Sem 1 Stewart Flashcards
function or not?
vertical line test
piecewise function
defined differently in different parts of the domain
even function or odd function
even f(-x)=f(x)
odd f(-x)=-f(x)
increasing functions
if f(x1)<f(x2) when x1<x2
vice versa for decreasing
polynomial
contains non-negative integers only
power function
f(x)=x^a
sec
1/cos
cosec
1/sin
cot
cos/sin = 1/tan
exponential functions
f(x)=b^x
log function
g(x)=logbx
lnx
logex
exponential properties
domain real numbers
range (0,infinity)
increasing if b>1, decreasing 0<b<1
secant
cuts a curve more than once
left hand limit
limit as approaches from the left
right hand limit
limit as approaches from the right
limit as x approaches a of f(x) only exists if…
Left hand limit=right hand limit
vertical asymptote if
lim= -ve or +ve infinity
LHL=-ve or +ve infinity
RHL=-ve or +ve infinity
lim as x approaches a (f(x)+/-g(x))=
lim as x approaches a f(x) +/- lim as x approaches a g(x)
lim as x approaches a (cf(x))=
c lim as x approaches a f(x)
lim as x approaches a (f(x)g(x))=
lim as x approaches a f(x) . lim as x approaches a g(x)
*same for quotient if g(x) does not equal 0**
lim as x approaches a nth root of f(x)=
nth root of lim as x approaches a f(x)
if f is a polynomial, a in domain, then
lim as x approaches a f(x) = f(a)
squeeze theorem
if f(x)</=g(x)</=h(x) x near a
lim as x approaches a f(x)=lim as x approaches a h(x) =L
then
lim as x approaches a g(x)=L
tangent = m = lim as x approaches a of
f(x)-f(a)/x-a
if differentiable at a,
continuous at a
differentiable on an open interval if
differentiable at every number in interval
can fail to be differentiable if:
- has corner/kink
- discontinuous
- has vertical tangent line
indeterminant form 0/0
form lim as x approaches a f(x)/g(x) where both f(x) and g(x) approach 0 as x approaches a
indeterminant form infinity/infinity
lim as x approaches a f(x)/g(x) where both f(x) and g(x) approach infinity as x approaches a
L’hospital’s rule
for type 0/0 or infinity/infinity
if f(x) and g(x) differentiable and g’(x) does not equal 0
then lim as x approaches a f(x)/g(x)
=lim as x approaches a f’(x)/g’(x)
valid for one-sided limits too
product rule
u’v+v’u
quotient rule
u’v-v’u/v^2
normal line
perpendicular to tangent line
d/dx b^x
b^xlnb
d/dxlogbx
1/xlnb
chain rule
dy/dx=dy/du du/dx
where y=f(u) and u=g(x)
chain rule for f(g(h(x)))
apply chain rule twice
implicit differentiation
differentiate wrt x then solve for y’
(pretend y is an x then add y’)
logarithmic differentiation
- take logs of both sides
- differentiate implicitly wrt x
- solve for y’
general form of taylor series
f(x)=f(a)+f’(a)/1!(x-a)+f’‘(a)/2!(x-a)^2+…
Maclaurin series
taylor series with a=0
sof(x)=f(0)+f’(0)x/1! + f’‘(0)x^2/2!+…
taylor series for f(x)g(x) or f(x)/g(x)
x or dividing the series of f(x) and g(x)
usually easier than calculating directly
one-to-one
never takes the same value twice
horizontal line test
f^-1(y)=x implies
f(x)=y
graph of inverse function
reflects the graph of f(x) about the line y=x
steps for finding inverse function
- write y=f(x)
- solve for x in terms of y
- swap x and y
inverse for trig functions
need to restrict domain since trig functions are not one-to-one
(f^-1)’(a)=
1/f’(f^-1(a))
sinhx=
(e^x-e^-x)/2
coshx=
(e^x+e^-x)/2
tanhx=
sinhx/coshx
sechx
1/coshx
cosechx
1/sinhx
cothx
1/tanhx
point (coshb,sinhb) always lies b=1
right branch of hyperbola x^2-y^2=1
x^2-y^2=cosh^2b-sinh^2b=1
sinh^-1x
ln(x+root(x^2+1))
cosh^-1x
ln(x+root(x^2-1))
tanh^-1x
1/2 ln(1+x/1-x)
critical numbers
f’(f)=0 or does not exist
rolle’s theorem
continuous [a.b]
differentiable (a,b)
f(a)=f(b)
then there exists a c in (a,b) such that f’(c)=0
mean value theorem
continuous [a,b]
differentiable (a,b)
then there exists a c in (a,b) such that f’(c)=f(b)-f(a)/b-a
increasing/decreasing test
f’(x)>0 increasing
f’(x)<0 decreasing
first derivative test
f’ +ve to -ve at critical number then f has a local max at c
f’ -ve to +ve at c, local min
concavity
if graph lies above all tangent, concave upward (smiley)
below, concave downward :(((
conditions for concave up/down
f’‘(x) > 0 for all x, concave upward :)
f’‘(x)<0 for all x, concave downward :(
second derivative test
if f’(c)=0 and f’‘(c)>0, local min at c
if f’(c)=0 and f’‘(x)<0, f has local max at c
position vector
origin to initial point
|a|
root a1^2+a2^2+…
unit vector
u=a/|a|
basis vectors
i,j,k
<1,0,0>
<0,1,0>
<0,0,1>
dot product
a.b=a1b1+a2b2+a3b3
a.b=|a||b|cos theta
properties of dot product
a.a = |a|^2
a.b=b.a
(ca).b=c(a.b)=a.(cb) where c is a constant
vectors orthogonal if
theta=pi/2
therefore a.b=0
alpha,beta and gamma are
angles a vector makes with the positive x,y, and z axes respectively.
cos alpha= a1/|a|
cos beta = a2/|a|
cos gamma = a3/|a|
where a =<a1,a2,a3>
compa b
a.b/|a|
proja b
(a.b/|a|)a/|a|=a.b/|a|^2 a
cross product
result is vecotr
axb is 3x3 matrix determinant
axb=|a||b|sin theta
properties of cross product
axb orthogonal to both a and b
axb=-bxa
(ca)xb=c(axb)=ax(cb) where c is a constant
ax(b+c)=axb+axc
(a+b)xc=axc+bxc
vectors parallel if
axb=0
length of axb
ie |axb|
area parallelogram
a.(bxc)=
(axb).c
Vparallelepiped
|a.(bxc)|
vector triple product ax(bxc)=
(a.c)b-(a.b)c
equation of line
r=ro+tv
where ro is position vector of Po
v is parallel to L
parametric equation of line
x=x0+at
y=y0+bt
z=z0+ct
symmetric equations of line
x-x0/a=y-y0/b=z-z0/c
lines are parallel if
direction vectors parallel
lines intersect if
components of vector equation of each line match at P
skew lines
do not intersect and are not parallel
vector equation of plane
n.(r-r0)=0
n.r=n.ro
ax+by+cz=d
where d=ax0+by0+cz0
two planes parallel if
normal vectors are parallel
if two planes are not parallel
they intersect in a straight line
to find angle between planes
- find normal vectors
- cos theta = n1.n2/|n1||n2|
(or use sin and the cross product)
