{"id":9073,"date":"2017-02-14t11:52:02","date_gmt":"2017-02-14t19:52:02","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=9073"},"modified":"2018-10-26t06:26:09","modified_gmt":"2018-10-26t13:26:09","slug":"ap-calculus-bc-exam-multiple-choice-practice-problems","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-bc-exam-multiple-choice-practice-problems\/","title":{"rendered":"ap calculus bc exam multiple choice practice problems"},"content":{"rendered":"<p>there are 45 problems in the ap calculus bc exam multiple choice section. let&#8217;s take a look at the kinds of questions you might see on the test. below, you can find some information about the format of the test and then a sampling of practice problems with solutions provided at the end.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/shutterstock_387742072.jpg\" alt=\"ap calculus bc exam multiple choice practice problems\" width=\"500\" height=\"334\" class=\"aligncenter size-full wp-image-9140\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/shutterstock_387742072.jpg 500w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/shutterstock_387742072-300x200.jpg 300w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><\/p>\n<h2>format of the exam<\/h2>\n<p>the <a href=\"https:\/\/apstudent.collegeboard.org\/apcourse\/ap-calculus-ab\/about-the-exam\" target=\"_blank\" rel=\"noopener noreferrer\">calculus ab<\/a> and <a href=\"https:\/\/apstudent.collegeboard.org\/apcourse\/ap-calculus-bc\/about-the-exam\" target=\"_blank\" rel=\"noopener noreferrer\">calculus bc<\/a> exams both have the same format.<\/p>\n<ul>\n<li>section i \u2014 multiple choice (1 hour and 45 minutes total)\n<ul>\n<li>part a \u2014 30 questions \u2014 60 minutes \u2014 no calculator.<\/li>\n<li>part b \u2014 15 questions \u2014 45 minutes \u2014 graphing calculator required.<\/li>\n<\/ul>\n<\/li>\n<li>section ii \u2014 free response (1 hour and 30 minutes total)\n<ul>\n<li>part a \u2014 2 questions \u2014 30 minutes \u2014 graphing calculator required.<\/li>\n<li>part b \u2014 4 questions \u2014 60 minutes \u2014 no calculator.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>ap calculus bc exam multiple choice practice<\/h2>\n<p>go find a nice quiet distraction-free location to work on these problems. in fact, you should treat this practice set like the real deal. even though there are not 45 questions in this practice, there are enough to get the idea of what could show up on the exam.<\/p>\n<p>the suggested time limits are scaled down appropriately. first see what you can get through in the time allotted. then after time runs out, you can always go back and complete the rest.<\/p>\n<p>remember, on the ap calculus bc exam multiple choice section, you don&#8217;t have to show your work. furthermore, there is no penalty for a wrong choice. so if you have no idea how to do a problem, try to eliminate a choice or two and then guess among the remaining answers.<\/p>\n<h3>part a<\/h3>\n<p><em>(time \u2014 20 minutes \u2014 10 questions)<\/em><\/p>\n<p><em>(a calculator may <strong>not<\/strong> be used on this part of the examination)<\/em><\/p>\n<ol>\n<li>the position of a particle at time <em>y<\/em> \u2265 0, in seconds, is given by <em>x<\/em> = <em>t<\/em><sup>3<\/sup> &#8211; 12<em>t<\/em><sup>2<\/sup> + 45<em>t<\/em> &#8211; 1 and <em>y<\/em> =<em>t<\/em><sup>3<\/sup> &#8211; 15<em>t<\/em><sup>2<\/sup> + 63<em>t<\/em> + 5. at what values of <em>t<\/em> is the particle at rest?