18.01 Practice Final

This exam is 3 hours long and worth 250 points. No books, calculators or notes. A table of integrals is attached.

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

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

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

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

(c) f(x) = sin^{−1}(\sqrt{x}).

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

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

x + sin x = 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.

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.

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

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

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.

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

∫_{0}^{3π/4} sin^{4} x dx

with three intervals.

10. (10 points) Solve the differential equation

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

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

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.

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

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

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

(b) ∫ sin^{3} x cos^{2} x dx.

(c) ∫ \sqrt{a^{2} − x^{2}} dx.

(d) ∫ \frac{1}{x^{3} + x} dx.

14. (10 points) Evaluate the following improper integral or show that it diverges:

∫_{0}^{∞} xe^{−x^{2}} dx.

Basic integration formulas

∫ u^{n} du = \frac{u^{n+1}}{n + 1} + C, if n ≠ 1

∫ \frac{du}{u} = ln |u| + C

∫ e^{u} du = e^{u} + C

∫ cos u du = sin u + C

∫ sin u du = − cos u + C

∫ sec^{2} u du = tan u + C

∫ csc^{2} u du = − cot u + C

∫ sec u tan u du = sec u + C

∫ csc u cot u du = − csc u + C

∫ \frac{du}{\sqrt{a^{2} − u^{2}}} = sin^{−1} \frac{u}{a} + C

∫ \frac{du}{a^{2} + u^{2}} = \frac{1}{a} tan^{−1} \frac{u}{a} + C

∫ tan u du = − ln | cos u| + C

∫ cot u du = ln | sin u| + C

∫ sec u du = ln | sec u + tan u| + C

∫ csc u du = − ln | csc u + cot u| + C