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Lecture 36: Worksheet, Lecture notes of Algebra

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
Lecture 36: Worksheet
This worksheet was authored by Sofia 1
1 Differentiate the following functions: Level 1
a) f(x) = x+โˆšx
b) f(x) = ex
c) f(x) = xโˆ’โˆšx
d) f(x) = โˆ’2x
e) f(x) = x+ sin(x)
2 Integrate the following functions: Level 1
a) f(x) = 5x4
b) f(x) = 1
c) f(x) = 3x2
d) f(x) = โˆ’2
x3
e) f(x) = 3x2
1Written 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. T he later is of course much more
sophisticated. This entire LaTeX file was generated automatically.
3 Differentiate the following functions: Level 2
a) f(x) = (xโˆ’sin(x)) sec(x)
b) f(x) = โˆ’x3
c) f(x) = 1 โˆ’3x
d) f(x) = eโˆ’xโˆšx
e) f(x) = 1
x
4 Integrate the following functions: Level 2
a) f(x) = โˆ’
2 sec2(1
x2)
x5โˆ’
2 tan (1
x2)
x3
b) f(x) = exโˆ’exsec2(ex)
c) f(x) = 1โˆ’
1
2โˆšx
xโˆ’
xโˆ’โˆšx
x2
d) f(x) = โˆ’1
x+ cos(x) + 1
e) f(x) = โˆ’2
5 Differentiate the following functions: Level 3
a) f(x) = eโˆ’x
b) f(x) = 1
x3/2log(โˆšx)
c) f(x) = xโˆ’log(x)
d) f(x) = x4โˆ’1
x3+ cos(x)
e) f(x) = x4+โˆšxโˆ’ex
pf2

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Math 1A: introduction to functions and calculus

Sofia, 2013

Lecture 36: Worksheet This worksheet was authored by Sofia

(^1) 1 Differentiate the following functions: Level 1โˆš a)^ f^ (x) =^ x^ +^

x x b) f (x) = eโˆš c) f (x) = x โˆ’ x d)^ f^ (x) =^ โˆ’^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) =^ โˆ’^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^ โˆ’^

sin(x)) sec(x) (^3) b) f (x) = โˆ’x c) f (x) = 1 โˆ’^3 x โˆšโˆ’x d) f (x) = e x (^1) e) f (x) = x 4 Integrate the following functions: Level 22 sec a) f (x) = โˆ’ 112 2 tan^ ( )(^ )^2 2 x^ xโˆ’^5 3 x^ x x^ b) f (x) = eโˆ’^ x^2 xesec(e) (^1) โˆš 1 โˆ’ 2 xc) f (x) = x โˆšxโˆ’xโˆ’ 2 x (^1) d) f (x) = โˆ’+ cos( x^ x) + 1 e)^ f^ (x) =^ โˆ’^25 Differentiate the following functions: Level 3โˆ’x a)^ f^ (x) =^ e^1 b)^ f^ (x) =^3 /^2 xlog(

โˆšx) c)^ f^ (x) =^ x^ โˆ’^ log(

x) (^4) d) f (x) = xโˆ’ 1 + cos(x)^3 x (^4) e) f (x) = x+^ โˆšxx^ โˆ’^ e

6 Integrate the following functions: Level 3x+1 a)^ f^ (x) =^ โˆ’^4 x

3(x+1) log(x)log( +^4 x

x) (^3) x (^2) โˆ’ 3 x+ b) f (x) = xxx^3 ex+eexโˆ’xโˆ’^2 xโˆ’^3 (xโˆ’3) c)^ f^ (x) = (2x^ โˆ’

(^2 1) cos(x) โˆ’ x ) โˆ’ x + 3^ sin(x) 3 xd) f (x) = (e 2 3 /^2 x^ x)โˆ’^ esin (e) 2 +sec^3 xe) f (x) = 21 (x)+1^3 (โˆ’^ +x^2 xโˆ’^3 x

+tan (x))^4 x 7 Differentiate the following functions: Level 4xx+e a)^ f^ (x) =^ e

sec(x)^ โˆ’^ tan(x) โˆšb) f (x) = x^ + 2(

โˆšx + 4) + x c)^ f^ (x) = tan(x

โˆš) (โˆ’x^ + 2 tan( x)^ โˆ’^ 3) d)^ f^ (x) = (x^ + 4)

x^ x + 2x โˆ’ e+ tan( x) โˆše) f (x) = โˆ’x (^3) + sin(x) + sin( x) csc(sin(x)) 8 Integrate the following functions: Level 4^2 a)^ f^ (x) = sec(

โˆ’^ x) + 1 (^3) b) f (x) = sin(x (sin() 2 cos(x) +^ e

)(x) cos(x)^ +3^ 2 sin (

)sin(x) (^) x) + eโˆ’ 4 (^2) sin(x

(^2) c) f (x) = โˆ’ +^3 x (^ )^1 โˆšโˆ’^3 sin(x) + (^2 x^

โˆšx^ โˆ’^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 โˆ’ e) tan(

xx) + (x โˆ’ e) sec 2 (x)^ โˆ’^1