calc chapter 3

what does f slope being + / - mean for f prime

f prime values are + / -

what does f being concave up or down mean for f prime

f prime slope is + / -

what does rel extrema of f mean for f prime

f prime has a zero

what Is the exception ^ ?

if f has a cusp, in which case f prime is undefined

what does rel extrema for f prime mean for f

f has point of inflection

why do zeroes for f prime not always mean f has rel extrema

it can bounce off, meaning values still stay positive for f prime and slope still stays positive for f (plateu point because the slope positive, decreases to 0, then keeps increasing and is never negative)

what are critical points

x values where f is defined and f prime is 0 or f is defined and f prime is undf

what are stationary points

when f prime equals 0

what are singular points

when f prime is undefined (cusp)

what are plateau points

point where it flattens and then keeps going in the same direction, like slope is positive and decreases to 0, then keeps increasing again and slope is never negative

what is the dumb ah square not a rect but rect a square statement from this chapter

all local extrema of f are critical points but not all critical points are local extrema

definition of a derivative for a specific value

lim x —> c of f(x) - f(c ) over x - c

definition of a derivative for all values (difference quotient)

lim h —> 0 f(x+h) - f(x) over h

what do you need to remember on antiderivatives :)

the + C

steps to showing how you got a particular antideriv

show general antideriv first, then sub in point

how does concavity of f relate to f double prime

when f is concave up/down, values of f double prime are + / -

distance/displacement scalar/vector

distance is a scalar and displacement is a vector

speed/velocity scalar/vector

speed is a scalar and velocity is a vector

!!!! REMEMBER how to tell if something is speeding up or slowing down

if a and v have the same sign, speed is going up, if different signs, speed is going down

deriv of e^x

e^x

derivative of ln(x)

1 / x

deriv of b^x

ln(b) * b^x

product rule

ddx f(x) times g(x) = g times f prime + f times g prime

dealing w constants in derivatives/chain rule

constant of outermost function in chain rule remains unaffected, just keep it there, but constant in inside function means x becomes a function and you have to multiply by the deriv (the constant)

deriv of f(x) times g(x) times h(x)

f prime(gh) + g prime(fh) + h prime(gh)

squeeze theorem (memorize exactly)

If g(x) ≤ f(x) ≤ h(x) for all x in the neighborhood of c, x ≠ c, and lim x —> c of g(x) = lim x —> c h(x) = L, then lim x —> c f(x) = L

quotient rule

ddx f/g is (g times f prime) - (f times g prime) / g²

does a derivative exist at a cusp

no

what is normal line

line perpendicular to tangent that passes through point of tangency

2 forms of chain rule

• ddx f(g(x)) = f’(g(x)) times g’(x)

• if y = f(t) and t = g(x), then ddx y equals dy/dt * dt/dx

what is the limit of (sinx)/x as x approaches 0

1

what is the limit of (1 - cosx)/x as x approaches 0

0