(a) \u2003 <em>t<\/em> = 3, 5\n<p>(b) \u2003 <em>t<\/em> = 3<\/p>\n<p>(c) \u2003 <em>t<\/em> = 3, 5, 7<\/p>\n<p>(d) \u2003 <em>t<\/em> = 5, 7<\/li>\n<li>what is the value of \u2002<img decoding=\"async\" class=\"alignnone size-full wp-image-9076\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/geometricseries_problem.gif\" alt=\"geometric series problem\" width=\"65\" height=\"51\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/geometricseries_problem.gif 65w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/geometricseries_problem-30x24.gif 30w\" sizes=\"(max-width: 65px) 100vw, 65px\" \/>\u00a0?(a) 2\/9\n<p>(b) 2\/3<\/p>\n<p>(c) 3<\/p>\n<p>(d) diverges<\/li>\n<li><img decoding=\"async\" class=\"alignnone size-full wp-image-9077\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/improper_integral_problem.gif\" alt=\"improper integral problem\" width=\"129\" height=\"43\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/improper_integral_problem.gif 129w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/improper_integral_problem-30x10.gif 30w\" sizes=\"(max-width: 129px) 100vw, 129px\" \/>(a) \u03c0\/4\n<p>(b) \u03c0\/3<\/p>\n<p>(c) \u03c0\/2<\/p>\n<p>(d) diverges<\/li>\n<li>the length of a curve from <em>x<\/em> = 1 to <em>x<\/em> = 5 is given by <img decoding=\"async\" class=\"alignnone size-full wp-image-9078\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/length_integral_problem.gif\" alt=\"length integral problem\" width=\"120\" height=\"25\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/length_integral_problem.gif 120w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/length_integral_problem-30x6.gif 30w\" sizes=\"(max-width: 120px) 100vw, 120px\" \/>. if the curve contains the point (1, 10), which of the following could be the curve?(a) \u2003 <em>y<\/em> = 64<em>x<\/em><sup>2<\/sup> &#8211; 54\n<p>(b) \u2003 <em>y<\/em> = 64<em>x<\/em><sup>2<\/sup> + 1<\/p>\n<p>(c) \u2003 <em>y<\/em> = 8<em>x<\/em><sup>2<\/sup> + 2<\/p>\n<p>(d) \u2003 <em>y<\/em> = 4<em>x<\/em><sup>2<\/sup> + 6<\/li>\n<li>the graph of the function <em>f<\/em> is shown below.<img decoding=\"async\" class=\"aligncenter size-full wp-image-9079\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/wavy_graph.png\" alt=\"graph of a wavy function\" width=\"300\" height=\"300\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/wavy_graph.png 300w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/wavy_graph-150x150.png 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/>\n<p>if \u2003 <img decoding=\"async\" class=\"alignnone size-full wp-image-9082\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/accumulation_function_problem1.gif\" alt=\"accumulation function problem\" width=\"144\" height=\"47\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/accumulation_function_problem1.gif 144w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/accumulation_function_problem1-30x10.gif 30w\" sizes=\"(max-width: 144px) 100vw, 144px\" \/>\u00a0, what is <img decoding=\"async\" class=\"alignnone size-full wp-image-9081\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/g_prime_2.gif\" alt=\"the derivative of g at 2\" width=\"36\" height=\"18\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/g_prime_2.gif 36w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/g_prime_2-30x15.gif 30w\" sizes=\"(max-width: 36px) 100vw, 36px\" \/>\u00a0?<\/p>\n<p>(a) 5<\/p>\n<p>(b) 8<\/p>\n<p>(c) 12<\/p>\n<p>(d) 16<\/li>\n<li>find the area inside one loop of the curve <em>r<\/em> = 3cos 2<em>\u03b8<\/em>.(a) \u03c0\/4\n<p>(b) 3\u03c0\/8<\/p>\n<p>(c) 9\u03c0\/8<\/p>\n<p>(d) \u03c0<\/li>\n<li>for what values of <em>x<\/em> does the series\u2003 <img decoding=\"async\" class=\"alignnone size-full wp-image-9083\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/power_series_problem.gif\" alt=\"power series problem\" width=\"68\" height=\"51\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/power_series_problem.gif 68w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/power_series_problem-30x23.gif 30w\" sizes=\"(max-width: 68px) 100vw, 68px\" \/> \u00a0 converge?