MATH 141, PRACTICE FINAL EXAM , SPRING 2010 ANSWER KEY Evaluate the following indeterminate forms using L’Hôspital’s rule. Show all major steps. State what type of indeterminate form it is. Express final answer in simplest exact form. 1.) EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 Ans: Form is EMBED Equation.3 . Rewrite as EMBED Equation.DSMT4 form. Using L. R., this becomes EMBED Equation.DSMT4 2.) EMBED Equation.DSMT4 Ans: Form is EMBED Equation.DSMT4 . Let EMBED Equation.DSMT4 . Taking logarithms gives EMBED Equation.DSMT4 , which is of the form EMBED Equation.DSMT4 . Rewrite this as EMBED Equation.DSMT4 , which is of the form EMBED Equation.DSMT4 . Using L.R. gives EMBED Equation.DSMT4 . Thus L= 1 Perform the following integrations. Show all major steps! Show all substitutions used. Express final answer in simplest exact form. 3.) EMBED Equation.DSMT4 EMBED Equation.DSMT4 Ans: Let EMBED Equation.DSMT4 . So EMBED Equation.DSMT4 4.) EMBED Equation.DSMT4 Ans: EMBED Equation.DSMT4 5.) EMBED Equation.DSMT4 Ans: Let EMBED Equation.DSMT4 , so the integral becomes EMBED Equation.DSMT4 6.) EMBED Equation.DSMT4 Ans: Let EMBED Equation.DSMT4 , so the integral becomes EMBED Equation.DSMT4 7.) EMBED Equation.DSMT4 Ans: EMBED Equation.DSMT4 8.) EMBED Equation.DSMT4 Use partial fractions to express EMBED Equation.DSMT4 . Thus EMBED Equation.DSMT4 9.) EMBED Equation.DSMT4 Use trig substitution letting EMBED Equation.DSMT4 . The integral becomes EMBED Equation.DSMT4 = EMBED Equation.DSMT4 , the last step using properties of logs and remembering that the +C can absorb arbitrary constants. 10.) EMBED Equation.DSMT4 Ans: First complete the square: EMBED Equation.DSMT4 . Then let u = x+4 , so x = u-4, and the integral becomes EMBED Equation.3 = EMBED Equation.3 11.) EMBED Equation.DSMT4 Ans: Factor the denominator as EMBED Equation.DSMT4 . The partial fraction decomposition is EMBED Equation.DSMT4 . Clear fractions to get EMBED Equation.DSMT4 . Let x=0 to get 3=A. Let x = -1 to get –1=-C, or C=1. Equate coefficients of EMBED Equation.DSMT4 to get 1=A+B, so B=-2. Then EMBED Equation.DSMT4 . Integrate to get EMBED Equation.DSMT4 12.) Set up, but do not evaluate, an integral that gives the arc-length of the curve EMBED Equation.DSMT4 EMBED Equation.DSMT4 . Ans: Use the formula EMBED Equation.DSMT4 EMBED Equation.DSMT4 to get EMBED Equation.3 13.) Sketch the following parametric curves. Show the direction of increasing parameter. Show at least 3 points on the curve, including the parameter value – may be placed in a table of values on the side. Finally, eliminate the parameter to obtain a Cartesian curve in x and y. a) EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 Ans: EMBED Equation.DSMT4 only, EMBED Equation.DSMT4 b) EMBED Equation.DSMT4 Ans: From the first equation EMBED Equation.DSMT4 and then plugging this into the second equation yields EMBED Equation.DSMT4 , going from (1,1) to (10,10). c) EMBED Equation.3 new Ans: Use EMBED Equation.3 , where EMBED Equation.3 , to get EMBED Equation.3 , an ellipse with vertices at (5,0), (0, 3), (-5,0), (0,-3).. The curve is traced in the clockwise direction, starting at (5,0). 14.) Find the arc length of the curve given parametrically by EMBED Equation.DSMT4 Ans: Use the formula EMBED Equation.DSMT4 . Since EMBED Equation.DSMT4 then EMBED Equation.DSMT4 . Then let EMBED Equation.DSMT4 to get EMBED Equation.DSMT4 15.) Find the Cartesian equation of the tangent line to the parametric curve EMBED Equation.3 at the point P(6,6) on the curve. Ans: First note that (6,6) occurs when t = 1. Use the formula EMBED Equation.3 to get the slope of the line. Since EMBED Equation.3 , then EMBED Equation.3 . Plugging in t = 1 gives m=5. Next use the point slope formula to get y-6=5(x-6), or y = 5x-24. 