Exam 4 SCORE. MA 114 Exam 4 Spring Section and/or TA:

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1 Exam 4 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used except for a onepage sheet of formulas and facts. You may use a graphing calculator during the exam, but NO calculator with a Computer Algebra System (CAS) or a QWERTY keyboard is permitted. Absolutely no cell phone use during the exam is allowed. The exam consists of multiple choice questions and 5 free response questions. Record your answers to the multiple choice questions on this page by filling in the circle corresponding to the correct answer. Show all work to receive full credit on the free response problems. Unsupported answers on free response problems will receive no credit. Multiple Choice Questions A B C D E A B C D E 3 A B C D E 4 A B C D E 5 A B C D E 6 A B C D E 7 A B C D E 8 A B C D E 9 A B C D E A B C D E SCORE Multiple (BONUS) Total Choice Score 5

2 Multiple Choice Questions. The solution to the initial value problem is: A. y = ln ( x ) ) B. y = ln ( x ) C. y = ln ( x D. y = e x / E. y = e x dy dx = xey, y() =. The equation 9x 8x + 4y = 7 is the equation of A. An ellipse with a = 3, b =, and foci at (, ± 5) B. An ellipse with a = 3, b =, and foci at (± 5, ) C. An ellipse with a = 3, b =, and foci at (, ± 5) D. An ellipse with a = 3, b =, and foci at (, ± 5) E. An ellipse with a =, b =, and foci at (, ) Page of 5

3 3. What is the radius of convergence of the series n= xn n? A. (the series converges for no nonzero x) B. (the series converges for all x) C. D. E Which of the following curves has the polar description r = 5 cos θ? A. A circle of radius 5 with center at (5, ) B. A circle of radius 5/ with center at (, 5/) C. A circle of radius 5/ with center at (5/, ) D. A circle of radius 5/ with center at ( 5/, ) E. A circle of radius 5/ with center at (, 5/) Page 3 of 5

4 5. Which of the following is the direction field for the differential equation y = sin x sin y? A B C D 6. A population grows according to the logistic equation ( dp dt =.4P P ), P() = 6. Which of the following is true about this model (there is only one correct choice)? A. The carrying capacity is 6 and the growth constant k is.4. B. The carrying capacity is and the growth constant k is 6 C. The carrying capacity is and the growth constant k is.4 D. The initial population is 6 and the carrying capacity is 5 E. As t the population goes to zero. Page 4 of 5

5 7. Below is the direction field for a differential equation of the form y = f (y). Which of the following statements is correct (there is only one correct choice)? A. y =, y =, and y = are all stable equilibria B. y = and y = are stable equilibria, while y = is an unstable equilibrium C. y = and y = are unstable equilibria, but y = is a stable equilibrium D. y = and y = are the only equilibria, and both are stable E. y = is the only equilibrium, and is unstable Page 5 of 5

6 8. What is the slope of the tangent line to the graph of r = cos θ at θ = π/3? A. B. C. 3 D. / 3 E. / 3 9. Which of the following is the correct form for the partial fraction decomposition of x 4 (x x + )(x + )? A. B. C. D. Ax + B x x + + Cx + D x + Ax + B x x + + Cx + D (Ex + F) x + + (x + ) Ax + B x x + + Cx + D x + + Ex + F (x + ) Ax + Bx + C x x + E. None of the above + Cx + D x + + Ex + F (x + ). A function has MacLaurin series Find f (). A. /7 B. 4/7 C. /7 D. 6/7 E. 6/7 f (x) = ( ) n n= n + xn = x 3 + x 5 x3 7 + Page 6 of 5

7 . The equation defines a hyperbola. Free Response Questions 9y 4x 36y 8x = 4 (a) (4 points) By completing the square in x and y, put this equation in standard form. Solution: 9(y 4y) 4(x + x) = 4 9(y 4y + 4) 4(x + x + ) = (y ) 4(x + ) = 36 (y ) 4 (x + ) 9 = (b) (3 points) Find the foci, vertices, and asymptotes of the curve. Solution: This curve is a hyperbola shifted by (, ). The standard form hyperbola y a x b = has foci (±c, ), vertices (±a, ), and asymptotes y = ±(a/b)x. In our case, a =, b = 3, and hence c = 3 + = 3 or c = 3. Incorporating the shift we get: Foci: (, ± 3) Vertices: (, ) and (, 4) Asymptotes: y = ±(/3)(x + ) Page 7 of 5

8 (c) (3 points) Sketch the curve on the axes provided. Label the foci and the vertices, and sketch the asymptotes. Solution: Vertices are plotted at (, ) and (, 4), and foci are shown at (, 3) and (, + 3), The asymptotes are plotted as blue dashed lines. Page 8 of 5

