Calculus Finals Game

Derivatives

100

y=5x³-6x^-3

d³y/dx³=30+360/x⁶

200

y=(3x²)(x³+4)

dy/dx=15x⁴+24x

300

y=5x⁵-3x²+9/x

dy/dx=24x³-3-9x^-2

400

y=4x^5/2-7x^3/2

dy/dx=10x^3/2-21/2x^1/2

500

y=3x⁷-5x⁵+4x^-2-7x^-4

d⁵y/dx⁵= 7560x²-600-2880/x⁷+20160/x⁹

Limits

100

            lim     (2x+5)

x->3

11

200

                  lim     x²-4/x-2

x->2

4

300

                    lim    (1+x)²−1−2x/x²

x→0  

1

400

                      lim     5x²-3x+1/2x²+7

x->∞

5/2

500

              lim |x-2|/x-2

x->2^-

-1

Chain Rules

100

y=(3x+2)⁵

dy/dx=15(3x+2)⁴

200

y=(2x³+x)⁴

dy/dx=4(2x³+x)³(6x²+1)

300

y=(x²-4x)⁷

dy/dx=7(x²-4x)⁶(2x-4)

400

y=(x²+1)^-3

dy/dx=-6(x²+1)^-4

500

y=(4x³-5x²+2)⁶

dy/dx=6(4x³-5x²+2)⁵(12x²-10x)

Related Rates

100

A circle's radius is increasing at a rate of 3 cm/s. How fast is the area of the circle increasing when the radius is 10cm?

dA/dt= 60π cm²/s

200

Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4cm/min. How fast is the area of the pool increasing when the radius is 5cm?

da/dt=40π

300

Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The area of the spill increases at a rate of 9π m²/min. How fast is the radius of the spill increasing when the radius is 10m?

9m/20min = dr/dt

400

Air is pumping into a spherical ballon at 8cm³/s. How fast is the radius increasing when the radius is 5cm?

dr/dt = 2/25π cm/s

500

Water is flowing into a cone at 12 cm³/s. The cone has a height of 10cm and a radius of 5cm. How fast is the water level rising when the water is 4cm deep?

dh/dt = 3/π cm/s

Mean Value Theorem

100

f(x)=x²              [1,3]

x=2

200

f(x)=x³-x²-2x      [-1,1]

x=-1/3 , 1

300

f(x)=x²/(2)-2x-1        [-1,1]

x=0

400

f(x)=x³-3x       [-2,2]

x=-1,1

500

f(x)=x³+x           [-1,2]

x=-√(2/3), √(2/3)