Calculus BC Formula Quiz 1

Definition of e

e = lim n->∞ ( 1 + 1/n)^n

Absolute value |x| =

x if x >= 0

-x if x < 0

Definition of e

e = lim n->∞ ( 1 + 1/n)^n

Absolute value |x| =

x if x >= 0

-x if x < 0

f'(x)

f'(a)

lim x->a f(x) - f(a) / x-a

Definition of continuity: f is continuous at c if

1) f(c) is defined

2) lim x->c f(x) exists

3) lim x->c f(x) = f(c)

Average rate of change of f(x) on [a, b]

[f(b)-f(a)]/ (b-a)

Average value of f(x) on [a, b]

1/(b - a) ∫(a to b) f(x)dx

Rolle's Theorem

if f is continuous on [a, b] and differentiable on (a,b), and if f(a) = f(b), then there is at least one number c on (a,b) such that f'(c)=0.

Mean Value Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), then there exists a number c on (a,b) such that f'(c) = [f(b)-f(a)]/b-a

Intermediate Value Theorem

If f is continuous on [a, b] and k is any number between f(a) and f(b), then there is at least one number c between a and b such that f(c) = k.

sin(2x)

2 sinx cosx

cos(2x)

cos^2 x - sin^2x

1 - 2sin^2x

2cos^2x - 1

sin^2x

(1-cos2x)/2

cos^2x

(1+cos2x)/2

d/dx [uv]

u'v + uv'

d/dx [u/v]

(u'v - uv')/v^2

d/dx f(g(x))

f'(g(x)) * g'(x)

Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f(x) has both a maximum and a minimum on [a,b].