Calculus Project

Find the derivative of a function using different rules!

100

Limit Definition:

f(x)=2x-3x²

^{lim h -> 0}

2-3x

200

Power Rule:

f(x)=5x⁴+10x²

60x⁵

300

Product Rule:

f(x)=5x²(2y⁴)

20xy⁴

400

Chain Rule

f(x)=(7x²+5x)⁵

5(7x²+5x)⁴(14x+5)

500

Quotient Rule

f(x)=(3x²+6x+10)/4x²

-24x²-80x

Finding the derivative of a function that includes...

100

lnx

f(x)=lnx

1/x

200

e^x

f(x)=x³(e^-2x)

3x²e^-2x-(2x³e^-2x)

300

sinx

f(x)=sinx

cosx

400

cosx

f(x)=3/4cosx

-3/4sinx

500

Combination of Product and Chain Rules

f(x)= (12x²+7x)²(4x+6)

2(12x²+7)(24x+7)(4x+6)+4(12x²+7x)²

slopes and tangent lines

100

second derivative of a function:

f(x)=3x⁴-4x³

36x²-24x

200

slope at a point

find the slope of f(x)=sinx where x=π/4

2^1/2/2

300

equation of tangent line

f(x)=3x³-5x²+2x+1, x=2

y-9=18(x-2)

400

equation of normal line

f(x)=4-x² at x=1

y-3=1/2(x-1)

500

implicit differentiation

3x²-y=5x

6x-5

related rates, derivative tests, velocity, etc.

100

related rates

A pebble is dropped in a pool of water causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 1 ft per sec. when the radius is 4ft at what rate is the total area A of the distributed water changing.

8pi ft per sec

200

marginal profit

A company manufactures widgets and the total cost c(x) in dollars associated with producing x widgets is given by c(x)=500+10x+0.1x². The revenue from selling x widgets is given by r(x)=15x-0.05x². Find marginal profit when 100 widgets are procured and sold.

-25

300

velocity

At time t=0 a diver jumps from a diving board that is 32ft high. the divers initial velocity is 16ft per sec so the position of the diver is given by S=16t²+16t+32.  when does the diver hit the water?

2

400

first derivative test

f(x)=x²-4x+5

dec(-infinity, 2), inc (2, infinity), min(2, 1)

500

second derivative test

f(x)=2x³-12x²

concave down (-infinity, 2), concave up (2, infinity), poi (2, 32)

optimization, rieman's sum, and integrals

100

indefinite integrals

s(t)=/(3t²-6t+2)dt

s(t)=t³-3t²+2t+C

200

definite integral

/(1 to -1) (x+1)/(x²+2x+2) dx

6/25

300

integration by substitution

/x⁴(x⁵-9)³dx

1/20 (x⁵-9)⁴+C

400

optimization

make an open box that has a square base and surface area of 108 in². find the dimensions that will produce a box with max volume.

height: 3 inches

width: 6 inches

500

Riemann's sum

find the area under the curve from [-2, 2] for the equation f(x)=-|x|+3 from the left side. use 4 rectangles.

8