![\"top](\"\/\/www.catharsisit.com\/hs\/files\/2016\/05\/pitfall-o.gif\")
top 10 ap calculus bc exam mistakes<\/h2>\n
in my experience, students who sign up for the bc exam are generally tops in their class. they are probably going to pursue a stem major at a prestigious college or university.<\/p>\n
with that being said, though, certain errors can creep up on the best of us.<\/p>\n
so here are some of the most common calculus bc exam mistakes.<\/p>\n
1. not showing enough work<\/h3>\n
you should be aware that the ap calculus bc exam has two sections, multiple choice (mc) and free response (fr). <\/p>\n
you must show your work<\/strong> on every fr question. otherwise, the graders will not give you full credit even if your answer is correct.<\/p>\n
if you use a major calculator function (like numerical integration<\/em>), write down the integral itself, the result, and a sentence explaining that you used your calculator to compute it.<\/p>\n
in addition, you might want to check out these helpful tips to conquering ap calculus free response questions<\/a><\/p>\n
2. rounding partial answers<\/h3>\nwhen using a calculator, be sure to keep all of the digits in your work until the very last step. and even then, don’t round too much.<\/p>\n
as a silly example, suppose the question asks for the value of 22\/7 – π.<\/p>\n
both 22\/7 and π are roughly equal to 3.14, so is the answer 0?<\/p>\n
let’s take a closer look: 22\/7 ≈ 3.1428571, while π ≈ 3.1415926. <\/p>\n
therefore 22\/7 – π is about 0.0012645. that’s a tiny number, but not equal to 0.<\/p>\n
3. sequences are not series<\/h3>\n
a sequence<\/strong> is just a list of numbers.<\/p>\n
a series<\/strong> is the sum<\/em> of a list of numbers.<\/p>\n
they are not the same concepts.<\/p>\n
however, some of the same terminology is used with sequences as with series, and this is where confusion may arise.<\/p>\n
we say a sequence (an<\/sub><\/em>) converges<\/strong> if the limit exists as n<\/em> → ∞.<\/p>\n
but a series converges<\/strong> only if its partial sums<\/em> approach a definite value.<\/p>\n
for example, the harmonic sequence, 1, 1\/2, 1\/3, 1\/4, 1\/5, …, converges to 0. but the harmonic series, 1 + 1\/2 + 1\/3 + 1\/4 + 1\/5 + …., diverges.<\/p>\n
4. powers versus derivatives<\/h3>\nkeep in mind that the n<\/em>th derivative of a function is often written with a superscript (n<\/em>) if n<\/em> > 3. some students mistake these higher-order derivatives for powers of the function.<\/p>\n
![\"higher-order](\"\/\/www.catharsisit.com\/hs\/files\/2017\/06\/higher-order_derivatives_powers.gif\")
<\/p>\n
5. derivatives, velocity, and speed<\/h3>\nyou should know this in your sleep: the derivative of position is equal to velocity<\/strong><\/em>.<\/p>\n
sometimes, however, students forget how to find the speed<\/em> of an object. <\/p>\n
\n- for a one-variable function, speed is equal to the absolute value<\/em> of velocity.<\/li>\n
- for a vector function, speed is equal to the magnitude<\/em> of the velocity vector.<\/li>\n<\/ul>\n
6. integration by parts issues<\/h3>\n
it’s essential to have memorized the integration by parts (ibp)<\/strong> formula:<\/p>\n
![\"integration](\"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/integration_by_parts.gif\")
<\/p>\n
not only that, but you must have experience using it. don’t forget that the minus sign in the middle applies to the entire<\/em> integral, ∫ v du<\/em>.<\/p>\n
this is especially important if you have to do multiple ibp. make sure to use parentheses to group all of the terms of ∫ v du<\/em> together once you have worked out that integral.<\/p>\n
7. messing up the logic in comparison test<\/h3>\nproving that a series converges or diverges often relies on subtle mathematical reasoning. the most common errors that i have come across involve the direction of the inequalities in comparison test.<\/p>\n
suppose we are trying to prove whether a series σ an<\/sub><\/em> converges or diverges by comparing its terms to another (simpler) series, σ bn<\/sub><\/em>.<\/p>\n
here’s a handy chart.<\/p>\n\n\n\n\n\n\n
<\/p>\nyou should know this in your sleep: the derivative of position is equal to velocity<\/strong><\/em>.<\/p>\n
sometimes, however, students forget how to find the speed<\/em> of an object. <\/p>\n
- \n
- for a one-variable function, speed is equal to the absolute value<\/em> of velocity.<\/li>\n
- for a vector function, speed is equal to the magnitude<\/em> of the velocity vector.<\/li>\n<\/ul>\n
6. integration by parts issues<\/h3>\n
it’s essential to have memorized the integration by parts (ibp)<\/strong> formula:<\/p>\n
<\/p>\nnot only that, but you must have experience using it. don’t forget that the minus sign in the middle applies to the entire<\/em> integral, ∫ v du<\/em>.<\/p>\n
this is especially important if you have to do multiple ibp. make sure to use parentheses to group all of the terms of ∫ v du<\/em> together once you have worked out that integral.<\/p>\n
proving that a series converges or diverges often relies on subtle mathematical reasoning. the most common errors that i have come across involve the direction of the inequalities in comparison test.<\/p>\n
suppose we are trying to prove whether a series σ an<\/sub><\/em> converges or diverges by comparing its terms to another (simpler) series, σ bn<\/sub><\/em>.<\/p>\n
here’s a handy chart.<\/p>\n\n\n\n\n\n\n
![\"average](\"\/\/www.catharsisit.com\/hs\/files\/2017\/06\/avg_velocity.gif\")
<\/p>\n![\"average](\"\/\/www.catharsisit.com\/hs\/files\/2017\/06\/average_value.gif\")
<\/p>\n