Math 151, Fall 2018 Common Exam 1 Version A

Transcription

1 Math 151, Fall 018 Common Exam 1 Version A LAST NAME (print): FIRST NAME (print): INSTRUCTOR: STUDENT ID NUMBER: Directions: 1. No calculators, cell phones, or other electronic devices may be used, and they must all be put away out of sight.. TURN OFF cell phones and put them away. If a cell phone is seen during the exam, your exam will be collected and you will receive a zero. 3. In Part 1 (Problems 1 0), mark the correct choice on your ScanTron using a No. pencil. The scantrons will not be returned, therefore for your own records, also record your choices on your exam! 4. In Part (Problems 1 4), present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 5. Be sure to write your name, section number, and version letter of the exam on the ScanTron form. THE AGGIE CODE OF HONOR On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic work. Signature: Page 1 of 11

2 Part 1: Multiple Choice (3 points each) 1. Find the vertical asymptotes of the function f(x) = x 3x 4 x + 3x 8. (a) x = 4 only (b) x = 7 only (c) x = 1 only (d) x = 4 and x = 7 only No vertical asymptotes. Find a vector of length 7 that points in the same direction as the vector from the point (3, 5) to the point (, 1). 35 (a), (b) (c) (d) 35 8, , , , Evaluate lim x 3 + x 7 x + x 15 (a) 0 (b) 1 7 (c) 15 (d) Page of 11

3 4. Find the cosine of ABC for the points A(1, 3), B(, 1), and C(4, ). 14 (a) 5 37 (b) (c) 5 34 (d) Which of the following statements is true regarding the equation x 3 + x = 8. (a) A solution to the equation exists on the interval (0, 1) by the Squeeze Theorem. (b) A solution to the equation exists on the interval (1, ) by the Squeeze Theorem. (c) A solution to the equation exists on the interval (0, 1) by the Intermediate Value Theorem. (d) A solution to the equation exists on the interval (1, ) by the Intermediate Value Theorem. A solution exists in the interval (, 3). 6. Which of the following statements about the function graphed below is FALSE? (a) lim f(x) = 1 x (b) f(1) = (c) f(x) is continuous from the right at x = 1. (d) f(x) is continuous at x = lim f(x) = x 1 Page 3 of 11

4 7. Evaluate lim x 5 x 6x + 5 x 5 (a) 4 (b) 6 (c) 4 (d) 6 8. Find the vector projection of the vector 1, 4 onto the vector 3,. (a) 4 17, 6 17 (b) 11 13, (c) 33 17, 17 (d) 11 17, , Simplify the expression sin (arctan x). (a) 1 x (b) x (c) (d) 1 x x x 1 x x 1 + x Page 4 of 11

5 10. A force F = 7, is used to move an object from the point (, 4) to the point (4, 1). How much work is done if the force is measured in Newtons and the distance in meters? (J = N m) (a) 4 J (b) J (c) 6 J (d) 48 J 74 J 7e x e 4x 11. Evaluate lim x 5e x + 3e 4x (a) 0 (b) 7 5 (c) 1 3 (d) 1. Suppose you are given the point P and the line L as shown in the figure. The vector u = 3, 1 is a vector along the line, and the vector v = 5, is a vector from a point on the line to the point P. Find the distance from P to L. 11 (a) 9 (b) (c) (d) Page 5 of 11

6 13. Two forces F 1 = 5, 3 and F = 3, 7 act on an object. Find the angle the resultant force makes with the positive x-axis. (a) arctan() (b) arctan( ) ( ) 1 (c) arctan ( (d) arctan 1 ) arctan ( 5) 14. Evaluate lim x (a) 0 (b) 9 5 9x + 5x (c) 3 5 (d) Find the average speed of a particle whose position is given by f(t) = x + from t = 1 to t = 7. (a) 3 (b) 1 3 (c) 5 7 (d) Page 6 of 11

7 16. Find the Cartesian equation for the vector function r(t) = + sin t, 3 + cos t. Also determine the direction in which the curve is traced. (a) (x ) + (y + 3) = 1; clockwise (b) (x ) + (y + 3) = 1; counter-clockwise (c) x 4 + y = 1; clockwise 9 (d) x 4 + y 9 (x ) 4 = 1; counter-clockwise + (y 3) 9 = 1; counter-clockwise 17. The following is the graph of f(x) Which of the following graphs is the graph of f (x)? (a) (b) (c) (d) Page 7 of 11

8 18. lim t (t + 3) (3t 5)(3 t) (a) 0 (b) 4 3 (c) 3 (d) 19. Find parametric equations of the line that passes through the point ( 5, 6) and is perpendicular to the line 3x + y = 4. (a) x = 5 + t, y = 6 3t (b) x = 3 + 5t, y = + 6t (c) x = 5t, y = 3 + 6t (d) x = 5 + 3t, y = 6 + t x = + 6t, y = 3 + 5t 0. Given that x + 4x 1 f(x) e x 1 on the interval [ 5, 5], which of the following is correct regarding lim x 1 f(x)? (a) The value of the limit cannot be determined. (b) lim f(x) = 0 by the Intermediate Value Theorem x 1 (c) lim f(x) = 0 by the Squeeze Theorem x 1 (d) lim f(x) = by the Intermediate Value Theorem x 1 lim f(x) = by the Squeeze Theorem x 1 Page 8 of 11

9 Part : Work out Directions: Present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 1. (15 pts) Evaluate the following limits. Do not use L Hospital s Rule. [ (a) ln(x 3) ln ( x )] lim x (b) lim x 5 ( ) x 9 4 x 5 ( ) 1 (c) lim x 4 x 4 6 x x 8 Page 9 of 11

10 . (7 pts) A pilot steers the plane in a direction 150 from the positive x-axis at a speed of 500 mph. The wind is blowing in a direction 45 from the positive x-axis at a speed of 30 mph. Find the true (resultant) velocity vector of the plane. (Your answer should be a vector.) ax 4, x < 3 3. (8 pts) Consider the function f(x) = bx + 3, x = 3 x + a, x > 3 (a) Find the value of a such that lim x 3 f(x) exists. (b) If a is chosen so that lim x 3 f(x) exists, then find the value of b such that f(x) is continuous at x = 3. Page 10 of 11

11 4. (10 pts) Use the definition of the derivative to find f (x) for f(x) = 7x 3x. No points will be given for any shortcut formulas used. FOR INSTRUCTOR USE ONLY Question Points Awarded Points Total Page 11 of 11