Math test 2

What does EVT guarantee?

If f is continuous on a closed interval [a,b], it attains an absolute maximum and minimum.

Conditions for Rolle's Theorem

Continuous on [a,b], differentiable on (a,b), and f(a)=f(b).

Conclusion of Rolle's Theorem

There is at least one c in (a,b) where f'(c)=0.

Conditions for MVT

Continuous on [a,b] and differentiable on (a,b).

MVT formula

f'(c)=(f(b)-f(a))/(b-a), for some c in (a,b).

How to solve an MVT problem

Check conditions, calculate average slope, set f'(c) equal to it, and solve for c.

Fermat's Theorem

If f has an interior local extremum and is differentiable there, f'(c)=0.

Critical number definition

A value c in the domain where f'(c)=0 or f'(c) does not exist.

Absolute extrema on [a,b]

Evaluate f at interior critical numbers and both endpoints; compare values.

Inflection point definition

A point on a continuous curve where concavity changes.

Derivative of e^u

e^u times u'.

Derivative of a^x

a^x ln(a) times x'

Derivative of ln(u)

u'/u

Derivative of log base a of u

u'/(u ln(a))

Derivative of arcsin(u)

u'/sqrt(1-u^2)

Derivative of arccos(u)

-u'/sqrt(1-u^2)

Derivative of arctan(u)

u'/(1+u^2)

Derivative of arccot(u)

-u'/(1+u^2)

Derivative of arcsec(u)

u'/(|u|sqrt(u^2-1))

Derivative of arccsc(u)

-u'/(|u|sqrt(u^2-1))

Sphere volume

V=(4/3)pi*r^3.

Cone volume

V=(1/3)pir^2h.

Circle area

A=pi*r^2.

When to substitute values in related rates

After differentiating with respect to time.

Similar triangles in a cone

r/h=R/H.

Linearization formula

L(x)=f(a)+f'(a)(x-a).

Differential formula

dy=f'(a)dx.

Concave down linearization error

Tangent-line estimate is an overestimate.

Parametric slope

dy/dx=(dy/dt)/(dx/dt), if dx/dt is nonzero.

Parametric curve direction

Direction follows increasing t.

Pythagorean trig identity

sin^2(t)+cos^2(t)=1.

Derivative of sin(u)

cos(u)u'.

Derivative of cos(u)

-sin(u)u'.

Derivative of tan(u)

sec^2(u)u'.

Derivative of sec(u)

sec(u)tan(u)u'.

Derivative of csc(u)

-csc(u)cot(u)u'.

Derivative of cot(u)

-csc^2(u)u'.