(a) -1\/4 &lt; <em>x<\/em> &lt; 1\/4\n<p>(b) -4 &lt; <em>x<\/em> &lt; 4<\/p>\n<p>(c) -1 &lt; <em>x<\/em> &lt; 1<\/p>\n<p>(d) all real numbers<\/li>\n<li><img decoding=\"async\" class=\"alignnone size-full wp-image-9084\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_problem.gif\" alt=\"integration by parts problem\" width=\"68\" height=\"21\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_problem.gif 68w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_problem-30x9.gif 30w\" sizes=\"(max-width: 68px) 100vw, 68px\" \/> =(a) <img decoding=\"async\" class=\"alignnone size-full wp-image-9085\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_a.gif\" alt=\"ibp answer a\" width=\"124\" height=\"22\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_a.gif 124w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_a-30x5.gif 30w\" sizes=\"(max-width: 124px) 100vw, 124px\" \/>\n<p>(b) <img decoding=\"async\" class=\"alignnone size-full wp-image-9086\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_b.gif\" alt=\"ibp answer b\" width=\"117\" height=\"22\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_b.gif 117w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_b-30x6.gif 30w\" sizes=\"(max-width: 117px) 100vw, 117px\" \/><\/p>\n<p>(c) <img decoding=\"async\" class=\"alignnone size-full wp-image-9087\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_c.gif\" alt=\"ibp answer c\" width=\"71\" height=\"23\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_c.gif 71w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_c-30x10.gif 30w\" sizes=\"(max-width: 71px) 100vw, 71px\" \/><\/p>\n<p>(d) <img decoding=\"async\" class=\"alignnone size-full wp-image-9088\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_d.gif\" alt=\"ibp answer d\" width=\"124\" height=\"22\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_d.gif 124w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_d-30x5.gif 30w\" sizes=\"(max-width: 124px) 100vw, 124px\" \/><\/li>\n<li>which of the following series converges?<img decoding=\"async\" class=\"aligncenter size-full wp-image-9089\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/series_convergence_problem.gif\" alt=\"series convergence problem\" width=\"347\" height=\"51\" \/>\n<p>(a) ii only<\/p>\n<p>(b) iii only<\/p>\n<p>(c) ii and iii<\/p>\n<p>(d) i and iii<\/li>\n<li>what is the coefficient of <em>x<\/em><sup>6<\/sup> in the taylor series for <img decoding=\"async\" class=\"alignnone size-full wp-image-9090\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/exp_problem.gif\" alt=\"exp problem\" width=\"20\" height=\"16\" \/> about <em>x<\/em> = 0?(a) 1\/720\n<p>(b) 1\/24<\/p>\n<p>(c) 1\/3<\/p>\n<p>(d) 1\/6<\/li>\n<\/ol>\n<h3>part b<\/h3>\n<p><em>(time \u2014 15 minutes \u2014 5 questions)<\/em><\/p>\n<p><em>(a graphing calculator is <strong>required<\/strong> for some questions on this part of the examination)<\/em><\/p>\n<ol start=\"11\">\n<li><img decoding=\"async\" class=\"alignnone size-full wp-image-9091\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/partial_fractions_problem.gif\" alt=\"partial fractions problem\" width=\"186\" height=\"42\" \/>(a) 4ln|2<em>x<\/em> + 1| &#8211; 3ln|<em>x<\/em> &#8211; 1| + <em>c<\/em>\n<p>(b) 8ln|2<em>x<\/em> + 1| &#8211; 3ln|<em>x<\/em> &#8211; 1| + <em>c<\/em><\/p>\n<p>(c) 8ln|2<em>x<\/em> + 1| + 3ln|<em>x<\/em> &#8211; 1| + <em>c<\/em><\/p>\n<p>(d) 4ln|2<em>x<\/em> + 1| + 3ln|<em>x<\/em> &#8211; 1| + <em>c<\/em><\/li>\n<li>the monthly costs that a certain company incurs can be modeled by\u2003<img decoding=\"async\" class=\"alignnone size-full wp-image-9092\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/cost_function_problem.gif\" alt=\"cost function for problem\" width=\"133\" height=\"38\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/cost_function_problem.gif 133w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/cost_function_problem-30x9.gif 30w\" sizes=\"(max-width: 133px) 100vw, 133px\" \/>\u00a0, where <em>t<\/em> is measured in months. what is the total cost incurred by the company in months 12 \u2264 <em>t<\/em> \u2264 24, to the nearest dollar?