16.) Sketch the following polar curves. Identify which family it belongs to. a) EMBED Equation.DSMT4 EMBED Equation.3 Ans: Graph is a circle, tangent to the x-axis at the origin, diameter = 3. b) EMBED Equation.DSMT4 Ans: Graph is a cardiod, pinched in from below on the negative y – axis. c) EMBED Equation.3 (also state which lines are tangent to the curve at the origin) Ans: Graph is a 4 – leaf rose, tangent to the origin at EMBED Equation.3 17.) Set up an integral that gives the arc length of the inner loop of the limaçon EMBED Equation.DSMT4 . Do not evaluate. Ans: To find the limits of the inner loop, first solve EMBED Equation.3 to get EMBED Equation.3 , or EMBED Equation.3 , or in radians EMBED Equation.3 . Thus the inner loop is sketched for EMBED Equation.3 . Next use the formula EMBED Equation.3 . So EMBED Equation.3 , then EMBED Equation.3 . This can be simplified by expanding out the inside and using a Pythagorean Identity to get EMBED Equation.3 . 18.) Set up an integral to find the area inside the lemniscate EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.DSMT4 and outside the circle r=2. Ans: The lemniscate is a figure 8 curve, along the x-axis. Due to symmetry, we will only find the area in the first quadrant, and multiply the result by 4. Use the formula EMBED Equation.DSMT4 . Find the intersection of the two curves by solving EMBED Equation.DSMT4 to get EMBED Equation.DSMT4 , so EMBED Equation.DSMT4 , and EMBED Equation.DSMT4 . The area in the first quadrant will be given by EMBED Equation.DSMT4 , so finally A= EMBED Equation.DSMT4 . Test the following series for convergence or divergence. State what test you are using, and support your conclusion. a) EMBED Equation.DSMT4 Ans: Use the Limit Comparison Test. Compare with the series EMBED Equation.DSMT4 , which is a divergent p=series with p=1/2. Since EMBED Equation.DSMT4 , then the original series diverges. . b) EMBED Equation.DSMT4 Ans: Use the Ratio test. Then EMBED Equation.DSMT4 c) EMBED Equation.3 Ans: Since EMBED Equation.3 , the series diverges by the divergence test. d) EMBED Equation.3 Ans: Since EMBED Equation.3 EMBED Equation.DSMT4 , and EMBED Equation.3 is monotonically decreasing, then by the alternating series test, the series converges (but only conditionally). e) EMBED Equation.DSMT4 Ans: Use the comparison test. Note that EMBED Equation.DSMT4 , then multiplying by 3 and then adding 5 gives EMBED Equation.DSMT4 . Next divide by n (a positive number) and summing from n = 1 to infinity gives EMBED Equation.DSMT4 . But since EMBED Equation.DSMT4 diverges, so does the oroginal series. f) EMBED Equation.DSMT4 Ans: Since EMBED Equation.DSMT4 is not always positive, we may not use the comparison test. Instead, check for absolute convergence. Since EMBED Equation.DSMT4 , then EMBED Equation.DSMT4 . So the series is absolutely convergent by the comparison test. Hence the series is convergent. 20.) Show the following series converges, and find the exact value of the sum: EMBED Equation.DSMT4 Ans: This is a geometric series, with EMBED Equation.DSMT4 , so the series converges. Use the formula EMBED Equation.DSMT4 21.) Find the interval of convergence for the following power series. Be sure to test the endpoints of the interval of convergence. Where does the series converge? Where does the series converge absolutely? Where does the series converge conditionally? EMBED Equation.DSMT4 EMBED Equation.DSMT4 EMBED Equation.3 Ans: Use the absolute ratio test to get EMBED Equation.DSMT4 EMBED Equation.DSMT4 . Thus the series is absolutely convergent for 6