Calc Jeopardy

Limits

100

limx→2 (x³+4x²-3)

21

200

lim x→1 (x⁴+x²-1)/(x²+5)

1/6

300

lim x→0 (tan x)/(x)

1

400

lim x→infinity (5x⁷-3x)/(16x⁶-3x)

0

500

lim x→infinity (xe^-2x)

0

Derivatives

100

y=x²⁰

dy/dx=20x¹⁹

200

y=7x^1/2

dy/dx=(7/2)x^-1/2

300

y=3x⁴+8x¹⁰

dy/dx=12x³+80x⁹

400

y=50x⁵+(3/x)-7x^-5/3

dy/dx=250x⁴-(3/x²)+((35/3)x^-8/3)

500

find d²y/dx² of y=√5x³+x

dy/dx=1/2(5x³+x)^-1/2(15x²+1)

d²y/dx²=1/2(5x³+x)^-1/2(30x)+(15x²+1)[-1/4(5x³+x)^-3/2(15x²+1)]

Integrals

100

∫(x³+x²-x)dx

(x⁴/4)+(x³/3)-(x²/2)+C

200

∫(dx/(9+x²))

1/3 tan^-1(x/3)+C

300

∫sec²3xdx

(tan3x)/3 +C

400

∫10x(5x²-3)⁶dx

((5x²-3)⁷)/7 +C

500

∫sinxdx [0, π/4] + ∫cosxdx [-π/4, 0]

1

Rates

100

A circular pool of water is expanding at the rate of 16π in²/sec. At what rate is the radius expanding when the radius is 4in?

A=πr²

dA/dt=2πr dr/dt

16π=2π(4)dr/dt

dr/dt=2in/sec

200

A 25 foot long ladder is leaning against a wall and sliding toward the floor. If the foot of the ladder is sliding away from the base of the wall at a rate of 15ft/sec, how fast is the top of the ladder sliding down the wall when the top of the ladder is 7 feet from the ground? 

x²+y²=25²

2x dx/dt+2y dy/dt=0

dx/dt=15, y=7, x=24

2(24)(15)+2(7)(dy/dt)=0

dy/dt=-360/7 ft/sec

300

A spherical balloon is expanding at a rate of 60π in³/sec. How fast is the surface area of the balloon expanding when the radius of the balloon is 4in?

V=4/3πr³

A=4πr²

dV/dt=4πr²dr/dt

60π=4π(4)²dr/dt

dr/dt=15/16in/sec

dA/dt=8πrdr/dt

dA/dt=8π(4)(15/16)=30π in²/sec

400

An underground conical tank, standing on its vertex, is being filled with water at the rate of 18π ft³/min. If the tank has a height of 30 feet and a radius of 15 feet, how fast is the water level rising when the water is 12 feet deep?

r=h/2

V=1/3π(h/2)

h=πh³/12

dV/dt=(π/12)3h² dh/dt

18π=(π/12)3(12)²dh/dt

dh/dt=1/2 ft/min

500

A circle is increasing in area at the rate of 16π in²/sec. How fast is the radius increasing when the radius is 2 in?

dA/dt=2πrdr/dt

16π=2π(2)dr/dt

dr/dt=4 in/sec

Miscellaneous

100

sin²θ+cos²θ=

1+tan²θ=

1+cot²θ=

1

sec²θ

csc²θ

200

sin(π/2 -θ)=

cos(π/2 -θ)=

tan(π/2 -θ)=

cosθ

sinθ

cotθ

300

steps to solving differential equations

1. separate variables

2. integrate

3. +C

3. solve for c, x, or y

4. use to approximate/ find tangent line

400

y'=

y"=

y'=0

y"=0

slope

concavity

a possible max/min

a possible point of inflection

500

Determine the area of the region bounded by        y=(1/(x+2)), y=(x+2)², x=−(3/2), x=1

(67/8)−ln(1/2)−ln(3)

=7.9695