𝑄𝑠 𝑄𝑑 = 30 − 2𝑃 𝑄𝑠 = −6 + 4𝑃
6. Textbook exercise 3.2.2.(b)
7. Textbook exercise 3.4.3.
8. Find solution of system of equations. If solution is not unique, then shortly explain why.
a. 𝑥 − 4𝑦 = −3 2𝑥 + 3𝑦 = 7 b. 3𝑥 + 2𝑦 = 5 4𝑥 + 8
3 𝑦 = 6
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9. Answer below matrix operations. If it cannot be defined, shortly explain why.
a. [ 1 2 3 4
] + [ −1 −2 5 0
] b. 3 [ 0 −3 3 0
] − 2 [ −1 −2 1 7
]
c. [ 1 2 3 4
] [ −2 3 0 5 −8 1
] d. [ 5 0 1 2 1 −3
] [ 3 0 2
−3 2 8 ]
e. [ −2 3 0 5 −8 1
] [ 1 2 3 4
]
10. Textbook exercise 4.2.6.
11. Check linear dependence of three vectors.
𝑢 = (−2, 3), 𝑣 = (5, 1), 𝑤 = (1, −2)
12. For two vectors 𝑢 = (3,2) and 𝑣 = (−1, 7), find scalars 𝑎, 𝑏, 𝑐, and 𝑑 satisfying below
equations. If you cannot find those scalars, then shortly explain why.
a. 𝑎𝑢 + 𝑏𝑣 = (1,0) b. 𝑐𝑢 + 𝑑𝑣 = (0,1)
13. Prove (𝐴′)−1 = (𝐴−1)′. You can prove this statement by utilizing matrix multiplication,
and properties of transpose and inverse. You do not have to introduce dimension of matrix 𝐴.
14. Find determinant of below matrices
a. [7] b. [ 1 2 3 4
]
c. [ 1 0 2
−1 5 5 −3 0 7
] d. [ 1 2 3 4 5 6 0 0 0
] e. [ 1 2 7
−6 5 0 −6 2 2
]
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15. Find rank of below matrices
a. [ 6 8 3
4 1] b. [
1 2 5 1 3 9 2 0 4
] c. [ 1 0 0 0 1 0 0 0 1
]
16. Check the statements below by each and verify whether they are TRUE or FALSE, and
shortly explain why.
a. If 𝐴 is a square matrix, then both of 𝐴𝐴′ and 𝐴′𝐴 are well defined.
b. If 𝐴 is a (2 × 2) nonsingular matrix, then its rank is equal to 2.
c. If determinant of (3 × 3) matrix 𝐴 is zero, then its rank cannot be 3.
d. For any (3 × 3) symmetric matrix 𝐴, 𝐴2 is also a symmetric matrix.
e. If 𝐴𝐵 is well defined, then 𝐴′𝐵′ is also well defined.
f. For a (2 × 2) matrix 𝐴, if 𝐴2 = 𝐼 then 𝐴 = 𝐼
17. Textbook exercise 5.4.2. only (a) and (b)
18. Textbook exercise 5.4.5. only (a) and (b)
19. Find 𝑥2 ∗ by applying Cramer’s rule. If you cannot find a unique value of 𝑥2
∗, then shortly
explain why.
a.
3𝑥1 + 2𝑥2 + 4𝑥3 = 2
2𝑥1 − 4𝑥2 + 𝑥3 = 3
2𝑥1 − 1𝑥2+ = −4
b.
𝑥1 + 2𝑥2 + 3𝑥3 = 1
5𝑥1 − 1𝑥2 + 𝑥3 = −3
2𝑥1 − 1𝑥2 + 3𝑥3 = 2
