Calculus I, Fall 2014 (http://www.math.nthu.edu.tw/~wangwc/)

Final Exam

Jan 08, 2015

1. (14 pts) Find the solutions for \frac{dy}{dx} = e^{x−y} and x\frac{dy}{dx} + y = sin x, respectively. Ans: (1)y = ln(e^{x} + C_{1}) for some constant C_{1}. (2)y = \frac{1}{x}(− cos x + C_{2}) for some constant C_{2}.

2. (12 pts) Find the volume and surface area of the object obtained by rotating the region {(x − 2)^{2} + y^{2} ≤ 1, x ≥ 2} around the y axis. Note the surface area consists of two parts, one generated by a half circle, the other generated by a line segment. Ans: Volume=∫_{−1}^{1} π((2 + \sqrt{1 − y^{2}})^{2} − 2^{2})dy = 2π^{2} + \frac{4}{3}π. Surface=∫_{−1}^{1} 2π(2 + \sqrt{1 − y^{2}})\sqrt{\frac{1}{1−y^{2}}} dy + 2π · 2 · 2 = 4π^{2} + 12π.

3. (8 pts) Order e^{x}, x^{x}, (ln x)^{x} and x^{e} from slowest to fastest growing rate as x → ∞. Explain. Ans: From slowest to fastest is x^{e}, e^{x}, (ln x)^{x}, x^{x}

4. (6 pts) Write down the form of partial fraction expansion for \frac{x^{7}}{(1 − x^{4})^{2}} . Need NOT find the undetermined coefficients. Ans: \frac{x^{7}}{(1−x^{4})^{2}} = \frac{A}{1−x} + \frac{B}{(1−x)^{2}} + \frac{C}{1+x} + \frac{D}{(1+x)^{2}} + \frac{Ex+F}{1+x^{2}} + \frac{Gx+H}{(1+x^{2})^{2}} . where A ∼ H are undetermined coefficients.

5. (64 pts) Evaluate

(1) ∫ \frac{1}{2 + sin x}dx (2) ∫_{1}^{e^{π}} sin(ln x) dx (3) ∫ \frac{1}{\sqrt{4x − x^{2}}} dx (4) ∫_{1}^{2} \frac{1}{e^{2x} − e^{−x}} dx

(5) ∫_{0}^{π/4} tan^{3} x sec^{3} x dx (6) ∫ x^{2}e^{−x}dx (7) ∫_{0}^{1} sechx dx (8) ∫_{0}^{1} \frac{1}{\sqrt{1 + e^{x}}} dx

Ans: (1)=\frac{2}{\sqrt{3}} tan^{−1}(\frac{2}{\sqrt{3}}(tan(\frac{x}{2}) + \frac{1}{2})) + C for some constant C.

(2)=\frac{e^{π}+1}{2} 1

(3)=sin^{−1}(\frac{x−2}{2}) + C for some constant C.

(4)=\frac{1}{3} ln(\frac{e^{2}−1}{e−1}) − \frac{1}{6} ln(\frac{e^{4}+e^{2}+1}{e^{2}+e+1}) − \frac{1}{\sqrt{3}}(tan^{−1}(\frac{2}{\sqrt{3}}(e^{2} + \frac{1}{2})) − tan^{−1}(\frac{2}{\sqrt{3}}(e + \frac{1}{2}))).

(5)=\frac{2\sqrt{2}+2}{15}.

(6)=−x^{2}e^{−x} − 2xe^{−x} − 2e^{−x} + C for some constant C.

(7)=tan^{−1}(\frac{e−e^{−1}}{2} ).

(8)=1 − 2 ln(\sqrt{1 + e} + 1) + 2 ln(\sqrt{2} + 1).