Calculus1.1-1.3

Ms. Hernandez

Calculus

• Syllabus & Assignments Hard Copy

Tangent and Derivative Pblm

• See Handout Calculus Concepts and Contexts, Stewart 3rd ed, 2.1 p92-98

• #1,3,5 QZ 1 due Wed 8-16

• GSP WS

Limits

• Limiting property is why they are called limits

• b/c a function f(x) gets really close to some value – its INDTENDED value

• Yet, sometimes may not really get there• Like vertical or horizontal asymptotes

y = log(x) y = ex

• Limits are not limited to asymptotes

Limits

• Occur at points of discontinuity

• Pg 48, 49

Formal Def of a Limit

• As function f(x) gets REALLY close to a number, we call this number L (or Bob) doesn’t matter what we call it

• The value the function gets really close to (techie term is arbitrarily close)

• Is the limit of the function f(x)

Formal Def of a Limit

• Need a place to go to (anchor – reference)• Let c be that place to go to on your

function• So then as your function f(x) gets closer to

c (somewhere on your function) from the left and from the right of c

• Then f(x) approaches a value L • L stands for Limit

Def of limit works for all lim

• Not all limits are created equal

• Some are nicer than othersy=f(x) and in nice f(x) then y=L

1.2 # 9,10, 16

piecewise f(x) 1.2 #11, 12

• So the def has to include the asymptotes

• So that is works for everyone

Limits can fail

• Different– Agree to disagree p50

• Unbounded– Can be confused w/infinite limits p50

• Oscillating– f(x) on crack p51

GSP WS

Evaluating Limits

• Properties of limits p57– Basic– Scalar multiple– Sum or Difference– Product– Quotient– Power

Theorem 1.1 Some Basic Limits

• Let b and c be real numbers and let n be a positive integer

limx c

b b

3

lim5 5x

limx c

x c

3

lim 3x

x

lim n n

x cx c

2 2

2lim 2 4x

x

Theorem 1.2 Properties of Limits• Let b and c be real numbers and let n be a positive integer,

AND let f and g be functions with the following limits.

lim ( )x c

f x L

lim ( )x c

g x K

lim[ ( )]x c

bf x bL

lim[ ( ) ( )]x c

f x g x LK

2 2

2lim[2 ] 2(2 ) 8x

x

2 2

2lim[ ] 4(4) 16x

x x

Thm 1.2 cont’d

lim[ ( ) ( )]x c

f x g x L K

2

2lim[ ] 2 4x

x x

( )lim , 0

( )x c

f x LK

g x K

lim[ ( )]n n

x cf x L

Direct Sub

• Direct Substitution property is valid for ALL polynomial & rational functions with nonzero denominators

• NO ZERO IN THE DENOMINATOR!!!

• Thm 1.3

Thm 1.4 f(x) with a radical

3 3

8lim 8 2x

x

lim n n

x cx c

As long as n is positive integer. If n is odd then it works for all values of c. If n is even, then it only works when c > 0.

1.5 Limit of Composite f(x)

• If f and g are functions with the following limits lim ( )

x cg x L

lim ( ) ( )

x Lf x f L

this means that you got a function g(x) as x c with a limit L and then you take another function f(x) as xL then f(x) limit is really just f(L).

lim ( ( )) lim ( ) ( )x c x c

f g x f g x f L

Example 4 pg.59

• 4 min discuss in groups

• In (a) which is f(x) and which is g(x)

• In (b) which is f(x) and which is g(x)

• Does it really matter?

Thm 1.6 Trig f(x)

• Six basic trig limits on pg 59

• Memorize them all

• Practice using them all

• And oh yeah, get to work on memorizing the trig identities in the back of the book

• All but co-function identities

Evaluating Limits

• Polynomial Limits– Just break it up!– p58

• Polynomial & Rational functions– Polynomial & Rational Functions– Ok as long as denominator is NOT zero!

Evaluating Limits

• Substitution

• Divide out (factor out)– Ex 7 pg 61 – Discuss 4 min.– What is an indeterminate form?– How can technology trip you up?

Evaluating Limits

• Rationale• Conjugate• Ex 8 pg 62

– Discuss 4 min– How do you get the conjugate?– How do you multiply by the conjugate?– Simplifying, what do you need to watch out

for?