Differentiation

Question 1

geometrical representation of f’(a)

Answer 1

f’(a) of f(x) at x = a is equal to the slope of the tangent to the graph of f at a

Question 2

definition of f’(a)

Answer 2

the limit of the slope of a line segment joining 2 nearby points on the graph of f
= lim h-> 0 ( f(a+h) - f(a) ) /h
if the limit exists we say f is differentiable at x = a

Question 3

differentiable

Answer 3

if f’(a) exists for all a in the domain of f we say f is differentiable and the function f’(x) is the derivative

Question 4

equation of tangent to f at (a,f(a)

Answer 4

y = f(a) + f’(a) (x-a)

Question 5

necessary condition for f’(a) to exists

Answer 5

is for f to be continuous at a but this is not sufficient (can be continuous and not differentiable)

Question 6

ways function can fail to be differentiable at a point where its continuous

Answer 6

• tangent line is vertical
• no tangent line at the point

Question 7

twice differentiable

Answer 7

f(x) is differentiable then f’(x) may be differentiable if it is we say its twice differentiable
(smooth means infinitely differentiable)
(f’(x))’ = f’‘(x)

Question 8

how to write nth derivative

Answer 8

f^(n)(x)
d^nf/dx^n
D^n f(x)
if its a variable of time we dot it to show differentiation

Question 9

leibniz rule

Answer 9

if f(x) and g(x) are differentiable n times so is there product fg(x)
D(fg) = (Df)g + f(Dg)
D^n(fg) = ∑ (nCk) (D^k f)(D^n-k g) from k=0 to n

Question 10

chain rule

Answer 10

for differentiating the composition of functions
fºg(x)
if g(x) is differentiable at x and f(x) is differentiable at g(x)
fºg’(x) = f’(g(x))g’(x)
or d/dx(f(g(x)) = df/dg dg/dx

Question 11

L’Hopitals rule

Answer 11

applies to limits in the form 0/0 or ♾️/♾️
if f(x) and g(x) are differentiable on I = (a-h,a) U (a,a+h) for some small h-> 0 with lim f(x) to a = lim g(x) to a = 0
and lim f’(x)/lim g’(x) exists
then lim f(x)/g(x) = lim f’(x)/g’(x)

Question 12

indeterminate form of the limit

Answer 12

lim x->a f(x) /g(x) doesnt exists if L=/= 0 and M = 0
if L= M = 0 then its indeterminate form
might or might not have a well defined value - we cant tell by knowing L and M
eg 0/0, ♾️/♾️, 0*♾️, ♾️-♾️, 0^0 , 1^♾️, ♾️^0

Question 13

proof of L’Hopitals rule

Answer 13

where f’(a) and g’(a) exist and the derivatives are continuous and g’(a) =/= 0
lim f(x)/g(x) = lim f(x) - 0/ g(x) -0 = lim f(x) - f(a) / g(x) -g(a) = lim (f(x)-f(a))/(x-a) / (g(x)-g(a))/(x-a) = f’(a)/g’(a) = lim f’(x)/g’(x)

Question 14

bounded above

Answer 14

f(x) is a function on an interval I
is bounded above by an upper bound k1 if f(x) <= k1 for all x in I
f(x) need not take the value k1 in I

Question 15

global maximal value

Answer 15

if there exists an x in I such that f(x) = k1 the upper bound is attained and k1 is the global max

Question 16

bounded

Answer 16

f(x) is bounded in I if its bounded above and below in I
there exists a k in R such that |f(x)| <= k for all x in I

Question 17

extreme value theorem

Answer 17

if f(x) is continuous on a closed interval [a,b] then its bounded on that interval and has upper/ lower bounds which are attained
(has global max/min)
there exists x1,x2 in [a,b] such that f(x2) <= f(x) <= f(x1) for all x in [a,b]

Question 18

monotonic increasing

Answer 18

if in [a,b] f(x1) <= f(x2) for all x1,x2 such that a<= x1 < x2 <= b
similar for decreasing

Question 19

strictly monotonic increasing

Answer 19

if in [a,b] f(x1) < f(x2) for all x1,x2 such that a <= x1 < x2 <= b
similar for decreasing

Question 20

local maximum (or minimum)

