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

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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) 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

 

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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