Math 143 Practice Final Exam Fall 2024 - PDF

Math 143 – Calculus III Name________________________________ Instructor: Michael Miller Date ________________________________ No Calculator Section: _______...

Math 143 – Calculus III Name________________________________ Instructor: Michael Miller Date ________________________________ No Calculator Section: _______ Your Score: Final Exam ______ out of 200 Clearly show all relevant work in order to receive full credit. Simplify all answers when possible and leave them in exact form, unless stated otherwise. Relax and good luck! 1. True or False (2 points each) Circle T if the statement is always true, otherwise circle F. You do not need to justify your answer. (a) T or F (b) T or F (c) T or F (d) T or F (e) T or F 1 2. Evaluating Series (12 points) Show whether each series converges or diverges using an appropriate test. If the series converges, then find the value of the sum if possible. (a) (b) 2 3. Absolute and Conditional Convergence (12 points) Determine if the series converges absolutely, converges conditionally, or diverges. Give a clear justification with the appropriate tests and check their necessary conditions. (a) (b) 3 4. Repeating Decimal Representation (8 points) Express the given number as a ratio of integers in simplest form by first representing it as a series, then evaluating the series. 5. Interval of Convergence (12 points) Find and state the values of x for which the series converges by giving the interval of convergence. Also state the radius of convergence. Radius of Convergence: 𝑅 =____________ Interval of Convergence: ____________________ 4 6. Integrating a Power Series (10 points) Express the general antiderivative as a power series. Find the radius of convergence. Radius of Convergence: 𝑅 =____________ 7. Power Series (10 points) Find a power series representation for the function at the given center. Determine the radius of convergence. Radius of Convergence: 𝑅 =____________ 5 8. Taylor Series (10 points) Find the first four non-zero terms of the Taylor series representation of the function at the given center. 9. Evaluating Series (12 points) Find the sum of each series. Simplify your answer if possible. (a) (c) 6 10. Parametric Equations (16 points) Consider the parametric curve given by the following equations. 𝜋 (a) Find the slope of the tangent line at 𝑡 = 3. (b) Find the area of the surface obtained by rotating the curve about the y-axis. 7 11. Converting Coordinates (6 points) Convert the polar equation to an equation of a circle in standard form in Cartesian coordinates, x and y. Identify the center and radius of the circle. Center:____________ Radius:____________ 12. Area of Polar Regions (12 points) Find the exact area of the cardioid, 𝑟 = 5 − 5sin𝜃. 8 13. Area Between Polar Curves (6 points) Set up an integral representing the shaded area between the two polar curves shown in the picture. Do not actually evaluate the integral! 14. Equation of a Sphere (6 points) Complete the square of the polynomial in each variable to put the equation of the sphere in standard form. Identify the center and radius of the sphere. Center: _______________________ Radius: _______________________ 9 15. Vector Projection (6 points) ________________ ________________ 16. Planes (12 points) Consider the three points in ℝ3 : (a) Find the equation of the plane in standard form that contains the three points. (b) Find the area of the triangle that has vertices equal to the given points. 10 17. Planes and Lines (10 points) Consider the plane in ℝ3 given by: (a) Find the distance between the plane and the point (1, −2, 4). (b) Find the point where the given line intersects the plane. 18. Tangent Line to a Vector Function (10 points) Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. 11 19. Arc Length (10 points) Find the exact length of the curve. 20. Motion in Space (10 points) Find the position function of a particle with the given acceleration and the specified initial velocity and position. Thank you for all your effort this quarter and have a great Spring Break! 12