Sem 1 Stewart

Question 1

function or not?

Answer 1

vertical line test

Question 2

piecewise function

Answer 2

defined differently in different parts of the domain

Question 3

even function or odd function

Answer 3

even f(-x)=f(x)
odd f(-x)=-f(x)

Question 4

increasing functions

Answer 4

if f(x1)<f(x2) when x1<x2

vice versa for decreasing

Question 5

polynomial

Answer 5

contains non-negative integers only

Question 6

power function

Answer 6

f(x)=x^a

Question 7

sec

Answer 7

1/cos

Question 8

cosec

Answer 8

1/sin

Question 9

cot

Answer 9

cos/sin = 1/tan

Question 10

exponential functions

Answer 10

f(x)=b^x

Question 11

log function

Answer 11

g(x)=logbx

Question 12

lnx

Answer 12

logex

Question 13

exponential properties

Answer 13

domain real numbers
range (0,infinity)
increasing if b>1, decreasing 0<b<1

Question 14

secant

Answer 14

cuts a curve more than once

Question 15

left hand limit

Answer 15

limit as approaches from the left

Question 16

right hand limit

Answer 16

limit as approaches from the right

Question 17

limit as x approaches a of f(x) only exists if…

Answer 17

Left hand limit=right hand limit

Question 18

vertical asymptote if

Answer 18

lim= -ve or +ve infinity
LHL=-ve or +ve infinity
RHL=-ve or +ve infinity

Question 19

lim as x approaches a (f(x)+/-g(x))=

Answer 19

lim as x approaches a f(x) +/- lim as x approaches a g(x)

Question 20

lim as x approaches a (cf(x))=

Answer 20

c lim as x approaches a f(x)

Question 21

lim as x approaches a (f(x)g(x))=

Answer 21

lim as x approaches a f(x) . lim as x approaches a g(x)

*same for quotient if g(x) does not equal 0**

Question 22

lim as x approaches a nth root of f(x)=

Answer 22

nth root of lim as x approaches a f(x)

Question 23

if f is a polynomial, a in domain, then

Answer 23

lim as x approaches a f(x) = f(a)

Question 24

squeeze theorem

Answer 24

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

Question 25

tangent = m = lim as x approaches a of

Answer 25

f(x)-f(a)/x-a

Question 26

if differentiable at a,

Answer 26

continuous at a

Question 27

differentiable on an open interval if

Answer 27

differentiable at every number in interval

Question 28

can fail to be differentiable if:

Answer 28

1. has corner/kink
2. discontinuous
3. has vertical tangent line

Question 29

indeterminant form 0/0

Answer 29

form lim as x approaches a f(x)/g(x) where both f(x) and g(x) approach 0 as x approaches a

Question 30

indeterminant form infinity/infinity

Answer 30

lim as x approaches a f(x)/g(x) where both f(x) and g(x) approach infinity as x approaches a

Question 31

L’hospital’s rule

Answer 31

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

Question 32

product rule

Answer 32

u’v+v’u

Question 33

quotient rule

Answer 33

u’v-v’u/v^2

Question 34

normal line

Answer 34

perpendicular to tangent line

Question 35

d/dx b^x

Answer 35

b^xlnb

Question 36

d/dxlogbx

Answer 36

1/xlnb

Question 37

chain rule

Answer 37

dy/dx=dy/du du/dx
where y=f(u) and u=g(x)

Question 38

chain rule for f(g(h(x)))

Answer 38

apply chain rule twice

Question 39

implicit differentiation

Answer 39

differentiate wrt x then solve for y’

(pretend y is an x then add y’)

Question 40

logarithmic differentiation

Answer 40

1. take logs of both sides
2. differentiate implicitly wrt x
3. solve for y’

Question 41

general form of taylor series

Answer 41

f(x)=f(a)+f’(a)/1!(x-a)+f’‘(a)/2!(x-a)^2+…

Question 42

Maclaurin series

Answer 42

taylor series with a=0

sof(x)=f(0)+f’(0)x/1! + f’‘(0)x^2/2!+…

Question 43

taylor series for f(x)g(x) or f(x)/g(x)

Answer 43

x or dividing the series of f(x) and g(x)

usually easier than calculating directly

Question 44

one-to-one

Answer 44

never takes the same value twice

horizontal line test

Question 45

f^-1(y)=x implies

Answer 45

f(x)=y

Question 46

graph of inverse function

Answer 46

reflects the graph of f(x) about the line y=x

Question 47

steps for finding inverse function

Answer 47

1. write y=f(x)
2. solve for x in terms of y
3. swap x and y

Question 48

inverse for trig functions

Answer 48

need to restrict domain since trig functions are not one-to-one

Question 49

(f^-1)’(a)=

Answer 49

1/f’(f^-1(a))