(a) $32,105\n<p>(b) $34,820<\/p>\n<p>(c) $35,271<\/p>\n<p>(d) $36,529<\/li>\n<li>use euler&#8217;s method with <em>h<\/em> = 0.25 to estimate <em>y<\/em>(1) if <em>y<\/em>\u00a0&#8216; = 3<em>y<\/em> &#8211; 1 and <em>y<\/em>(0) = 1.(a) 5.873\n<p>(b) 6.013<\/p>\n<p>(c) 6.586<\/p>\n<p>(d) 7.312<\/li>\n<li>a particle moves along the <em>x<\/em>-axis so that its acceleration at any time <em>t<\/em> \u2265 0 is given by <em>a<\/em>(<em>t<\/em>) = 12sin 2<em>t<\/em> &#8211; 8cos 2<em>t<\/em> m\/s<sup>2<\/sup>. if its initial velocity is 12 m\/s and its initial position is <em>x<\/em> = 8, what is its position, in meters, at time <em>t<\/em> = 2 seconds?(a) 42.96\n<p>(b) 51.09<\/p>\n<p>(c) 102.17<\/p>\n<p>(d) 115.22<\/li>\n<li>let <em>r<\/em> be the region in the first quadrant between the graphs of <em>y<\/em> = <img decoding=\"async\" class=\"alignnone size-full wp-image-9093\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/exponential.gif\" alt=\"exponential, e to the -2x\" width=\"31\" height=\"16\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/exponential.gif 31w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/exponential-30x15.gif 30w\" sizes=\"(max-width: 31px) 100vw, 31px\" \/>, <em>y<\/em> = sin <em>x<\/em>, and the <em>y<\/em>-axis. the volume of the solid that results when <em>r<\/em> is revolved about the <em>x<\/em>-axis is(a) -0.565\n<p>(b) 0.565<\/p>\n<p>(c) 0.395<\/p>\n<p>(d) -0.395<\/li>\n<\/ol>\n<h2>solutions<\/h2>\n<p><em>answer key:<\/em><\/p>\n<table id=\"tablepress-93\" class=\"tablepress tablepress-id-93 tablepress-responsive\">\n<thead>\n<tr class=\"row-1 odd\">\n<th class=\"column-1\"><\/th>\n<th class=\"column-2\"><\/th>\n<th class=\"column-3\"><\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n<td class=\"column-1\">1. b<\/td>\n<td class=\"column-2\">6. c<\/td>\n<td class=\"column-3\">11. d<\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n<td class=\"column-1\">2. a<\/td>\n<td class=\"column-2\">7. b<\/td>\n<td class=\"column-3\">12. b<\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n<td class=\"column-1\">3. b<\/td>\n<td class=\"column-2\">8. a<\/td>\n<td class=\"column-3\">13. c<\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n<td class=\"column-1\">4. d<\/td>\n<td class=\"column-2\">9. c<\/td>\n<td class=\"column-3\">14. a<\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n<td class=\"column-1\">5. c<\/td>\n<td class=\"column-2\">10. d<\/td>\n<td class=\"column-3\">15. b<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!-- #tablepress-93 from cache --><\/p>\n<p>short explanations are given below for each ap calculus bc exam multiple choice problem.<\/p>\n<h3>problem 1<\/h3>\n<p>(b) \u2003 <em>t<\/em> = 3<\/p>\n<p>first, find the derivative of each component. then set them both equal to zero and solve.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9095\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem1_solution.gif\" alt=\"problem 1 solution\" width=\"317\" height=\"47\" \/><\/p>\n<p>the only <em>t<\/em>-value that makes both components equal to 0 is <em>t<\/em> = 3.<\/p>\n<h3>problem 2<\/h3>\n<p>(a) 2\/9<\/p>\n<p>re-express the sum as a geometric series. then, because the common ratio has absolute value less than 1, the series converges and you may use the sum of a geometric series formula.