SAMPLE MIDTERM 2 MAT 142 Midterm will be Thursday, March 30, 2000

1. Place the letter corresponding to the correct answer in the box next to each question.

(i) Which of the following grows fastest as x → ∞/ (a) x^{2} (b) e^{x} (c) e^{x^{2}} (d) x^{100} (e) ln x (f) xe^{x}

(ii) cos(arctan(x)) =(a) 1/\sqrt{1 + x^{2}} (b) 1/\sqrt{1 − x^{2}} (c) 1+x^{2} (d) \sqrt{1 − x^{2}} (e) 1−x^{2} (f) none of these.

(iii) ∫ \frac{du}{\sqrt{1−u^{2}}} = (a) arctan(u)+C (b) arcsin(u)+C (c) arccos(u)+C (d) ln |1−u^{2}|+C (e) arcsec(u) + C (f) none of these.

(iv) cosh x = (a) e^{x} (b) e^{x} − e^{−x} (c) \frac{1}{2}(e^{x} + e^{x}) (d) 1 − sinh^{2} x (e) \frac{1}{2}(e^{−x} − e^{x}) (f) none of these.

(v) Which of the following is a solution of y′ = y^{2} (a) y = x (b) y = 1/x (c) y = e^{x} (d) y = −1/x (e) y = e^{−x} (f) none of these.

(vi) Use Euler’s method with step size Δx = 1, on the inital value problem y′ = y+x, y(0) = 0 to estimate y(2). (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 (f) none of these.

(vii) Which of the following equations is separable? (a) \frac{dy}{dx} = y + sin x (b) \frac{dy}{dx} = sin(y + x) (c) \frac{dy}{dx} = y sin x (d) \frac{dy}{dx} = sin(xy) (e) \frac{dy}{dx} = x + y (f) none of these.

(viii) Suppose \frac{x^{2}+1}{(x−1)(x−2)(x−3)} = \frac{A}{x−1} + \frac{B}{x−2} + \frac{C}{x−3}. Then A = (a) −2 (b) −1 (c) 0 (d) 1 (e) 2 (f) none of these.

(ix) Which of the following improper integrals diverges? (a) ∫_{0}^{∞} e^{−x}dx (b) ∫_{0}^{∞}(1 + x^{2})^{−1}dx (c) ∫_{1}^{∞} \frac{x}{1+x^{2}} dx (d) ∫_{1}^{∞} \frac{x}{1+x^{3}} dx (e) ∫_{0}^{∞} \frac{sin x}{1+x^{2}} dx (f) none of these.

(x) If we use the substitution x2 tan θ, the integral ∫ \frac{dx}{\sqrt{4+x^{2}}} becomes (a) ∫ sec θdθ (b) ∫ tan θdθ (c) ∫ sin^{2} θdθ (d) ∫ sec θ tan θdθ (e) ∫ sec^{2} θdθ (f) none of these.

2. Find each of the following integrals.

(i) ∫ \frac{xdx}{\sqrt{8x^{2}+1}}

(ii) ∫ x^{2} ln xdx (iii) ∫ \frac{x}{x^{2}−3x+2}dx

(iv) ∫_{−∞}^{∞} \frac{4}{x^{2}+16}

(v) ∫ \frac{1}{1+sin x}dx (Hint: multiply and divide by 1 − sin x)

3. Solve the differential equation (1 + x)y′ + y = \sqrt{x}.

4. Find a positive number a such that ∫_{0}^{a} \frac{dx}{1+x^{2}} = ∫_{a}^{∞} \frac{dx}{1+x^{2}} .

5. Find the length of the curve y = ln x from x = 1 to x = e.