Final exam review

Limits

100

If lim_x->a f(x) exists, then lim_x->a-f(x) 

and lim_x->a+f(x) must this

What is exist

200

This is the value of lim_x->3(x²+x-12)/(x²+3x+2)

What is 0

300

lim_x->5[sqrt(x-1)-2]/[x-5]

What is 1/4

400

These are all the horizontal asymptotes of the function 

f(x)=(x²-x-12)/(x²+5x+6)

What is x=-2

500

These are all the vertical asymptotes of the function 

f(x)=(x²-x-12)/(x²+5x+6)

What is y=1

Derivatives

100

This is the derivative of 

f(x)=sin(x)+cos(x)+tan(x)+4x³

What is f'(x)=cos(x)-sin(x)+sec²(x)+12x²

200

This is the derivative of 

f(x)=(x²+2)(x³+4)

What is f'(x)=5x⁴+6x²+8x

300

This is the derivative of

f(x)=e^xsqrt(x)

what is e^x(2x+1)/2sqrt(x)

400

This is the derivative of f(x)=(x+1)/(x²e^x)

What is f'(x)=(-x-2x-2)/(x³e^x)

500

This is the derivative of 

f(x)=ln(sqrt(x⁴-2x²+3)

What is f'(x)=2x(x²-1)/(x⁴-2x²+3)

Integrals

100

d/dx∫ ^t₄ e^x dx 

f'(x)=2xsqrt(x⁸+5)

200

This is the value of ∫ ¹₀(x+3)(x-5)dx

What is -47/3

300

This is the value of 

∫²₁(x+1)/sqrt(x) dx

what is [10sqrt(2)-8]/3

400

This is the value of ∫3x^2/3-1/sqrt(x) dx

what is 9/5*x^5/3-2x^1/2+C

500

This is the value of ∫²₀|3x-1|dx

13/3

u-sub

100

This is the value of ∫sin(x)/cos²(x) dx

sec(x)+C

200

∫^{e^4}_e dx/xsqrt(ln(x)) dx

What is 2

300

This is the value of

 ∫ ³₁ 1/x²*sqrt(1-1/x) dx

What is -(2/3)^5/2

400

∫sin(lnx)/x + sec² (x) tan³ (x) dx

-cos(lnx)+1/4(tan^4(x))+C

500

∫1/[cos²(x)sqrt(1+tan(x)]

2sqrt(1+tan(x))+C

random

100

Use implicit differentiation to find the derivative of x⁴+ 3x²y + y ⁴ = 1.

y'=(6xy-4x³)/(4y³-3x²)

200

Use the limit definition of the derivative to find f '(x) where f (x) = 1 -4x² .

f'(x=-8x

300

f (x) = sin(e^x/(1 + e^x))

cos(e^x/(1+e^x))(e^x/(1+e^x)^2)

400

Use the second derivative test to find any local max or local min values of each of the given functions:  f(x) = 6x ² − x ³ 

local max is f(4)=32

local min is f(0)=0

500

Use symmetry to evaluate
∫²_-2(x ² + x ³ ) dx

16/3