Unit 2 Flashcards
what does it mean when the function is differentiable at a point?
the derivative of the function exits at that point
what is a differentiable function?
a function that is differentiable at every point
what are derivatives used for?
used to measure the rates at which things change
what is the first derivative of a function?
the slope of the tangent to the curve
also the instantaneous rate of change
derivatives are in other word what of average changes?
limiting values of average changes
(slope of the secant line approaches the slope of the tangent line)
what four cases do not have a derivative?
corner, cusp, vertical tangent, discontinuity
what is a corner
one sided derivatives differ
what is a cusp?
slopes of secant lines approach infinity from one side and negative infinity from the other
what is a vertical tangent?
where the slopes of the secant line approach either infinity or negative infinity from both sides
what is a discontinuity?
one or both of the one-sided derivatives are nonexistent
what does differentiability imply?
local linearity and continuity
what does continuity not imply?
differentiability
d/dx [cosx]
-sinx
d/dx [sinx]
cosx
d/dx [tanx]
sec^2(x)
d/dx [cotx]
-csc^2(x)
d/dx [secx]
secxtanx
d/dx [cscx]
-cscxcotx
product rule
d/dx (uv)= uv’+vu’
quotient rule
d/dx (u/v)= vu’-uv’/v2
chain rule
d/dx (f(g(x))) = f’(g(x)) (g’(x))
dy/dx =
(dy/du) (du/dx)
average velocity on the interval [t1,t2]
Δs/Δt = [s(t1)-s(t2)]/ (t1-t2)
average acceleration on the interval [t1,t2]
Δv/Δt = [v(t1)-v(t2)]/ (t1-t2)
velocity
v(t) = s’(t) = ds/dt
what does it mean when the sign changes for velocity
particle reverses direction
when is v(t) +/-?
+ when s(t) is increasing
- when s(t) is decreasing
accerleration
a(t) = v’(t) = dv/dt= s”(t)= d2s/dt
speed
magnitude of velocity: |v(t)|
when is a particle speeding up?
v and a have the same sign
when is a particle slowing down?
v and a have different signs
jerk
a’(t)
at what velocity is a particle at max height?
0
when does particle hit ground?
h=0
at what velocity does a particle hit ground?
find zero and plug into velocity equation
when is a particle moving in positive direction?
v(t)>0
when is a particle at rest?
v(t)=0
given f(x) is differntiable, g(x) = f-1(x) , and f’(x) does not equal 0 the,
g’(f(x)) = 1/f’(x)
pi/2-arcsinx=
arccosx
pi/2-arctanx=
arccotx
pi/2-arcsecx=
arccscx
d/dx arcsinu
(1/√1-u^2)(du/dx)
d/dx arctanu
(1/1+u^2)(du/dx)
d/dx arcsecu
(1/|u|√u^2-1)(du/dx)
d/dx arccosu
(-1/√1-u^2)(du/dx)
d/dx arccotu
(-1/1+u^2)(du/dx)
d/dx arccscu
(-1/|u|√u^2-1)(du/dx)
d/dx (e^x)
e^x
d/dx (lnx)
1/x
d/dx (a^x)
(a^x)(lna)
d/dx (logax)
1/xlna
d/dx (e^u)
e^u(du/dx)
d/dx (lnu)
1/u(du/dx)
d/dx (a^u)
(a^u)(lna)(du/dx)
d/dx (logau)
(1/ulna)(du/dx)
Δx =
dx=(x+Δx)-(x); x2-x1
Δy=
f(x+Δx)-f(x); y2-y1
dy=
f’(x)(dx)
diff bw Δy and dy
Δy is the exact change
