In Pre Calculus H we’ve worked on some tough problems (that probably have no relation to the real world) like the following.

6a - Absolute Value and Transformations_1

The smiley face represents the fact that we don’t know what sign goes there because we don’t know if we’re multiplying by a positive or negative number. A student came up with that strategy, and I really like it.

Bonus, in Calculus BC we’re working on the chain rule and trig derivatives. To show that it gets messy fast, try finding the 10th derivative of this function: y=(cot x)/(cos (x^2)).

Better yet. Don’t. Let wolframalpha do it.

( d^10 y)/( dx^10) = 90 cot(x) (354560 x^8 sec^9(x^2)+819840 x^4 sec^7(x^2)+4871168 x^8 tan^2(x^2) sec^7(x^2)+4963840 x^6 tan(x^2) sec^7(x^2)+4677120 x^8 tan^4(x^2) sec^5(x^2)+11149824 x^6 tan^3(x^2) sec^5(x^2)+6437760 x^4 tan^2(x^2) sec^5(x^2)+819840 x^2 tan(x^2) sec^5(x^2)+8400 sec^5(x^2)+418816 x^8 tan^6(x^2) sec^3(x^2)+1946112 x^6 tan^5(x^2) sec^3(x^2)+2405760 x^4 tan^4(x^2) sec^3(x^2)+779520 x^2 tan^3(x^2) sec^3(x^2)+30240 tan^2(x^2) sec^3(x^2)+256 x^8 tan^8(x^2) sec(x^2)+3584 x^6 tan^7(x^2) sec(x^2)+13440 x^4 tan^6(x^2) sec(x^2)+13440 x^2 tan^5(x^2) sec(x^2)+1680 tan^4(x^2) sec(x^2)) csc^2(x)-10 (512 sec(x^2) tan^9(x^2) x^9+2516992 sec^3(x^2) tan^7(x^2) x^9+51796992 sec^5(x^2) tan^5(x^2) x^9+105613312 sec^7(x^2) tan^3(x^2) x^9+25866752 sec^9(x^2) tan(x^2) x^9+12764160 sec^9(x^2) x^7+9216 sec(x^2) tan^8(x^2) x^7+15077376 sec^3(x^2) tan^6(x^2) x^7+168376320 sec^5(x^2) tan^4(x^2) x^7+175362048 sec^7(x^2) tan^2(x^2) x^7+48384 sec(x^2) tan^7(x^2) x^5+26272512 sec^3(x^2) tan^5(x^2) x^5+150522624 sec^5(x^2) tan^3(x^2) x^5+67011840 sec^7(x^2) tan(x^2) x^5+4919040 sec^7(x^2) x^3+80640 sec(x^2) tan^6(x^2) x^3+14434560 sec^3(x^2) tan^4(x^2) x^3+38626560 sec^5(x^2) tan^2(x^2) x^3+30240 sec(x^2) tan^5(x^2) x+1753920 sec^3(x^2) tan^3(x^2) x+1844640 sec^5(x^2) tan(x^2) x) csc^2(x)+(353792 cot(x) csc^10(x)+1841152 cot^3(x) csc^8(x)+1304832 cot^5(x) csc^6(x)+128512 cot^7(x) csc^4(x)+512 cot^9(x) csc^2(x)) sec(x^2)+20 x (-7936 csc^10(x)-137216 cot^2(x) csc^8(x)-185856 cot^4(x) csc^6(x)-31616 cot^6(x) csc^4(x)-256 cot^8(x) csc^2(x)) sec(x^2) tan(x^2)+45 (7936 cot(x) csc^8(x)+24576 cot^3(x) csc^6(x)+7680 cot^5(x) csc^4(x)+128 cot^7(x) csc^2(x)) (4 x^2 sec^3(x^2)+4 x^2 tan^2(x^2) sec(x^2)+2 tan(x^2) sec(x^2))+120 (-272 csc^8(x)-2880 cot^2(x) csc^6(x)-1824 cot^4(x) csc^4(x)-64 cot^6(x) csc^2(x)) (12 x sec^3(x^2)+40 x^3 tan(x^2) sec^3(x^2)+8 x^3 tan^3(x^2) sec(x^2)+12 x tan^2(x^2) sec(x^2))+210 (272 cot(x) csc^6(x)+416 cot^3(x) csc^4(x)+32 cot^5(x) csc^2(x)) (80 x^4 sec^5(x^2)+288 x^4 tan^2(x^2) sec^3(x^2)+240 x^2 tan(x^2) sec^3(x^2)+12 sec^3(x^2)+16 x^4 tan^4(x^2) sec(x^2)+48 x^2 tan^3(x^2) sec(x^2)+12 tan^2(x^2) sec(x^2))+252 (-16 csc^6(x)-88 cot^2(x) csc^4(x)-16 cot^4(x) csc^2(x)) (32 sec(x^2) tan^5(x^2) x^5+1856 sec^3(x^2) tan^3(x^2) x^5+1952 sec^5(x^2) tan(x^2) x^5+800 sec^5(x^2) x^3+160 sec(x^2) tan^4(x^2) x^3+2880 sec^3(x^2) tan^2(x^2) x^3+120 sec(x^2) tan^3(x^2) x+600 sec^3(x^2) tan(x^2) x)+210 (16 cot(x) csc^4(x)+8 cot^3(x) csc^2(x)) (3904 x^6 sec^7(x^2)+3600 x^2 sec^5(x^2)+30656 x^6 tan^2(x^2) sec^5(x^2)+29280 x^4 tan(x^2) sec^5(x^2)+11456 x^6 tan^4(x^2) sec^3(x^2)+27840 x^4 tan^3(x^2) sec^3(x^2)+12960 x^2 tan^2(x^2) sec^3(x^2)+600 tan(x^2) sec^3(x^2)+64 x^6 tan^6(x^2) sec(x^2)+480 x^4 tan^5(x^2) sec(x^2)+720 x^2 tan^4(x^2) sec(x^2)+120 tan^3(x^2) sec(x^2))+120 (-2 csc^4(x)-4 cot^2(x) csc^2(x)) (128 sec(x^2) tan^7(x^2) x^7+69504 sec^3(x^2) tan^5(x^2) x^7+398208 sec^5(x^2) tan^3(x^2) x^7+177280 sec^7(x^2) tan(x^2) x^7+81984 sec^7(x^2) x^5+1344 sec(x^2) tan^6(x^2) x^5+240576 sec^3(x^2) tan^4(x^2) x^5+643776 sec^5(x^2) tan^2(x^2) x^5+3360 sec(x^2) tan^5(x^2) x^3+194880 sec^3(x^2) tan^3(x^2) x^3+204960 sec^5(x^2) tan(x^2) x^3+8400 sec^5(x^2) x+1680 sec(x^2) tan^4(x^2) x+30240 sec^3(x^2) tan^2(x^2) x)+cot(x) (51733504 x^10 sec^11(x^2)+223372800 x^6 sec^9(x^2)+1099281408 x^10 tan^2(x^2) sec^9(x^2)+1164003840 x^8 tan(x^2) sec^9(x^2)+1996556288 x^10 tan^4(x^2) sec^7(x^2)+4752599040 x^8 tan^3(x^2) sec^7(x^2)+18446400 x^2 sec^7(x^2)+3068835840 x^6 tan^2(x^2) sec^7(x^2)+558432000 x^4 tan(x^2) sec^7(x^2)+553207808 x^10 tan^6(x^2) sec^5(x^2)+2330864640 x^8 tan^5(x^2) sec^5(x^2)+2946585600 x^6 tan^4(x^2) sec^5(x^2)+1254355200 x^4 tan^3(x^2) sec^5(x^2)+144849600 x^2 tan^2(x^2) sec^5(x^2)+1844640 tan(x^2) sec^5(x^2)+15111168 x^10 tan^8(x^2) sec^3(x^2)+113264640 x^8 tan^7(x^2) sec^3(x^2)+263854080 x^6 tan^6(x^2) sec^3(x^2)+218937600 x^4 tan^5(x^2) sec^3(x^2)+54129600 x^2 tan^4(x^2) sec^3(x^2)+1753920 tan^3(x^2) sec^3(x^2)+1024 x^10 tan^10(x^2) sec(x^2)+23040 x^8 tan^9(x^2) sec(x^2)+161280 x^6 tan^8(x^2) sec(x^2)+403200 x^4 tan^7(x^2) sec(x^2)+302400 x^2 tan^6(x^2) sec(x^2)+30240 tan^5(x^2) sec(x^2))



September 20, 2013

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