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9096\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem2_solution.gif\" alt=\"problem 2 solution\" width=\"235\" height=\"52\" \/><\/p>\n<h3>problem 3<\/h3>\n<p>(b) \u03c0\/3<\/p>\n<p>use the antiderivative formula involving arctan. and don&#8217;t forget about the infinite limit.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9099\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem3_solution.gif\" alt=\"solution to problem 3\" width=\"357\" height=\"192\" \/><\/p>\n<h3>problem 4<\/h3>\n<p>(d) \u2003 <em>y<\/em> = 4<em>x<\/em><sup>2<\/sup> + 6<\/p>\n<p>comparing the arc-length formula to the given integral, it must be the case that [<em>f<\/em>\u00a0&#8216;(<em>x<\/em>)]<sup>2<\/sup> = 64<em>x<\/em><sup>2<\/sup>. this implies that <em>f<\/em>\u00a0&#8216;(<em>x<\/em>) = \u00b18<em>x<\/em>. therefore, <em>f<\/em>(<em>x<\/em>) = \u00b14<em>x<\/em><sup>2<\/sup> + <em>c<\/em>.<\/p>\n<p>however none of the answer choices have a negative coefficient on the square term, so we know that <em>f<\/em>(<em>x<\/em>) = 4<em>x<\/em><sup>2<\/sup> + <em>c<\/em>. finally, using the given information that <em>f<\/em>(1) = 10, solve for <em>c<\/em> and put together the final answer:<\/p>\n<p><em>f<\/em>(<em>x<\/em>) = 4<em>x<\/em><sup>2<\/sup> + 6.<\/p>\n<h3>problem 5<\/h3>\n<p>(c) 12<\/p>\n<p>using the fundamental theorem of calculus (the part that deals with the derivative of an accumulation function),<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9100\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem5_solution.gif\" alt=\"solution to problem 5\" width=\"274\" height=\"72\" \/><\/p>\n<p>according to the graph, <em>f<\/em>(4) = 3, so the final answer is 3(4) = 12.<\/p>\n<h3>problem 6<\/h3>\n<p>(c) 9\u03c0\/8<\/p>\n<p>a quick sketch of the graph indicates that the region is between -\u03c0\/4 and \u03c0\/4.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9101\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/polar_area_graph.png\" alt=\"polar area graph\" width=\"300\" height=\"300\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/polar_area_graph.png 300w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/polar_area_graph-150x150.png 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>using the integral formula for area of a polar region,<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9103\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem6_solution.gif\" alt=\"problem 6 solution\" width=\"249\" height=\"272\" \/><\/p>\n<h3>problem 7<\/h3>\n<p>(b) -4 &lt; <em>x<\/em> &lt; 4<\/p>\n<p>use the ratio test.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9104\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem7_solution_part1.gif\" alt=\"part 1 of problem 7 solution\" width=\"369\" height=\"94\" \/><\/p>\n<p>so the series will converge if |<em>x<\/em>|\/4 &lt; 1, or |<em>x<\/em>| &lt; 4. since the series diverges at both <em>x<\/em> = 4 and <em>x<\/em> = -4, the interval of convergence is -4 &lt; <em>x<\/em> &lt; 4.<\/p>\n<h3>problem 8<\/h3>\n<p>(a) <img decoding=\"async\" class=\"alignnone size-full wp-image-9085\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_a.gif\" alt=\"ibp answer a\" width=\"124\" height=\"22\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_a.gif 124w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/ibp_answer_a-30x5.gif 30w\" sizes=\"(max-width: 124px) 100vw, 124px\" \/><\/p>\n<p>integration by parts.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9105\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem8_solution.gif\" alt=\"problem 8 solution\" width=\"286\" height=\"83\" \/><\/p>\n<h3>problem 9<\/h3>\n<p>(c) ii and iii<\/p>\n<p>series i is a <em>p<\/em>-series with <em>p<\/em> = 3\/4. since 3\/4 &lt; 1, this series diverges. series ii is also a <em>p<\/em>-series, but with <em>p<\/em> = 4\/3 &gt; 1. thus, series ii converges. series iii is alternating and since 1\/\u221an \u2192 0 as n \u2192 \u221e, series iii converges.