Answer 20

f(x) has a local max (or min) at x = a if there exists h>0 such that f(a) >= f(x) (or f(a)<= f(x))
on (a-h,a+h)

Question 21

local extremum

Answer 21

either a local max or a local min

Question 22

max/ min and derivative

Answer 22

if f(x) has a maximum (or minimum) at x = a and is differentiable at x = a
f’(a) = 0

Question 23

stationary point

Answer 23

f(x) has one at x = a if its differentiable at x = a and f’(a) = 0

Question 24

critical point

Answer 24

an interior point x = a of the domain of f(x) is a critical point if either f’(a) = 0 or f’(a) does not exist

Question 25

point of inflection

Answer 25

if f(x) is twice differentiable in an open interval around x = a with f''(a) = 0 and f''(x) changes sign at x = a then x = a is a point of inflection

Question 26

1st derivative test

Answer 26

f(x) is continuous at a critical point x = a - if there exists h>0 such that f'(x) <0 for all x in (a-h,a) and f'(x) >0 for all x in (a, a+h) then x = a is a local min - opposite for local max - if f'(x) is constant in sign then not a local extremum at x = a

Question 27

2nd derivative test

Answer 27

suppose f(x) is twice differetiable at x = a with f'(a) = 0 - if f''(a) >0 then x = a is a local min - if f''(a) <0 then x = a is a local max - if f''(a) = 0 then the test tells us nothing

Question 28

end points

Answer 28

if c is in the domain of f and there exists a small h >0 such that f(x) is defined on [c,c+h) but not (c-h,c] (or vice versa) then x = c is an endpoint of f(x)

Question 29

endpoint max (or min)

Answer 29

if x = a is an endpoint of f(x) then endpoint max (or min) at x = a if f(x)<= f(a) (or f(x)>= f(a)) for x close to a

Question 30

sign of deritvative + nature of endpoint

Answer 30

sign of derivative can determine the nature of the endpoint if f(x) is continuous and differentiable

Question 31

global extrema and critical points or end points

Answer 31

if f(x) is continuous on [a,b] then all global extrema in this interval are attained at either critical points or endpoints

Question 32

rolles theorem

Answer 32

if f(x) is continuous and differentiable on [a,b] and f(a) = f(b) then there exists c in (a,b) such that f'(c) = 0

Question 33

corollary of rolles theorem

Answer 33

if f(x) is differentiable on an open interval I then each pair of zeros is seprated by at least one zero fo the derivative (rolles theorem if f(a) = f(b) = 0) can be used to find bound on no. zeros of a function

Question 34

mean value theorem

Answer 34

if f(x) is continuous on [a,b] and differentiable on (a,b) then there exists c in (a,b) such that f'(c) = f(b)-f(a)/b-a geometrically - at least one point where tangent is parallel to line joining end points can be used to prove results relative monoticity to sign of derivative

Question 35

relative monoticity to sign of derivative

Answer 35

use MVT f(x) is continuous on [a,b] and differentiable on (a,b) with f'(x) >= 0 for all x in (a,b) then f(x) if monotonic increasing

Question 36

inverse function rule

Answer 36

f(x) is continuous and differentiable on (a,b) with f'(x) >0 for all x in (a,b) its inverse g(y) exists and is differentiable on (f(a), f(b)) g'(y) = 1/f'(g(y)) similarly if f'(x) <0 then g(y) is differentiable for all f(b)

Question 37

partial derivatives idea

Answer 37

can consider functions of 2 or more variables wither f(x,y) is the height of a function over the x-y plane we now think of the rate of change of f(x,y) as we vary x or y keeping the other fixed

Question 38

partial derivative f(x,y) with respect to x

Answer 38

∂f/∂x(x,y) = lim (h->0) (f(x+h,y) - f(x,y)) / h same for ∂f/∂y but add h to the y differentiate with respect to one variable and treat the other as constant fx = ∂f/∂x fy = ∂f/∂y fxy = (fx)y = ∂/∂y (∂f/∂x)

Question 39

clairauts theorem

Answer 39

∂^2f/∂x∂y = ∂^f/∂y∂x and this is true if f and all its 1st and 2nd partial derivatives are continuous