National Taiwan Normal University Division of Preparatory Programs for Overseas Chinese Students
Academic Year 114 Spring and Fall Semester Final Mock Exam Paper
Academic Year 114 Spring and Fall Semester Final Mock Exam Paper
Subject: Mathematics
Group 2 & 3, Class (8)
Student ID:
Seat No.:
Name:
$\displaystyle \circledast$ Note: For the Mathematics exam (1) No Calculator (2) No Calculation Papers allowed.
I. Single Choice Questions (80%) (Total 20 questions, please fill in the correct answers on the answer card according to the question number)
((1) $\displaystyle \sim$ (10) 5 points each; (11) $\displaystyle \sim$ (20) 3 points each)
((1) $\displaystyle \sim$ (10) 5 points each; (11) $\displaystyle \sim$ (20) 3 points each)
(1)
Evaluate $\displaystyle \lim_{x \to -1} \frac{x - 1}{x^2 + 1} = ?$
| (1) $\displaystyle \frac{1}{2}$ | (2) $\displaystyle \frac{-1}{2}$ | (3) $\displaystyle 1$ | (4) $\displaystyle -1$ | (5) Does not exist |
(2)
Evaluate the definite integral $\displaystyle \int_{0}^{1} 3^{2x} dx = ?$
| (1) $\displaystyle \frac{4}{\ln 3}$ | (2) $\displaystyle \frac{8}{\ln 3}$ | (3) $\displaystyle 4 \ln 3$ | (4) $\displaystyle 8 \ln 3$ | (5) $\displaystyle 8$ |
(3)
Assume that $\displaystyle f(x) = \frac{\ln(ex)}{e + e^x}$. Find $\displaystyle f'(1) = ?$
| (1) $\displaystyle 0$ | (2) $\displaystyle \frac{1}{4e}$ | (3) $\displaystyle \frac{1}{2e}$ | (4) $\displaystyle \frac{1}{e}$ | (5) $\displaystyle \frac{2}{e}$ |
(4)
Assume that $\displaystyle f'(x) = x^2(x - 1)(x + 2)$. At what number of $\displaystyle x$, $\displaystyle f(x)$ has a local minimum value?
| (1) $\displaystyle x = -2$ | (2) $\displaystyle x = 0$ | (3) $\displaystyle x = 1$ | (4) $\displaystyle x = 3$ | (5) none of the above |
(5)
Find the slope of the tangent line to the curve $\displaystyle y = \sin^2 x \cos^2 x$ at $\displaystyle x = \frac{\pi}{6}$.
| (1) $\displaystyle \frac{5\sqrt{3}}{2}$ | (2) $\displaystyle 2\sqrt{3}$ | (3) $\displaystyle \frac{3\sqrt{3}}{2}$ | (4) $\displaystyle \sqrt{3}$ | (5) $\displaystyle \frac{\sqrt{3}}{4}$ |
(6)
Evaluate $\displaystyle \left. \frac{d}{dx} \left( \int_{0}^{5x} e^{t^2} dt \right) \right|_{x=0} = ?$
| (1) $\displaystyle 2$ | (2) $\displaystyle 1$ | (3) $\displaystyle 0$ | (4) $\displaystyle 5$ | (5) $\displaystyle 10$ |
(7)
Assume that $\displaystyle f(x) = \begin{cases} 7x - 3, & x \geq 1 \\ 3x^2 + x, & x < 1 \end{cases}$. Find $\displaystyle f'(1) = ?$
| (1) $\displaystyle 7$ | (2) $\displaystyle 5$ | (3) $\displaystyle 3$ | (4) $\displaystyle 1$ | (5) Does not exist |
(8)
Evaluate the definite integral $\displaystyle \int_{0}^{\pi} \cos x dx = ?$
| (1) $\displaystyle 0$ | (2) $\displaystyle 1$ | (3) $\displaystyle -1$ | (4) $\displaystyle 2$ | (5) $\displaystyle -2$ |
(9)
Assume that $\displaystyle f(x) = (x^2 - 3x + 4)^5$. Find $\displaystyle f'(2) = ?$
| (1) $\displaystyle -80$ | (2) $\displaystyle -40$ | (3) $\displaystyle 32$ | (4) $\displaystyle 40$ | (5) $\displaystyle 80$ |
(10)
Assume that $\displaystyle \frac{df}{dx} = 3x^{\frac{-2}{3}} + 2x$ and $\displaystyle f(-1) = -5$. Find $\displaystyle f(x) = ?$
