Math 175 Common Exam 2A Spring 2018

Transcription

1 Math 175 Common Exam 2A Spring 2018 Part I: Short Form The first seven (7) pages are short answer. You don t need to show work. Partial credit will be rare and small. 1. (8 points) Suppose f(x) is a function with these features: f(4) = 2, f (4) = 4, f (4) = 0, f (4) = 1 2 Write the third degree Taylor polynomial approximation of f(x), centered at x = 4. Use the standard form with fully simplified coefficients. 2. (6 points) Suppose that the Taylor series for a function is given by f(x) = (a) Write the fourth degree Taylor polynomial that approximates f(x). (b) Write the third degree Taylor polynomial that approximates f (x). n=0 ( 1) n 2 n+1 xn. 3. (4 points) Use the fourth degree Taylor polynomial of e x, centered at x = 0, to approximate e 2. Rounding the approximation to four (4) decimal digits is sufficient. 4. (6 points) A 5 meter long beam has a distributed force density given by ω(x) = 250e 0.2x N/m, where x is the distance from the left end of the beam as shown. A typical slice located at x is shown. Use correct notation to write an appropriate expression for each quantity below. Do not compute any integrals. Units are not needed. (a) Write an expression for the total force on this beam. (b) Write an expression for the moment of a typical slice, located at x, about the point B. (c) Write an expression for the total moment of this beam about the origin, O. 1

2 5. (6 points) A plate is in the shape of the region shown at right, bounded by the coordinate axes and one quarter of the circle x 2 + y 2 = 16. The shaded region has a radial weight density given by ρ(r) = 10/(1 + r 2 ) g/in 2, where r is the distance from the center of the circle. (a) Draw a typical radial slice of thickness dr in the provided figure, at a distance 0 < r < 4 from the origin. (b) Write an expression for the area of such a typical slice. (c) Write an integral for the total mass of the plate. Do not compute the integral. 6. (14 points) The shape at right shows the curve y = 4 x 2, for 0 x 2, rotated about the y-axis, with a cone of radius 1 inch and a height of 4 inches removed from the center. The volume of the resulting shape can be found by slicing this object along the y-axis. A typical 3-D slice and a 2-D xy plane cross section appear below. (a) Write the volume of the slice in terms of the dimensions r 1, r 2 and dy labeled on the slice. dv = (b) Label the dimensions r 1, r 2 and dy on the 2-D xy-plane cross section. (c) Write expressions for r 1 and r 2 in terms of the variable of integration. 2

3 r 1 = r 2 = 7. (8 points) A plate has the shape of the region shown at right, bounded by the curves y = x 3 and x = y 4. The plate has a variable area density given by ρ = 3 y kg/m 2. Determine if each of the following English or notational statements (using either x-axis or y-axis of integration) are true or false. Circle either T (true) or F (false) for each statement. ( T F ) da = (y 4 x 3 )dx. ( T F ) The mass of the plate is (3 y)(y 1/3 y 4 )dy. ( T F ) A = 1 0 (x 1/4 x 3 )dx. ( T F ) The area of a typical slice along the x-axis is x 1/4 x 3. ( T F ) The mass of the plate is 1 0 (3 y)(x 1/4 x 3 )dx. ( T F ) The mass of a typical slice is (3 y)(y 1/3 y 4 )dy. ( T F ) dm = 1 0 (3 y)(y 1/3 y 4 )dy. ( T F ) The area of a typical slice along the y-axis is (y 4 y 1/3 )dy. 8. (10 points) A function f(x) is graphed below, followed by questions about the second degree Taylor polynomial that approximates f(x) at x = 3. 3

4 (a) Write the standard form of T 2 (x), centered at x = 3. (b) Use the graph of f(x) to estimate the Taylor coefficients, as follows: i. Provide the exact value of a 0. ii. Provide an approximate value for a 1. iii. Provide the sign of a 2. Circle one: POSITIVE NEGATIVE (c) Use your results to add a rough sketch of T 2 (x) to the above graph. Part II: Long Form Show all work. Unsupported answers will not receive full credit. Present your work cleanly and clearly. Neatness counts. All computed results require correct units on the final answer. Integral problems require a clearly communicated slicing strategy: A stated axis of integration. Picture(s) of your slice(s). Correct and consistent labeling. Stated formula(s) for the size of the slice(s). Stated formula(s) are required for any relevant feature(s) of your slice(s), such as force or moment. Include properly written integral(s) for the total amount. If you compute an integral, show or explain how you did it. If you solve an equation, show or explain how you did it. You may use calculators to compute integrals or solve equations. Such use must include detailed statements about your calculator use: Name the calculator feature(s) used. Write the exact calculator entry. Provide the screen appearance, labeled as such. I used my calculator is insufficient. 4

5 9. (10 points) The shaded region in the sketch at right is bounded by the lines y = 4, x = 2, and the curve y = 4 x 2. Lengths are measured in centimeters. Compute the area of the plate. Be sure to show all work. See page (10 points) A triangular plate with vertices at (0, 0), (3, 0), and (0, 5) is shown in the figure at right. The plate has an area density of 12 kg/m 2. Write an integral to express the moment of this plate about the axis y = 2, as shown in the figure. Show all work involved in setting up the integral. See page 8. Do not compute the integral. 11. (10 points) The object shown in the sketch at right is created by rotating the curve y = (x 2) 3 around the line x = 2, for 2 x 3. Lengths are measured in feet. What is the volume of the described object? Be sure to show all work. See page 8. Turn the page for the exam s last problem. 5

6 12. (8 points) A 20 foot beam carries a distributed load in its first ten feet of length. That load is given by w(x) = k(10 x) lbs/ft, where x is the distance from the left end of the beam, as shown. Also, a concentrated force of F = 60 pounds is imposed at the right end of the beam. Find the value of k that makes the beam balance at the pivot labeled A, which is located at the center of the beam. The measurement unit for k is lbs/ft 2. 6