<\/p>\n<h3>problem 10<\/h3>\n<p>(d) 1\/6<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9106\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem10_solution.gif\" alt=\"problem 10 solution\" width=\"267\" height=\"87\" \/><\/p>\n<p>the coefficient of <em>x<\/em><sup>6<\/sup> is 1\/(3!) = 1\/6.<\/p>\n<h3>problem 11<\/h3>\n<p>(d) 4ln|2<em>x<\/em> + 1| + 3ln|<em>x<\/em> &#8211; 1| + <em>c<\/em><\/p>\n<p>this integral requires the method of partial fractions. omitting the algebraic steps that determine the numerator values, we have<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9107\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem11_solution.gif\" alt=\"problem 11 solution\" width=\"451\" height=\"87\" \/><\/p>\n<h3>problem 12<\/h3>\n<p>(b) $34,820<\/p>\n<p>we can use a definite integral to find the total cost over a period of time. however, the integral would be too difficult to work out by hand, so use your calculator&#8217;s numerical integration function.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9108\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem12_solution.gif\" alt=\"problem 12 solution\" width=\"199\" height=\"47\" \/><\/p>\n<h3>problem 13<\/h3>\n<p>(c) 6.586<\/p>\n<p>the table below records the steps of euler&#8217;s method.<\/p>\n<table id=\"tablepress-94\" class=\"tablepress tablepress-id-94 tablepress-responsive\">\n<thead>\n<tr class=\"row-1 odd\">\n<th class=\"column-1\"><em>n<\/em><\/th>\n<th class=\"column-2\"><em>x<\/em><sub>n<\/sub><\/th>\n<th class=\"column-3\"><em>y<\/em><sub>n<\/sub><\/th>\n<th class=\"column-4\"><em>y<\/em><sub>n<\/sub>&#039;<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-hover\">\n<tr class=\"row-2 even\">\n<td class=\"column-1\">0<\/td>\n<td class=\"column-2\">0<\/td>\n<td class=\"column-3\">1<\/td>\n<td class=\"column-4\">3(1) &#8211; 1 = 2<\/td>\n<\/tr>\n<tr class=\"row-3 odd\">\n<td class=\"column-1\">1<\/td>\n<td class=\"column-2\">0.25<\/td>\n<td class=\"column-3\">1 + 0.25(2) = 1.5<\/td>\n<td class=\"column-4\">3(1.5) &#8211; 1 = 3.5<\/td>\n<\/tr>\n<tr class=\"row-4 even\">\n<td class=\"column-1\">2<\/td>\n<td class=\"column-2\">0.5<\/td>\n<td class=\"column-3\">1.5 + 0.25(3.5) = 2.375<\/td>\n<td class=\"column-4\">3(2.375) &#8211; 1 = 6.125<\/td>\n<\/tr>\n<tr class=\"row-5 odd\">\n<td class=\"column-1\">3<\/td>\n<td class=\"column-2\">0.75<\/td>\n<td class=\"column-3\">2.375 + 0.25(6.125) = 3.906<\/td>\n<td class=\"column-4\">3(3.906) &#8211; 1 = 10.718<\/td>\n<\/tr>\n<tr class=\"row-6 even\">\n<td class=\"column-1\">4<\/td>\n<td class=\"column-2\">1<\/td>\n<td class=\"column-3\">3.906 + 0.25(10.718) = 6.586<\/td>\n<td class=\"column-4\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><!-- #tablepress-94 from cache --><\/p>\n<h3>problem 14<\/h3>\n<p>(a) 42.96<\/p>\n<p>integrate acceleration twice to find the position function. after each integration, use the initial data to solve for the constant of integration.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9112\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem14_solution_part11.gif\" alt=\"part 1 of problem 14 solution\" width=\"452\" height=\"41\" \/><\/p>\n<p>since the given initial velocity is 12, we find <em>c<\/em> = 18.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9111\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem14_solution_part2.gif\" alt=\"part 2 of problem 14 solution\" width=\"545\" height=\"41\" \/><\/p>\n<p>now the initial position <em>x<\/em><sub>0<\/sub> = 8 implies that <em>c<\/em> = 6. thus, the position function is:<\/p>\n<p><em>x<\/em>(<em>t<\/em>) = -3 sin 2<em>t<\/em> + 2 cos 2<em>t<\/em> + 18<em>t<\/em> + 6.