Pre Calc Final

Limits

100

g(x)=x³ 

lim g(x)=

x->-4

g(x)=(-4)³

= -64

200

Lim x⁴

^x->-6

Lim (-6)⁴

^{=  -1296}

300

f(x)= √(5 -x )

lim f(x)=

x->-3

Lim f(x)=√(5 - (-3) )

= 2.828

400

lim ( x+4 )/(x²-16)

x->-4

lim ( (-4)+4 )/((-4x)²-16)

=1/-8

500

find: lim (x²-6x+8)/(x²-4)

x-> 2

 lim ((2)²-6(2)+8)/((2)²-4)

= - (1/2)

Derivatives(Power Rule)

100

y=10x³

y´=30x²

200

y=2x¹²+6x⁷+x⁴

y´=24x¹¹+42x⁶+4x³

300

y=8x⁶+2x¹⁷

y´=48x⁵+34x¹⁶

400

y=1/2x^-2

y´=-1/x³

500

y=1/2x^3/2-22/7x^-5/2+x^3/7

y´=3/4x^1/2+55/7x^-7/2+3/7x^-4/7

Equation of Tangent Line

100

y=3x³-6 @ (2,18)

y-18=36(x-2)

200

y=x²+4x² @ (7,14)

y-14=70(x-7)

300

y=5x²+7 @ (2,5)

y-5=20(x-2)

400

f(x)=x²+4x³ @ x=1

y=14x-9

500

f(x)=2x³+x²+5x @ x=2

y-30=33(x-2)

Derivatives(Chain Rule)

100

f(x)=(6x²+7x)⁴

f´(x)=4(12x+7)(6x²+7x)³

200

g(t)=(4t²-3t+2)^-2

g´(t)=-2(8t-3)(4t²-3t+2)^-3

300

y=(x³+3)⁵

y´=15x²(x³+3)⁴

400

y=(-3x⁵+1)³

y´=-45x⁴(-3x⁵+1)²

500

y=(-x⁴-3)^-2

y´=8x³/(-x⁴-3)³

Derivatives(Product Rule)

100

f (x) = (2x^4− 3)(x^2+ 1)

f '(x) = (2x⁴− 3) ⋅ 2x + (x²+ 1) ⋅ 8x3

        = 12x⁵+ 8x³− 6x

200

y = (−2x⁴− 3)(−2x²+ 1)

= (−2x⁴−3) ⋅ −4x + (−2x²+ 1) ⋅ −8x³

= 24x⁵−8x³+12x

300

f (x) = (5x⁵+ 5)(−2x⁵− 3)

f '(x) = (5x⁵+ 5) ⋅ −10x⁴+ (−2x⁵− 3) ⋅ 25x⁴

= −100x⁹− 125x⁴

400

f (x) = (−3 + x-3)(−4x3+ 3)

f '(x) = (−3 + x^-3) ⋅ −12x²+ (−4x³+ 3) ⋅ −3x^-4

= 36x²−(9/x⁴)

500

y = (x⁴+ 3)(−4x⁵+ 5x⁴+ 5)

dy/dx= (x⁴+ 3)(−20x⁴+ 20x³) + (−4x⁵+ 5x⁴+ 5) ⋅4x³

= −36x⁸+ 40x⁷− 60x⁴+ 80x³