Question 50

sinhx=

Answer 50

(e^x-e^-x)/2

Question 51

coshx=

Answer 51

(e^x+e^-x)/2

Question 52

tanhx=

Answer 52

sinhx/coshx

Question 53

sechx

Answer 53

1/coshx

Question 54

cosechx

Answer 54

1/sinhx

Question 55

cothx

Answer 55

1/tanhx

Question 56

point (coshb,sinhb) always lies b=1

Answer 56

right branch of hyperbola x^2-y^2=1

x^2-y^2=cosh^2b-sinh^2b=1

Question 57

sinh^-1x

Answer 57

ln(x+root(x^2+1))

Question 58

cosh^-1x

Answer 58

ln(x+root(x^2-1))

Question 59

tanh^-1x

Answer 59

1/2 ln(1+x/1-x)

Question 60

critical numbers

Answer 60

f’(f)=0 or does not exist

Question 61

rolle’s theorem

Answer 61

continuous [a.b]
differentiable (a,b)
f(a)=f(b)

then there exists a c in (a,b) such that f’(c)=0

Question 62

mean value theorem

Answer 62

continuous [a,b]
differentiable (a,b)

then there exists a c in (a,b) such that f’(c)=f(b)-f(a)/b-a

Question 63

increasing/decreasing test

Answer 63

f’(x)>0 increasing
f’(x)<0 decreasing

Question 64

first derivative test

Answer 64

f’ +ve to -ve at critical number then f has a local max at c

f’ -ve to +ve at c, local min

Question 65

concavity

Answer 65

if graph lies above all tangent, concave upward (smiley)
below, concave downward :(((

Question 66

conditions for concave up/down

Answer 66

f’‘(x) > 0 for all x, concave upward :)

f’‘(x)<0 for all x, concave downward :(

Question 67

second derivative test

Answer 67

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

Question 68

position vector

Answer 68

origin to initial point

Question 69

|a|

Answer 69

root a1^2+a2^2+…

Question 70

unit vector

Answer 70

u=a/|a|

Question 71

basis vectors

Answer 71

i,j,k
<1,0,0>
<0,1,0>
<0,0,1>

Question 72

dot product

Answer 72

a.b=a1b1+a2b2+a3b3

a.b=|a||b|cos theta

Question 73

properties of dot product

Answer 73

a.a = |a|^2
a.b=b.a
(ca).b=c(a.b)=a.(cb) where c is a constant

Question 74

vectors orthogonal if

Answer 74

theta=pi/2

therefore a.b=0

Question 75

alpha,beta and gamma are

Answer 75

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>

Question 76

compa b

Answer 76

a.b/|a|

Question 77

proja b

Answer 77

(a.b/|a|)a/|a|=a.b/|a|^2 a

Question 78

cross product

Answer 78

result is vecotr
axb is 3x3 matrix determinant

axb=|a||b|sin theta

Question 79

properties of cross product

Answer 79

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

Question 80

vectors parallel if

Answer 80

axb=0

Question 81

length of axb
ie |axb|

Answer 81

area parallelogram

Question 82

a.(bxc)=

Answer 82

(axb).c

Question 83

Vparallelepiped

Answer 83

|a.(bxc)|

Question 84

vector triple product ax(bxc)=

Answer 84

(a.c)b-(a.b)c

Question 85

equation of line

Answer 85

r=ro+tv

where ro is position vector of Po
v is parallel to L

Question 86

parametric equation of line

Answer 86

x=x0+at
y=y0+bt
z=z0+ct

Question 87

symmetric equations of line

Answer 87

x-x0/a=y-y0/b=z-z0/c

Question 88

lines are parallel if

Answer 88

direction vectors parallel

Question 89

lines intersect if

Answer 89

components of vector equation of each line match at P

Question 90

skew lines

Answer 90

do not intersect and are not parallel

Question 91

vector equation of plane

Answer 91

n.(r-r0)=0
n.r=n.ro

ax+by+cz=d
where d=ax0+by0+cz0

Question 92

two planes parallel if

Answer 92

normal vectors are parallel

Question 93

if two planes are not parallel

Answer 93

they intersect in a straight line

Question 94

to find angle between planes

Answer 94

1. find normal vectors
2. cos theta = n1.n2/|n1||n2|

(or use sin and the cross product)