| (1) $\displaystyle -9x^{\frac{1}{3}} + x^2 + 3$ | (2) $\displaystyle -9x^{\frac{1}{3}} + 2x^2 + 2$ |
| (3) $\displaystyle 9x^{\frac{1}{3}} + x^2 + 3$ | (4) $\displaystyle 9x^{\frac{1}{3}} + 2x^2 + 3$ |
| (5) $\displaystyle -9x^{\frac{1}{3}} - x^2 + 3$ |
(11)
Assume that $\displaystyle f(x) = 2x^2, x \geq 0$. Find $\displaystyle (f^{-1})'(5000) = ?$
| (1) $\displaystyle \frac{1}{5}$ | (2) $\displaystyle \frac{1}{10}$ | (3) $\displaystyle \frac{1}{20}$ | (4) $\displaystyle \frac{1}{40}$ | (5) $\displaystyle \frac{1}{200}$ |
(12)
Evaluate the definite integral $\displaystyle \int_{0}^{2} \left( \frac{x^3}{\sqrt{x^4 + 9}} \right) dx = ?$
| (1) $\displaystyle 0$ | (2) $\displaystyle 1$ | (3) $\displaystyle 2$ | (4) $\displaystyle 3$ | (5) $\displaystyle 4$ |
(13)
Assume that $\displaystyle x^2 - \tan^{-1} y = y - \frac{\pi}{3}$. Find $\displaystyle \left. \frac{dy}{dx} \right|_{x=1, y=\sqrt{3}} = ?$
| (1) $\displaystyle \frac{8+\pi}{10}$ | (2) $\displaystyle \frac{4+\pi}{10}$ | (3) $\displaystyle \frac{8}{5}$ | (4) $\displaystyle \frac{4}{5}$ | (5) $\displaystyle \frac{4}{3}$ |
(14)
Evaluate the definite integral $\displaystyle \int_{\frac{\pi}{2}}^{\pi} (\sin^3 x \cos^2 x) dx = ?$
| (1) $\displaystyle \frac{2}{15}$ | (2) $\displaystyle \frac{1}{15}$ | (3) $\displaystyle \frac{-2}{15}$ | (4) $\displaystyle \frac{-1}{15}$ | (5) $\displaystyle 0$ |
(15)
Evaluate $\displaystyle \lim_{x \to 1^+} \left( \frac{4x}{x-1} - \frac{4}{\ln x} \right) = ?$
| (1) $\displaystyle -1$ | (2) $\displaystyle \frac{-1}{2}$ | (3) $\displaystyle \frac{1}{2}$ | (4) $\displaystyle 1$ | (5) $\displaystyle 2$ |
(16)
Evaluate the definite integral $\displaystyle \int_{0}^{\frac{\sqrt{3}}{2}} (\sin^{-1} x) dx = ?$
| (1) $\displaystyle \frac{\sqrt{3}}{2}\pi - \frac{1}{2}$ | (2) $\displaystyle \frac{1}{2\sqrt{3}}\pi - \frac{1}{2}$ | (3) $\displaystyle \frac{\sqrt{3}}{2}\pi + \frac{1}{2}$ |
| (4) $\displaystyle \frac{1}{2\sqrt{3}}\pi + \frac{1}{2}$ | (5) $\displaystyle 2\sqrt{3}\pi + \frac{1}{2}$ |
(17)
Evaluate the definite integral $\displaystyle \int_{1}^{3} \left( \frac{(\log_2(x+1))^2}{x+1} \right) dx = ?$
| (1) $\displaystyle \frac{4}{3} \ln 2$ | (2) $\displaystyle \frac{5}{3} \ln 2$ | (3) $\displaystyle 2 \ln 2$ | (4) $\displaystyle \frac{7}{3} \ln 2$ | (5) $\displaystyle \frac{8}{3} \ln 2$ |
(18)
Evaluate the definite integral $\displaystyle \int_{1}^{5} \left( \frac{x}{\sqrt{2x-1}} \right) dx = ?$
| (1) $\displaystyle \frac{1}{3}$ | (2) $\displaystyle \frac{2}{3}$ | (3) $\displaystyle \frac{13}{3}$ | (4) $\displaystyle \frac{14}{3}$ | (5) $\displaystyle \frac{16}{3}$ |
(19)
Evaluate the definite integral $\displaystyle \int_{\frac{\sqrt{2}}{2}}^{1} \left( \frac{(\sqrt{1-x^2})^3}{x^6} \right) dx = ?$
| (1) $\displaystyle 1$ | (2) $\displaystyle \frac{1}{2}$ | (3) $\displaystyle \frac{1}{3}$ | (4) $\displaystyle \frac{1}{5}$ | (5) $\displaystyle \frac{1}{7}$ |
(20)
Assume that the indefinite integral $\displaystyle \int \frac{3x^2+x+4}{x(x^2+1)} dx = A_1 \cdot \tan^{-1} x + A_2 \cdot \ln|x| + A_3 \cdot \ln|x^2+1| + C$, where $\displaystyle A_1, A_2, A_3$ are the nonzero real numbers and $\displaystyle C$ is an arbitrary constant.
