AP Calc AB

lim x→0 sin(x)/x=

1

lim x→0 cos(x)-1/x=

0

lim x→0 sin(ax)/x

a

limx→0 sin(ax)/sin(bx)

a/b

IVT conditions

f(x) continuous on [a,b]

f(c ) is between f(a) and f(b)

IVT conclusion

since f(x) is continuous on [a,b] and f(c ) is between f(a) and f(b), by the IVT there is a x in (a,b) such that f(x)=c

definition of f(x) at x=a when x→0

lim x→0 f(a+h)-f(a)/h

definition of f(x) at x=a when x→a

lim x→a f(x)-f(a)/x-a

dy/dx tan(x)=

sec²(x)

dy/dx cot(x)=

-csc²(x)

dy/dx sec(x)=

sec(x)tan(x)

dy/dx csc(x)=

-csc(x)cot(x)

dy/dx a^x=

a^x ln(a)

dy/dx log_a(x)=

1/xln(a)

dy/dx sin^-1(x)=

1/√(1-x²)

dy/dx cos^-1(x)=

-1/√(1-x²)

dy/dx tan^-1(x)=

1/(1+x²)

dy/dx sin^-1(x)=

inverse function derivative equation

1/f’(f^-1(x))

linearization

L(x)=f’(a)(x-a)+f(a)

MVT conditions

continuous on [a,b]

differentiable on interval (a,b)

MVT conclusion

since f(x) is continuous on [a,b] and differentiable on (a,b), by MVT there exists a c in (a,b) where f’(c )= f(b)-f(a)/b-a

rolle’s theorem conditions

continuous on [a,b]

differentiable on interval (a,b)

f(a)=f(b)

rolle’s theorem conclusion

since f(x) is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), there is at least one value on (a,b), call it c, that f’(c )=0

EVT conditions

f(x) is continuous on [a,b]

EVT conclusion

since f(x) is continuous on [a,b], by the EVT, there exists at least one local maximum and local minimum on (a,b)

midpoint riemann sum

(x₂-x₁)(f(midpoint of x values)

trapezoid riemann sum

(width)(f(a)+f(b)/2)

∫1/x dx=

ln|x| + C

∫e^x dx=

e^x + C

∫sin(x) dx=

-cos(x)+C

∫cos(x) dx=

sin(x)+C

∫sec²(x) dx=

tan(x)+C

∫csc²(x) dx=

-cot(x)+C

∫sec(x)tan(x)=

sec(x)+C

∫csc(x)cot(x)=

-csc(x)+C

exponential equation

P(t)=ce^kt

average value

1/b-a ∫_a^b f(x)² dx

disc method

π ∫_a^b f(x)² dx

washer method

π ∫_a^b f(x)²-g(x)² dx

square cross section

∫_a^b f(x)² dx

rectangle cross section

∫_a^b f(x) (height) dx

semicircle cross section

π/8 ∫_a^b f(x)² dx

iso. triangle with leg on base cross section

½ ∫_a^b f(x)² dx

iso. triangle with hyp. on base cross section

¼ ∫_a^b f(x)² dx

equilateral triangle cross section

√3/4 ∫_a^b f(x)² dx