Math 141 Spring 2017 Final Exam PDF

## Final Examination - MATH 141 - May 12, 2017 ### Instructions: - Answer each of the 9 numbered problems on a separate answer sheet. - Each answer sheet must have your name, your TA's name, and the problem number (=page number). - Show all your work for each problem clearly on the answer sheet f...

## Final Examination - MATH 141 - May 12, 2017 ### Instructions: - Answer each of the 9 numbered problems on a separate answer sheet. - Each answer sheet must have your name, your TA's name, and the problem number (=page number). - Show all your work for each problem clearly on the answer sheet for that problem. - You must show enough written work to justify your answers. - Finally, please copy and sign the Honor Pledge on page 1. ### NO CALCULATORS OR OTHER ELECTRONIC DEVICES ### 1. **(a)** (10) Let the surface *S* have its base *R* on the *xyz* plane, where *R* is the region bounded by the graph of *f(x) = x^1/2* and the positive *x* axis for *0 ≤ x ≤ 3*. Assume that the cross sections perpendicular to the base and perpendicular to the *x* axis are semi-circles. Set up the integral for the volume *V* of *S*, and then find the numerical value of *V*. Include a sketch of the region and a nontrivial cross-section. **(b)** (10) Let *g(x) = 4 / (3^3/2x^3/2)* for *1 ≤ x ≤ 2*. Find the length *L* of the graph of *g*. ### 2. **(a)** (15) A bowl is in the shape obtained by revolving about the *y* axis the graph of *y = x²* for 0 ≤ x ≤ 3. The bowl has water (weight 62.5 lbs per cubic foot) to a depth of 7 feet. Write down the integral for the work *W* needed to pump all but a depth of 3 feet of water, raising the water to 4 feet above the top of the tank. Draw a picture of the bowl, including pertinent information. DO NOT EVALUATE THE INTEGRAL. **(b)** (10) Let *R* denote the region in the *xy*-plane between the graphs of *f(x) = x²* and *g(x) = √x*. Write down the integrals necessary in order to find the center of gravity of *R*. Draw a picture of *R* in the *xy*-plane. ### 3. **(a)** (15) Let *f(x) = x³*cosx + 5x. Find the largest interval *I* containing *x = π* for which *f* has an inverse, showing your work. Then compute *(f^-1)'(-1)*. **(b)** (10) Let *g(x) = cos^x(x)*. Find the domain of *g*, and calculate *g'(x)*. ### 4. **(a)** (10) Evaluate lim_x→0 *ln(x + e^2x)* / *x* **NOTE**: In problems 4b, 5a, 5b, 6a, and 6b, if you make a substitution, indicate the substitution. If you integrate by parts, then indicate *u* and *dv*. **(b)** (10) Evaluate ∫ tan³(2t) sec³(2t) dt ### 5. **(a)** (10) Evaluate ∫₀² √(4 - x²) dx. **(b)** (10) Evaluate ∫₂^∞ (2x + 5) / (x + 2)² dx. ### 6. **(a)** (10) Evaluate ∫_-2¹ 1 / (y² + 4y + 7) dy **(b)** (10) Determine whether the improper integral ∫₁^∞ e^x / (1 + x²)³ dx converges, or diverges. Support your answer. ### 7. **(a)** (10) Find lim_n→∞ (√(n + 4) - √n) / (√n). **(b)** (15) Determine whether the series ∑_n=1^∞ (-1)ⁿ⁺¹ (2/n²) converges absolutely, or converges conditionally. Then find an approximation of the sum of the series with an error less than 1/8. ### 8. **(a)** (10) Let g(x) = ∑_n=1^∞ (n + 1) / ((3n + 1)n!) x^-2n. Find the radius of convergence *R* of the power series, and then find the power series for *g'(x)*. **(b)** (10) Let f(x) = x / (1 + 2x³). Assume that 0 < c < R₁, radius of convergence of the power series. Write down an appropriate power series for ∫₀^c f(t) dt. ### 9. **(a)** (15) Consider the circle r = sinθ and the cardioid r = 1 - sinθ. On one graph, draw the circle and the cardioid. Indicate all intercepts and the values of θ for points of intersection of the circle and the cardioid. Then set up the integral for the area A of the region inside the circle and outside the cardioid. DO NOT EVALUATE THE INTEGRAL. **(b)** (10) Let z = -16. Write each of the 4th roots of z in the form re^iθ or r(cosθ + isinθ). Then display each of the roots in the complex plane, and identify each of them. ### END OF EXAM - GOOD LUCK!