18.01 Practice Final, Answers

1. (10 points) State the definition of the derivative of a function f(x).

Answer: The derivative of a function f(x) is the function:

f′(x) = lim_{Δx→0} \frac{f(x + Δx) − f(x)}{Δx}.

2. (10 points each) Find the derivatives of the following functions:

(a) f(x) = \frac{e^{x} + 1}{x^{2} + 1}.

Answer: f′(x) = \frac{e^{x}(x^{2} + 1) − 2x(e^{x} + 1)}{(x^{2} + 1)^{2}} = \frac{e^{x}(x − 1)^{2} − 2x}{(x^{2} + 1)^{2}}

(b) f(x) = cos x ln x.

Answer: f′(x) = \frac{cos x}{x} − sin x ln x.

(c) f(x) = sin^{−1}(\sqrt{x}).

Answer: f′(x) = \frac{1}{2\sqrt{x − x^{2}}} .

(d) f(x) = ∫_{0}^{x^{2}}(sin t)^{1/3} dt.

Answer: f′(x) = 2x(sin(x^{2}))^{1/3}.

3. (10 points) Find \frac{dy}{dx} if

x + sin x = y^{3}e^{y}.

Answer: \frac{dy}{dx} = \frac{1 + cos x}{y^{2}(y + 3)e^{y}}.

4. (10 points) Find the equation of the line tangent to the graph of

y = (x + 1)^{1/3}

at the point where x = 0.

Answer: y = \frac{1}{3}x + 1.

5. (30 points) Sketch the graph of

f(x) = x^{1/3}(x − 4),

indicating the intervals where the function is increasing, decreasing, concave up or concave down. Label all relative maxima and minima and inflection points.

Answer:

\begin{matrix}f′(x) = & \frac{4}{3}x^{1/3} − \frac{4}{3}x^{−2/3} = & \frac{4(x − 1)}{3x^{2/3}}; \\ f′′(x) = & \frac{4}{9}x^{−2/3} + \frac{8}{9}x^{−5/3} = & \frac{4(x + 2)}{9x^{5/3}}. \end{matrix}

Decreasing on (−∞, 1) except x = 0, increasing on (1, ∞). Concavity: up on (−∞, −2), down on (−2, 0), up on (0, ∞). Minimum at (1, −3), inflection points at (−2, 6 ^{3} \sqrt{2}) and (0, 0).

6. (20 points) Find the maximum volume of a cylinder inscribed in a sphere of radius 1 foot.

Answer: V = 2πx^{2}y with x^{2} + y^{2} = 1, so V = 2π(y − y^{3}), 0 ≤ x ≤ 1.

\frac{dV}{dy} = 2π(1 − 3y^{2}) = 0 when y = \sqrt{1/3}. \frac{d^{2}V}{dy^{2}} = −12πy < 0,

so the maximum volume is V = \frac{4π}{3\sqrt{3}} units.

7. (10 points) Find lim_{x→0} \frac{e^{−x} − 1}{tan^{−1} x}.

Answer: lim_{x→0} \frac{e^{−x} − 1}{tan^{−1} x} = lim_{x→0} \frac{−e^{−x}}{1/(1 + x^{2})} = −1.

8. (10 points) Give the recursive formula for x_{n} if Newton’s method is applied

to solve the equation

x^{5} + x^{2} + 1 = 0.

Answer: x_{n+1} = x_{n} − \frac{x_{n}^{5} + x_{n}^{2} + 1}{5x_{n}^{4} + 2x_{n}} = \frac{4x_{n}^{5} + x_{n}^{2} − 1}{5x_{n}^{4} + 2x_{n}}.

9. (10 points) Find the Riemann sum with left-hand points for

∫_{0}^{3π/4} sin^{4} x dx

with three intervals.

Answer: f(x) = sin^{4} x, Δx = π/4, ao f(0) = 0, f(π/4) = 1/4, f(π/2) = 1.

The Riemann sum is \frac{π}{4} (0 + \frac{1}{4} + 1) = \frac{5π}{16} .

10. (10 points) Solve the differential equation

\frac{dy}{dx} = \frac{y^{2}}{x + 1}

with initial value y = −1/2 when x = 0.

Answer: \frac{−1}{y} = ln |x + 1| + C, C = −2 (and x > −1),

so y = \frac{1}{2 + ln(x + 1)}.

11. (20 points) The region under the arch of y = sin x between x = 2π and x = 3π is rotated about the y-axis. Find the volume of the resulting solid of revolution.

Answer: dV = 2πx sin x dx, so V = ∫_{2π}^{3π} 2πx sin x dx.

Integration by parts (u = x, dv = sin x dx)

gives ∫ x sin x dx = sin x − x cos x + C.

So V = 2π(3π + 2π) = 10π^{2}.

12. (20 points) Find the arclength of the curve y = x^{3/2}, 1 ≤ x ≤ 9.

Answer: ds = \sqrt{1 + (9x/4)} dx, so s = ∫_{1}^{9} \frac{1}{2}\sqrt{4 + 9x} dx.

Substituting u = 4 + 9x, du = 9 dx gives

s = \frac{1}{18} ∫_{13}^{85} u^{1/2} du = \frac{1}{27}(85^{3/2} − 13^{3/2}).

13. (10 points each) Find the following indefinite integrals:

(a) ∫ \frac{cos x dx}{\sqrt{1 + sin x}}.

Answer: Let u = 1 + sin x, du = cos x dx. This gives ∫ \frac{cos x dx}{\sqrt{1 + sin x}} = 2\sqrt{1 + sin x} + C.

(b) ∫ sin^{3} x cos^{2} x dx.

Answer: Let u = cos x, du = − sin x dx, and replace sin^{2} x by 1 − cos^{2} x. This gives ∫ sin^{3} x cos^{2} x dx = \frac{1}{5} cos^{5} x − \frac{1}{3} cos^{3} x + C.

(c) ∫ \sqrt{a^{2} − x^{2}} dx. Answer: Let x = a sin θ, dx = a cos θ dθ. This gives ∫ \sqrt{a^{2} − x^{2}} dx = ∫ a^{2} cos^{2} θ dθ = \frac{a^{2}}{2} ∫ (1 + cos 2θ) dθ,

which is \frac{a^{2}}{2} (θ + \frac{1}{2} sin 2θ) + C = \frac{a^{2}}{2} sin^{−1} \frac{x}{a} + \frac{x}{2}\sqrt{a^{2} − x^{2}} + C.

(d) ∫ \frac{1}{x^{3} + x} dx.

Answer: “Partial fractions” gives \frac{1}{x^{3} + x} = \frac{1}{x} − \frac{x}{x^{2} + 1},

so ∫ \frac{1}{x^{3} + x} dx = ln |x| − \frac{1}{2} ln(x^{2} + 1) + C.

14. (10 points) Evaluate the following improper integral or show that it di-

verges:

∫_{0}^{∞} xe^{−x^{2}} dx.

Answer: u = −x^{2}, du = −2x dx gives ∫ xe^{−x^{2}} dx = −\frac{1}{2}e^{−x^{2}} + C,

so ∫_{0}^{∞} xe^{−x^{2}} dx = −\frac{1}{2} lim_{t→∞}(e^{−t^{2}} − 1) = \frac{1}{2}.