9 . Newton s law of cooling states that the rate of heat loss of a body is proportional to the difference between its temperature and the temperature of its surroundings. Thus if T(t) is the temperature of the object and T is the temperature of its surroundings, dt dt = k (T(t) T ) where k is a rate constant. A cup of coffee at 95 C is placed in a C room and reaches a temperature of 57.5 after / hour. (a) ( points) Set up the initial value problem (differential equation and initial condition) for the temperature of the coffee as a function of time in hours. Solution: dt dt = k (T(t) ), T() = 95 (b) (4 points) Using the method of separation of variables, solve this equation for T(t) up to a constant of integration and the constant k. Solution: dt T = kdt ln(t ) = kt + C T = Ce t T = + Ce kt (c) (4 points) Find the rate constant k and the constant of integration, and state the solution formula. Solution: At time, the coffee is at 95 C, so 95 = + C C = 75 Page 9 of 5

10 After / hour, the coffee s temperature is 75 C so The solution formula is then 57.5 = + 75e (/)k 37.5 = 75e (/)k = e (/)k ln = k k = ln ( ln )t T(t) = + 75e (d) ( points) How long does it take for the coffee to cool to C? Solution: We solve the equation ( ln )t = + 75e ( ln )t 8.75 = 75e ln )t = e ( 4 ln 4 = ( ln )t t = Page of 5

11 3. Find the radius of convergence and the interval of convergence for the series ( ) n n5 n= n x n. Solution: Using the ratio test lim n a n+ a n a n+ a n so the radius of convergence of the series is 5. = n5 n (n + )5 n+ x = x 5 To find the interval of convergence, we check the endpoints x = 5 and x = 5. (a) x = 5: The series n= is a harmonic series which diverges. (b) x = 5: The series n= ( ) n n5 n ( 5) n = ( ) n n5 n (5) n = n= n= n ( ) n is an alternating series which converges. Hence, the interval of convergence is ( 5, 5]. n Page of 5

12 4. The goal of this problem is to find the area that lies inside the curves r = 3 sin θ and r = 3 cos θ. (a) (4 points) Find the equation of each of these curves in Cartesian coordinates, and identify the curves. Solution: The first curve has Cartesian equation x + y = 3y x + y 3y = ( x + y 3 ) = 9 4 while the second curve has Cartesian equation x + y = 3x x 3x + y = ( x 3 ) + y = 9 4 Hence, the first curve is a circle of radius 3/ with center (, 3/), and the second curve is a circle of radius 3/ with center (3/, ). (b) ( points) Graph the two curves on the axes provided, and find their points of intersection in Cartesian coordinates. Solution: From the graph, one of the points of intersection is (, ). The other point of intersection has sin θ = cos θ = π/4 so r = 3/ and θ = π/4. Hence (x, y) = ( 3, 3 ). (c) (4 points) Set up, but do not evaluate, an integral for the area that lies inside of both curves. Solution: The area that lies inside both curves is bounded for θ π/4 by the curve r = 3 sin θ and for π/4 θ π/ by the curve r = 3 cos θ. So, the total area is given by π/4 π A = 9 sin θ dθ + 9 cos θ dθ π/4 Page of 5

13 By symmetry each of these integrals is equal so we also have A = π/4 9 sin θ dθ. Page 3 of 5

14 5. Suppose R is the region bounded by the curves y = x 3, y = x with x. (a) Sketch the curves and the region R on the axes provided, and use the given equations to solve for the points of intersection. Solution: The points of intersection are (, ) and (, ). (b) Set up an integral for the volume of the region obtained by rotating R about the x-axis. Be sure to state whether you are using the disc method, the shell method, or the washer method. Solution: V = π ( x x 6) dx (c) Compute the integral. Solution: [ x 3 V = π 3 x7 7 ] = π ( 3 ) = 4π 7. Page 4 of 5

15 6. (OPTIONAL BONUS QUESTION) ( points) This problem concerns using Taylor series to find limits. Solutions using L Hospital s rule will receive no credit! (a) Use the Taylor series e x = + x + x + 6 x to find lim x e x e x. x Solution: e x = + x + x + 6 x e x = x + x 6 x e x e x = x + 3 x e x e x = + x 6 x +... e x e x Hence, lim x =. x (b) Use the Taylor series to find cos x = x + x cos(x) lim x x. Solution: cos(x) = 4x + 6x cos(x) x = ) ( 4x x + 6x = + 3 x +... so cos(x) lim x x =. Page 5 of 5

SCORE. Exam 3. MA 114 Exam 3 Fall 2016

SCORE. Exam 3. MA 114 Exam 3 Fall 2016 Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing