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Math 1A: introduction to functions and calculus. Sofia, 2013. Lecture 36: Worksheet. This worksheet was authored by Sofia 1.
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Math 1A: introduction to functions and calculus
Sofia, 2013
(^1) 1 Differentiate the following functions: Level 1โ a)^ f^ (x) =^ x^ +^
x x b) f (x) = eโ c) f (x) = x โ x d)^ f^ (x) =^ โ^2 x e)^ f^ (x) =^ x^ + sin(
x) 2 Integrate the following functions: Level 1^4 a)^ f^ (x) = 5x b)^ f^ (x) = 1^2 c)^ f^ (x) = 3x^2 d)^ f^ (x) =^ โ^3 x^2 e)^ f^ (x) = 3x^1 Written in the academic year 2003/2004, thanks to a grant from the Harvard Provost together with
Johnny
Carlsson,^ Andrew Chi
and^ Mark Lezama. Sofia was a chat bot which would use computer algebra systems to solve calculus problems while chatting, similar to Wolfram Alpha now.
The later is of course much more
sophisticated. This entire LaTeX file was generated automatically.
3 Differentiate the following functions: Level 2 a)^ f^ (x) = (x^ โ^
sin(x)) sec(x) (^3) b) f (x) = โx c) f (x) = 1 โ^3 x โโx d) f (x) = e x (^1) e) f (x) = x 4 Integrate the following functions: Level 22 sec a) f (x) = โ 112 2 tan^ ( )(^ )^2 2 x^ xโ^5 3 x^ x x^ b) f (x) = eโ^ x^2 xesec(e) (^1) โ 1 โ 2 xc) f (x) = x โxโxโ 2 x (^1) d) f (x) = โ+ cos( x^ x) + 1 e)^ f^ (x) =^ โ^25 Differentiate the following functions: Level 3โx a)^ f^ (x) =^ e^1 b)^ f^ (x) =^3 /^2 xlog(
โx) c)^ f^ (x) =^ x^ โ^ log(
x) (^4) d) f (x) = xโ 1 + cos(x)^3 x (^4) e) f (x) = x+^ โxx^ โ^ e
6 Integrate the following functions: Level 3x+1 a)^ f^ (x) =^ โ^4 x
3(x+1) log(x)log( +^4 x
x) (^3) x (^2) โ 3 x+ b) f (x) = xxx^3 ex+eexโxโ^2 xโ^3 (xโ3) c)^ f^ (x) = (2x^ โ
(^2 1) cos(x) โ x ) โ x + 3^ sin(x) 3 xd) f (x) = (e 2 3 /^2 x^ x)โ^ esin (e) 2 +sec^3 xe) f (x) = 21 (x)+1^3 (โ^ +x^2 xโ^3 x
+tan (x))^4 x 7 Differentiate the following functions: Level 4xx+e a)^ f^ (x) =^ e
sec(x)^ โ^ tan(x) โb) f (x) = x^ + 2(
โx + 4) + x c)^ f^ (x) = tan(x
โ) (โx^ + 2 tan( x)^ โ^ 3) d)^ f^ (x) = (x^ + 4)
x^ x + 2x โ e+ tan( x) โe) f (x) = โx (^3) + sin(x) + sin( x) csc(sin(x)) 8 Integrate the following functions: Level 4^2 a)^ f^ (x) = sec(
โ^ x) + 1 (^3) b) f (x) = sin(x (sin() 2 cos(x) +^ e
)(x) cos(x)^ +3^ 2 sin (
)sin(x) (^) x) + eโ 4 (^2) sin(x
(^2) c) f (x) = โ +^3 x (^ )^1 โโ^3 sin(x) + (^2 x^
โx^ โ^3 x) cos(x
( d) f (x) = 3 x^ + )sin(x) e^ cos(x) + sin(
(sin(x)^ x) ecos (x
e)^ f^ (x) = cos(x
x) + (1 โ e) tan(
xx) + (x โ e) sec 2 (x)^ โ^1