AP Calculus AB

Derivative

100

Find the derivative of f(x)= -x+tan(x)

f'(x)= -1+sec²(x)

200

Find the derivative of f(x)=sin(x)cos(x)

f'(x)=cos²(x)-sin²(x)

300

Find the derivative of f(x)= 2xsin(x) + x²cos(x)

f'(x) = 4xcos(x) + 2sin(x) - x²sin(x)

400

Find the derivative of f(x)= 5xsec(x) + xtan(x)

f'(x) = 5sec(x)+5xsec(x)tan(x) + tan(x) + xsec²(x)

500

Find the dy/dx: sin(x) + 2cos(2y) = 1

dy/dx = cos(x)/4sin(2y)

Integrals

100

Integrate: dx/√1 − 4x²

1/2 arcsin(2x) + C

200

Integrate: 12/(1+9x²) dx

4arctan(3x) + C

300

Integrate: t/(√1 − t⁴) dt

1/2arcsint² + C

400

Integrate 3/(2(√y)(1 + y)) dy

3arctan(tan(x)/5) + C

500

Integrate (x-3)/(x²+1) dx

1/2ln(x²+1) - 3arctan(x) + C

Trig Substitution

100

∫1/(√x²-25) dx

ln|x+(√x²-25)| + C

200

∫x/(√x²-25) dx

(√x²-25) + C

300

∫x⁶/(√1-x¹⁴) dx

1/7 sin^-1(x⁷) + C

400

∫ 1/x²(√x²+4) dx

-(√x²+4)/4x + C

500

∫ x²/(1+9x²)^5/2 dx

x³/3(1+9x²)^3/2 + C

Trig Substitution 2

100

∫x/(√x²+1) dx

(√x²+1) + C

200

∫arcsin(x) dx

xarcsin(x) + (√1-x²) + C

300

∫x/(√16-x²) dx

-(√16-x²) + C

400

∫ (√4-x²)/x dx

2ln|2/x - (√4-x²)/x| + (√4-x²) + C

500

∫ x³/(√1-x²) dx

1/5(1-x²)^5/2 - 1/3(1-x²)^3/2 + C

Random

100

y = (x⁴+2)²  What is dy/dx?

2(4x³+2)

200

Where is the graph of f(x) = xe^2x concave up?

x>-1

300

f(x) = ln(e^-4x + 5x + 4)  Find f'(0)

1/5

400

The derivative of function g is given by g'(x)= 4/x² - x.  When is g increasing?

(-infinity, 4^1/3)

500

Function f is defined by the equation f(x) = x³ - 2x. If g(x) = f^-1(x) and g(4)=2, what is g'(4)?

1/10