Then $\displaystyle A_1 + A_2 + A_3 = ?$
Then $\displaystyle A_1 + A_2 + A_3 = ?$
| (1) $\displaystyle \frac{1}{2}$ | (2) $\displaystyle \frac{5}{2}$ | (3) $\displaystyle \frac{7}{2}$ | (4) $\displaystyle \frac{9}{2}$ | (5) $\displaystyle \frac{11}{2}$ |
II. Multiple Choice Questions (10%) (5 points each, total 2 questions, please fill in the correct answers on the answer card according to the question number)
(21)
Assume that \( \displaystyle f(x) = \begin{cases} \frac{4x^2 - x}{(x+2)(x+3)}, & x > 0 \\ 0, & x \leq 0 \end{cases} \).
Which of the following options is/are correct?
Which of the following options is/are correct?
- (1) \( \displaystyle y = 1 \) is a horizontal asymptote of the graph of \( \displaystyle f \)
- (2) \( \displaystyle \lim_{x \to 0} f(x) = 0 \)
- (3) \( \displaystyle f \) is continuous at \( \displaystyle x = 0 \)
- (4) \( \displaystyle f'(0) = 0 \)
- (5) \( \displaystyle x = -2 \) is a vertical asymptote of the graph of \( \displaystyle f \)
(22)
Assume that $\displaystyle f(x) = \frac{3}{5}x^{\frac{5}{3}} - 3x^{\frac{2}{3}}$. Which of the following options is/are correct? (Comprehensive of 4.1-4.4, 5.4)
- (1) $\displaystyle f(x)$ has two critical numbers
- (2) $\displaystyle f(x)$ is increasing on the interval $\displaystyle (-\infty, -1)$
- (3) $\displaystyle f(x)$ is concave up on the interval $\displaystyle (-\infty, -1)$
- (4) $\displaystyle \int_{0}^{1} f(x) dx = \frac{-63}{40}$
- (5) $\displaystyle f(x)$ has two inflection points
III. Fill-in-the-Blank Questions (10%) (5 points each, total 2 questions, please fill in the correct answers on the answer card. Answers for each question must be given in simplest fraction form and filled in the designated question number spaces, otherwise no points will be awarded.)
A.
Assume that $\displaystyle y = (2-x)^3 (5x^2 - 4)^{\frac{1}{3}} (3x^2 - 2)^{\frac{2}{5}}$. Then $\displaystyle \left. \frac{dy}{dx} \right|_{x=1} = \frac{(23)(24)}{(25)(26)}$
B.
Evaluate the definite integral $\displaystyle \int_{0}^{3/2} \sqrt{9 - 4x^2} dx = \frac{(27)}{(28)} \pi$
Answer Key (အဖြေမှန်များ)
Part I & II
Q1: (4)
Q2: (1)
Q3: (2)
Q4: (3)
Q5: (5)
Q6: (4)
Q7: (1)
Q8: (1)
Q9: (5)
Q10: (3)
Q11: (5)
Q12: (2)
Q13: (3)
Q14: (1)
Q15: (5)
Q16: (2)
Q17: (4)
Q18: (5)
Q19: (4)
Q20: (4)
Q21: (1), (2), (3)
Q22: (1), (2), (4)
Part III (Fill-in-the-Blank)
A:
23=4, 24=1
25=1, 26=5
(Ans: 41/15)
23=4, 24=1
25=1, 26=5
(Ans: 41/15)
B:
27=9, 28=8
(Ans: 9/8)
27=9, 28=8
(Ans: 9/8)
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