<\/p>\n<p>plugging in <em>t<\/em> = 2, using a calculator, we find <em>x<\/em>(2) = 42.96 seconds.<\/p>\n<h3>problem 15<\/h3>\n<p>(b) 0.565<\/p>\n<p>a graph of the two curves helps to locate the bounds of integration.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9113\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/region_between_curves.jpeg\" alt=\"region between curves in problem 15\" width=\"315\" height=\"318\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/region_between_curves.jpeg 315w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/region_between_curves-297x300.jpeg 297w, \/\/www.catharsisit.com\/hs\/files\/2017\/02\/region_between_curves-150x150.jpeg 150w\" sizes=\"(max-width: 315px) 100vw, 315px\" \/><\/p>\n<p>the left bound is 0, but the right bound is the <em>x<\/em>-coordinate of the point of intersection of the two curves. you your calculator to numerically approximate this point, which is roughly 0.4336.<\/p>\n<p>then set up the volume integral using the washer method. finally, find the volume using the numerical integration function on your calculator.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9114\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/02\/problem15_solution.gif\" alt=\"problem 15 solution\" width=\"351\" height=\"46\" \/><\/p>\n<h2>what&#8217;s next?<\/h2>\n<p>now that you&#8217;ve gotten some practice with multiple choice questions, check out these <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-bc-exam-free-response-practice-problems\/\">ap calculus bc exam free response practice problems<\/a>. practice makes perfect!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>there are 45 problems in the ap calculus bc exam multiple choice section. let&#8217;s take a look at the kinds of questions you might see on the test.<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 (yoast seo v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>ap calculus bc exam multiple choice 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succeed!","sameas":["http:\/\/valdosta.academia.edu\/shaunault","https:\/\/twitter.com\/shaunaultmath"],"url":"\/\/www.catharsisit.com\/hs\/author\/shaunault\/"}]}},"authors":[{"term_id":24932,"user_id":223,"is_guest":0,"slug":"shaunault","display_name":"shaun ault","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/28c36635b8fed6717752755f46cda239?s=96&d=mm&r=g","user_url":"http:\/\/valdosta.academia.edu\/shaunault","last_name":"ault","first_name":"shaun","description":"shaun earned his ph. d. in mathematics from the ohio state university in 2008 (go bucks!!). he received his ba in mathematics with a minor in computer science from oberlin college in 2002. in addition, shaun earned a b. mus. from the oberlin conservatory in the same year, with a major in music composition.  shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass.  shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!"}],"_links":{"self":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/9073"}],"collection":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/users\/223"}],"replies":[{"embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/comments?post=9073"}],"version-history":[{"count":0,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/posts\/9073\/revisions"}],"wp:attachment":[{"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/media?parent=9073"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/categories?post=9073"},{"taxonomy":"post_tag","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/tags?post=9073"},{"taxonomy":"author","embeddable":true,"href":"\/\/www.catharsisit.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=9073"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}