calculus

Question 1

continuity

Answer 1

no jumps at x∈D

Question 2

smoothness

Answer 2

has a tangent at x∈D

Question 3

f(x) is smooth at x => continuous?

Answer 3

continuous at x

Question 4

f(x) is continuous at x => smooth?

Answer 4

not necessarily

Question 5

limit def

Answer 5

the value that a function approaches as the argument approaches some value

Question 6

when does a limit not exist

Answer 6

left limit ≠ right limit

Question 7

f(x) is continuous at a => lim(x->a) f(x) = ?

Answer 7

f(a)

Question 8

f(x) has a horizontal asymptote at y=a => lim(x->∞) f(x) = ?

Answer 8

a … constant

Question 9

lim(x->a) b = ?

a is a real number or +/- ∞, b is a constant

Answer 9

b
limit of a constant is constant

Question 10

lim(x->a) b(f(x)) = ?

a is a real number or +/- ∞, b is a constant

Answer 10

b lim(x->a) f(x)

Question 11

lim(x->a) (f(x) ± g(x)) =

a is a real number or +/- ∞, b is a constant

Answer 11

lim(x->a) f(x) ± lim(x->a) g(x)

Question 12

lim(x->a) (f(x) x g(x)) =

a is a real number or +/- ∞, b is a constant

Answer 12

lim(x->a) f(x) x lim(x->a) g(x)

Question 13

lim(x->a) (f(x) / g(x)) =

a is a real number or +/- ∞, b is a constant

Answer 13

lim(x->a) f(x) / lim(x->a) g(x)

Question 14

0/0 =

Answer 14

indeterminate form

Question 15

∞ / ∞ =

Answer 15

indeterminate form

Question 16

∞ - ∞ =

Answer 16

indeterminate form

Question 17

1^∞ =

Answer 17

indeterminate form

Question 18

∞^0

Answer 18

indeterminate form

Question 19

0^0 =

Answer 19

indeterminate form

Question 20

0 x ∞ =

Answer 20

indeterminate form

Question 21

∞ + ∞ =

Answer 21

∞

Question 22

∞^∞ =

Answer 22

∞

Question 23

∞^(-∞) =

Answer 23

0

Question 24

0^∞ =

Answer 24

0

Question 25

derivative def

Answer 25

gradient of the curve/tangent to a curve rate of change of f(x) at x = x1

Question 26

differentation by first principle | y = kx + b ... tangent

Answer 26

k = lim (h->0) (f(x+h) - f(x))/h = f'(x)

Question 27

the normal

Answer 27

line perpendicular to the tangent

Question 28

c' = ? | c is a constant

Answer 28

0

Question 29

(ax + b)' = ?

Answer 29

a

Question 30

(x^n)' = ?

Answer 30

nx^(n-1)

Question 31

((c)(f(x)))' = ? | c is a constant

Answer 31

cf'(x)

Question 32

(g(x) ± f(x))' = ?

Answer 32

g'(x) ± f'(x)

Question 33

(f(x) x g(x)) = ?

Answer 33

(f'(x) x g(x)) + (f(x) x g'(x))

Question 34

(f(x) / g(x))' = ?

Answer 34

((f'(x) x g(x)) - (f(x) x g'(x))/(g^2(x))

Question 35

f(g(x)) = ?

Answer 35

f'(g(x)) x g'(x) inside function into outside differentiated -> multiply by inside'

Question 36

turning point | def, when

Answer 36

the function changes from increasing to decreasing or vice versa local extremum turning point => f'(x) = 0; not <=>

Question 37

lim(x->0) (cosx-1)/x = ?

Answer 37

0

Question 38

lim(x->0) (sinx/x) =

Answer 38

1

Question 39

sin'x = ?

Answer 39

cosx

Question 40

cos'x = ?

Answer 40

-sinx

Question 41

stationary point | condition, types

Answer 41

the tangent is horizontal => f'(x) = 0 local extrema (max/min), inflection point

Question 42

f'(x) > 0

Answer 42

f(x) is increasing at x

Question 43

f'(x) < 0

Answer 43

f(x) is decreasing at x

Question 44

f'(x) = 0

Answer 44

f(x) is neither increasing nor decreasing (stationary point)

Question 45

f''(x) > 0

Answer 45

f(x) concave-up at x

Question 46

f''(x) < 0

Answer 46

f(x) is concave-down at x

Question 47

f'(x) = 0 and f''(x) > 0

Answer 47

local min at x

Question 48

f'(x) = 0 and f''(x) < 0

Answer 48

local max at x

Question 49

inflection point on f(x) => ? on f'(x)

Answer 49

local maxima/minima at x

Question 50

odd f(x) => ? f'(x)

Answer 50

even f'(x)

Question 51

even f(x) => ? f'(x)

Answer 51

odd f'(x)

Question 52

f(x) is steeper at x => f'(x) ?

Answer 52

f'(x) is greater (closer to infinity)

Question 53

inflection point at x on f(x) < = > ? f'(x) < = > ? f''(x)

Answer 53

turning point on f'(x) at x 0 at f''(x) at x

Question 54

finding inflection points

Answer 54

sign chart of f''(x) sign change => inflection point

Question 55

concave-up | tangents, line segments, f''(x)

Answer 55

tangents are below the graph line segments are above the graph f''(x) > 0

Question 56

concave-down | tangents, line segments, f''(x)

Answer 56

tangents are above the graph line segments are below the graph f''(x) < 0