Math 31A midterm

limit defintion

suppose there is a number L = R such that

• you can make f(x) as close to L as you want

• by choosing x close enough to c (but x doesnt not =c)

then: as x approches c, f(x) approaches the limit L

(you can make f(x)-L small as you want by making x-c small enough)

three basic scenarios for limits

• DNE

  • limit is infinite

  • from left does not equal right

  • does smth wild as x approaches c

• exists but lim f(x) does not equal f (c )

• lim exists and f(x) = f(c )

limit laws

comtinuity

• function f(x) is continous at x=c if

  • the limit lim x→c f(x) exists

  • the limit is equal to the value of the fuction lim x→c f(x) = f(c )

if continous simply sub in c

no gaps or jumps

removable discontinuity

both lim x→c f(x) and f( c) exists but they are not equal, or not defined

jump discontinuity

• both one sided limits exist but are not equal

• right continuous at x-c if lim x→+c f(x) = f(c )

• and left

infinite discontinuity

one or both of the one-sided limits are infinite

continuity of basic functions

• any polynomial

• if P and Q are polynomial then P/Q

• sinx and cosx

• x^(1/n)

  • n even D = R

  • n odd D=R

• b^x, b = R

laws of continuity

if f(x) and g(x) are continous at x=c, then the following are also conitnous at x=c:

• f+g, f-g

• kf, where k is any constant

• fg

• f/g if g(c ) doesnt equal 0

composition of functions

first do one on the right, then do the one on the left

the function f composed with g is the function

(f ° g): A → B → D

(f ° g)(x) = f(g(x))

if g and f are continous than f of g is continous

IVT

the graph must hit 0 in the in between

plug in both ends and then if ones negative and ones positive state that 0 is in between the two

indeterminate

0/0

• use algebra to transform f(x) into an expression that us defined and continuous at x=c

• evaluate the limit by substitution

sin theta/ theta =

1

1-cos theta / theta

0

pythagorean identity

sin²(x) = 1 - cos²(x)

definition of limits at infinity

• suppose there is a # L c R such

  • you can make f(x) as close to L (resp M) as you like

  • by choosing x large (resp small) enough

basic limits at infinity: exponential functions

for all a > 1

lim x approaches infinity a^x = infinity

lim x approches negative infinity a^x = 0

0<a<1

lim x approches infinity a^x =0

lim x approches negative infinity a^x = infinity

limits at infinity: power functions

for all n>0

lim as x approaches infinity x^n = infinity

lim as x approaches infinity x^-n = 0

for all n > 0 with n a whole #

lim x → -infinity x^n ( infity if n is even, negative infinity if n is odd)

lim → -infinity x^-n = 0

limits at infinity of rational functions

a/b lim +- inifinity x^n-m

secant line

let f: R → R be a function

• pick any two points on the graph of f

• P = (a, f(a)), Q = (x, f(x))

• the line through P and Q is called a secant line

• its slope is given by delta f/delta x, rise/run, f(x) - f(a)/ x-a

expression on right is called the difference quotitent

tangent line definition

the tangent line to the graph of y=f(x) at P = (a, f(a)) is the line through P with slope

lim x→a f(x) - f(a)/ x-a

it is the best straight line approximation to y = f(x) at a

derivative defintion

the derivative of f(x) at a is the limit

f’(a) = df/dx|a = lim x→a f(x) - f(a)/x-a

it is the instantaneuous rate of change of f at a

equation of a line

slope of a line = rise over run

one of the points its fixed (A, B)

one of the points is variable (x,y)

let m be the slope of the line

rise/run = (y-B)/(x-A) = m

point slope form y = m(x-A) + B

slope intercept form y= mx + (B-mA)

equation of the tangent line

y = f’(a)(x-a) +. f(a)

derivative definition restated

f’(a) = lim h→ 0 f(a+h) - f(a)/h

derivative as a function

the derivative f"‘: D(f’) → R of f is the function

f’(x) = lim h→0 f(x+h) - f(x)/h

if f’(a) exists, we say that f is differentiable at a

Leibniz Notation

df/dx|x=a = f’(a)

df/dx = lim h→ 0 f(x+h) - f(x)/h = lim delta x→ 0 delta f/delta x

power rule

put exponent in fron then subtract by 1 (basic derivative)

derivative of e^x

e^x