Calculus I Chapter 4

Front 1

If f(x1)<f(x2) and x1<x2, then the function f(x) is __________

the graph is________

Back 1

increasing, uphill

Front 2

If f(x1)>f(x2) and x1< x2, then the function f(x) is __________

the graph is________

Back 2

decreasing, downhill

Front 3

If f(x1)>=(x2) for every x1, x2, then the function f(x) is _________

the graph is________

Back 3

constant, horizontal

Front 4

F(c) is a maximum value of f if...

Back 4

f(x)<_ f(c) for all x on the interval I

Front 5

F(c) is a minimum value of f if...

Back 5

f(x)>_ f(c) for all x on the interval I

Front 6

F(c) is a local maximum of f if there is an open interval containing c such that:

Back 6

f(x)<_ f(c) for all x on the interval (a,b)

Front 7

F(c) is a local minimum of f if there is an open interval containing c such that:

Back 7

f(x)>_f(c) for all x on the interval (a,b)

Front 8

steps for finding critical numbers

Back 8

1) differentiate the function
2) find the c values for which f'(c)= 0 or f'(c) dne

Front 9

Steps for finding the local extrema of f on (a,b)

Back 9

1) find the critical numbers
2)calculate f(c) for each critical number
3)calculate f(a) and f(b)
4) the absolute max and min are the largest values calculated in steps 2 and 3

Front 10

Rolle's theorem

Back 10

IF
-f is continuous on [a,b]
-f is differentiable on (a,b)
-f(a)= f(b)
THEN
there must be a critical number on the interval (a,b) such that f'(c)=0

Front 11

Strategy for using Rolle's theorem

Back 11

1) show that the function satisfies the conditions of continuity, differentiability, and equality of f(a) and f(b)
2)differentiate the function
3)set f'= 0 and solve for x
4) use the x value above and solve for f(x)

Front 12

Mean Value Theorem

Back 12

IF
-f is continuous on [a,b]
-f is differentiable on (a,b)
THEN
there exists a number c such that:
f(b)-f(a)= f'(c)(b-a)

Front 13

Strategy for using Mean Value theorem

Back 13

1) show that the function satisfies the conditions of continuity and differentiability
2) calculate f(a), f(b), and f'(c)
3) plug values into f(b)-f(a)= f'(c)(b-a)
4) solve for c
5) find f(c)

Front 14

First Derivative test

f(c) is a local max if:

Back 14

Let f be continuous at c and differentiable on an open interval containing c and let e>0.

f'(c-e)>0 and f'(c+e)<0
aka: f' changes from positive to negtive

Front 15

First Derivative test

f(c) is a local min if:

Back 15

Let f be continuous at c and differentiable on an open interval containing c and let e>0.

f'(c-e)<0 and f'(c+e)>0
aka: f' changes from negative to positive

Front 16

1st derivative test

f(c) is not a local extremum

Back 16

Let f be continuous at c and differentiable on an open interval containing c and let e>0.

f'(x)>0 or f'(x)<0 for every x except x=c

Front 17

Strategy for 1st derivative test

Back 17

1)prove continuity and differentiability
2) differentiate
3)find CN
4)determine intervals
5)determine signs
6)determine local extremum

Front 18

Concavity:
graph is concave up if...

Back 18

f''(c)>0
tan line is below graph

Front 19

Concavity:
graph is concave down if...

Back 19

f''(c)<0
tan line is above graph

Front 20

Second Derivative test
x has a local max at c if...

Back 20

f is differentiable on an open interval containing c and F(C)=0!!
f''(c)<0

Front 21

Second Derivative test
x has a local min at c if...

Back 21

f is differentiable on an open interval containing c and F(c)=0!!
f''(c)>0

Front 22

Second Derivative test
the test is inconclusive if...

Back 22

f is differentiable on an open interval containing c and
f''(c)0

Front 23

Strategy for second derivative test

Back 23

1) differentiate
2) find CN
3)find f''(x)
4) find f''(c)
5)decide concave up or down
6) determine local max and min

Front 24

definition: inflection point

Back 24

point at which the function changes from concave up to concave down

Front 25

how to find inflection points

Back 25

set the second derivative equal to 0 and solve
check with a sign chart

Front 26

Strategy for an application problem

Back 26

1)read
2)draw picture
3)write down facts and relationships
4) determine which variable is to be maximized or minimized and write it as a function of one other variable
5)find critical values of function found in 4 and test to see if they are max or min
6) check to see if extremes occur at the endpoints of function obtained in four

Front 27

Strategy for an infinity limit if degrees are the same

Back 27

multiply top and bottom by a form of one, a form which has x to the highest degree in the denominator.
Cancel values that go to zero

Front 28

lim as x-> infinity of c/ (x^k)=

Back 28

0

Front 29

lim as x-> negative infinity of c/ (x^k)=

Back 29

0

Front 30

Strategy for an infinity limit if degrees are different

Back 30

Divide function and rewrite as a product
cancel values that go to zero