Part II: Problem-Solving Problems ( 計算與證明題 Show all work)

Student ID number:

Guidelines for the test:

• Put your name or student ID number on every page.

• There are 13 problems: 3 problems in Part I and 10 problems in Part II .

• The exam is closed book; calculators are not allowed.

• There is no partial credit for problems in the Part I (選擇^, 填充及是非^).

• For problems in the Part II (problem-solving (計算與證明題) problems), please show all work, unless instructed otherwise. Partial credit will be given only for work shown. Print as legibly as possible - correct answers may have points taken off, if they’re illegible.

• Mark the final answer.

Part I: 單選 ^, 填充 ^, 是非題 (5 points for each problem)

1. (5 pts) The plot below shows the growth of the population on an small island.

Choose a better model from the options for this data? (Hint: find critical points, inflection points and asymptotes)

(A) x

x + 8 (B) x²

x² − 8² (C) x

x− 8 (D) x²

x²+ 8²

0 10 20 30 40

0 0.2 0.4 0.6 0.8 1

year (x)

population (y thousands)

2. (5 pts) f (x) is a continuous function on (−∞, ∞) and the graph of its derivative, f^′(x), is shown in the figure below. (Note: lim_x→−∞f^′(x) = 0; lim_x→∞f^′(x) =∞)

Answer the following True/False questions (True ⇒ ⃝ ; False ⇒ ×).

• f (x) has a horizontal asymptote.

• f (x) has a vertical asymptote.

• (1, f (1)) is an inflection point.

• (2, f (2)) is an inflection point.

• f has a local minimum at x = 0.

3. (5 pts)A Region is bounded by two curves: y = x³ and y = x. Set up a definite integral representing the area of the region

A) ∫₁

−1x− x³dx B) ∫₀

−1x− x³dx +∫₁

0 x³− x dx, C) ∫₀

−1x³− x dx +∫₁

0 x− x³dx D) ∫₁

−1x³− x dx

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Part II: Problem-Solving Problems ( 計算與證明題 Show all work)

4. (10 pts) The sequence {xn} is recursively defined.

x_n+1 = 4x²_n 4 + x²_n

(a) (2 pts) Find all equilibria (fixed points) of{xn}.

(b) (8 pts) Determine the stability of the equilibria (fixed points).

5. (10 pts) Given that F (x) =

∫ _x²

1

√1 + t²dt, for x≥ 0,

(a) F^′(x) =?

(b) (F⁻¹)^′(0) =?

Note: since √

1 + t² > 0, F (x) is monotone and F⁻¹(x) is well-defined.

6. (5 pts) Use formulas for indefinite integrals to evaluate

∫ 1

x²− 4x + 11dx.

7. (5 pts) d

dx(x^{cos x}).

8. (10 pts) f (x) = ln x

x , x > 0.

(a) Find the maximum value of f (x).

(b) Prove that π^e < e^π

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9. (5 pts) Evaluate

∫ 1

xe^{199+ln x}dx

10. (10 pts) Evaluate

∫ _e²

e

√ln x x dx

11. (10 pts) Evaluate

∫

ln x dx

12. (10 pts) Evaluate the indefinite integrals (a)

∫ 2t + 1 t²+ 2tdt

(b)

∫ cos³x

sin²x + 2 sin xdx

13. (10 pts) Compute: (Be sure to check whether l’Hospital’s rule can be applied before you use it.)

(a) lim

x→∞

x e^x

(b) lim

x→∞(1 + 2 x)^x

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