Theorums

Question 1

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.

Answer 1

Extreme Value Theorem

Question 2

If f is continuous on [a,b] and k is a number between f(a) and f(b), then there exists at least one number c such that f(c)=k

Answer 2

Intermediate Value Theorem

Question 3

a point near which the function values oscillate too much for the function to have a limit

Answer 3

Oscillating Discontinuity

Question 4

a point of discontinuity where one or both of the one-sided limits are infinite

Answer 4

Infinite Discontinuity

Question 5

a point of discontinuity where the one-sided limits exist, but have different values

Answer 5

Jump Discontinuity

Question 6

a discontinuity c of f for which f(c) can be redefined so that the limit as x approaches c of f(x)=f(c)

Answer 6

Removable Discontinuity

Question 7

if f is integrable on [a,b] its average value on [a,b] is

Answer 7

Definition of Average (mean) Value

Question 8

a point where the graph of a function has a tangent line and where the concavity changes

Answer 8

Definition of a Point of Inflection

Question 9

the graph of a differentiable function y=f(x) is a) concave up on an open interval I if y’ is increasing on I b) concave down on an open interval I if y’ is decreasing on I

Answer 9

Definition of Concavity

Question 10

a point in the interior of the domain of a function f at which f’=0 or f’ does not exist is a critical point of f

Answer 10

Definition of a Critical Point

Question 11

if f’(c)=0 and f’‘(c)<0 then f has a local maximum at x=c if f’(c)=0 and f’‘(c)>0 then f has a local minimum at x=c

Answer 11

Second Derivative Test for Local Extrema

Question 12

if a function f has a local maximum value or a local minimum value at an interior point c of its domain and if f’ exists at c then f’(c)=0

Answer 12

Local Extreme Value Theorem

Question 13

Let f be a function with domain D. Then f(x) is the a) absolute maximum value on D if and only if f(x)<=f(c) for all x in D b) absolute minimum value on D if and only if f(x)<=f(c) for all x in D

Answer 13

Definition of Absolute Extreme Values

Question 14

the derivative of velocity with respect to time

Answer 14

Definition of Acceleration

Question 15

the derivative of the position function s=f(t) with respect to time

Answer 15

Definition of Instantaneous Velocity

Question 16

the absolute value of velocity

Answer 16

Definition of Speed

Question 17

if and only if it is continuous at every point of the interval

Answer 17

a function f is continuous at point c

Question 18

the slope of a curve y=f(x) at the point (a,f(a)) is f’(a) provided f is differentiable at a

Answer 18

Definition of the slope of a curve at a point

Question 19

the derivative of the function f with respect to x is the function f’ whose value at x is

Answer 19

Definition of a Derivative

Question 20

xzAa function y=f(x) is differentiable on a closed interval [a,b] if it has a derivative at every interior point of the interval and if h exists as h approaches 0 from the left or the right

Answer 20

One-Sided Derivative

Question 21

F(x) becomes arbitrarily close to a particular, finite value L as x becomes arbitrarily close to some value a.

Answer 21

Definition of a limit

Question 22

If f(x) is continuous at c then the limit is lim x–>c f(x) = f(c)

Answer 22

Direct Substitution

Question 23

If f(x) < g(x) < h(x) and the lim x–>c f(x) = lim x–>c h(x) = L then lim x–>c = L

Answer 23

Sandwich/Squeeze Theorem

Question 24

The line y=b is a ____ if the graph of f(x) is either lim x–> 00 f(x) = b or lim x–> -00 f(x) = b

Answer 24

Horizontal Asymptote

Question 25

The line x=a is a ____ of f(x) if either lim x–> a- f(x) = +/- 00 or lim x–> a+ f(x) = +/- 00

Answer 25

Vertical asymptote

Question 26

(f(b) - f(a))/ b-a

Answer 26

Average rate of change

Question 27

lim h–> 0 f(x+h) -f(x) / h aka derivative

Answer 27

Instantaneous rate of change

Question 28

A line perpendicular to the tangent line at the point of tangency

Answer 28

Normal line

Question 29

Slope of tangent line

Answer 29

derivative

Question 30

lim x–>a f(x) - f(a)/ x-a

Answer 30

Alternate definition of the derivative

Question 31

f is continue at x=c and the derivative on the left and right of the c are equal

Answer 31

Derivative Piecewise

Question 32

No line/point/jump discontiuity

Answer 32

Continuous

Question 33

Non corners/cusps/vertical tangents/discontinuities

Answer 33

Differnitability

Question 34

If a function is differentiable then it is continuous

Answer 34

Differentiability Implies Continuity

Question 35

The process of using the tangent line to estimate the value of a funtions

Answer 35

Linearization

Question 36

When a limit results in indeterminate form (0/0) by direct substitution, then one is able to take ether derivative of the numerator and the derivative separately

Answer 36

L’Hoptial’s rule