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Calculus1
Calculus Q and A
41
Mathematics
12th Grade
04/11/2013

Additional Mathematics Flashcards

 


 

Cards

Term

 

 

The Mean Value Theorem

Definition
[image]
Term

 

Given an equation for (x),

determine where has relative extrema.

Definition

 

Find where has a critical number

f ' (x) = 0 or undefined)

and look for sign changes about those numbers to indicate whether it's a max or min.

Term
The Intermediate Value Theorem
Definition
Suppose f is continuous on [a,b]. Then for any number (k) between f(a) and f(b), there must be an x value between a and b such that f(x)=k.
Term
[image]
Definition
[image]
Term

 

 

Given a graph of  f '' , how do you locate inflection points of  ?

Definition

f  has an inflection point where  f '' changes sign (why?)

 

 

By definition, changes concavity (and so has an inflection point) whenever its 1st derivative has a relative extremum.  Sign changes in the second derivative indicate exactly this, and so transitively indicate inflection points

Term
Given a graph of f'', how do you locate relative extrema of f?
Definition
Not enough information.
f'' only tells whether f' is increasing or decreasing. This is not enough to draw conclusions about the relative extrema of f.
Term
Given a graph of f', how do you locate relative extrema of f?
Definition
Look for sign changes. f has a relative max where f' changes from (+) to (-), and a relative min where f' changes from (-) to (+).
Term

 

 

Given a graph of  f ', how do you locate inflection points of  ?

Definition

Look for where f ' has relative extrema; has an inflection point where f ' has has a peak or a valley.


 

Term
Find the equation of the line tangent to the graph of f(x) at x = a.
Definition
You need a point and a slope.
The point is (a,f(a)).
The slope is f'(a).
Term
Average rate of change
Definition
[image]
Term

 

Given a graph of  f ', what information

can be determined about f ?

Definition

---Net changes in f can be computed by finding net signed areas between f '.


-- f  has relative extrema where f ' changes sign.


- f  has inflection points where  f ' has extrema.

Term

The derivative of this function

[image]

is positive/negative  and  increasing/decreasing

Definition

 

 

Positive (because it is increasing)

and

increasing ( because it curves up)

Term

The derivative of this function

[image]

is positive/negative  and  increasing/decreasing

Definition

 

Negative (because it is decreasing)

and

positive (because its slope going from largely negative to less negative)

Term

The derivative of this function

[image]

is positive/negative and increasing/decreasing

Definition

 

Negative (it is decreasing)

and

decreasing (slope getting more negative)

Term

The derivative of this function

[image]

 

is positive/negative  and  increasing/decreasing

Definition

 

Positive (it is increasing)

and

decreasing (it is becoming less steep)

Term

 

 

What are the conditions for a function to be continuous at x = a ?

Definition
[image]
Term

Why is this function

[image]

 

discontinuous at x = a?

Definition

 

 

(a) does not exist

Term

Why is this function

 

[image]

discontinuous at x = a?

Definition

 

 [image]


left- and right-handed limits disagree

Term

 

[image]

Definition

 

1


The graphs of  y = x  and  y = sin x  are indistinguistable near the origin, so their

ratio is close to 1.

Term

 

[image]

Definition

 

 

The average value of  (x) on [a, b]

Term

 

[image]

Definition

 

= f (x)

 

The derivative of a function defined by an integral is the function being integrated.

Term

How is 

[image]

 

 evaluated without a graph?

Definition


F(b) - F(a),

where F is any any antiderivative of  .

 

This is one part of the Fundamental Theorem of Calculus

 

Term

 

[image]

Definition


(p(x))· p'(x)

 

 

Chain rule is needed.  The outside function is the integral.  The inside function is p(x).

Term

 

[image]

Definition

0

 

It can be derived from the special trig

limit for sine.

Term

 

Definitions of a

derivative as a limit

 

 f ' (x) =

Definition
[image]
Term

 

 

Under what conditions will a function

not be differentiable at a point?

Definition

 

Vertical Tangents

 

Corners

 

Discontinuities

Term

 

 

Volume of a solid using

a definite integral

Definition

[image], where A(x) is area of the cross sections of the solid as a function of xA(x) could be the area of a square, annulus, semicircle, etc.

Term

What function has differential equation:

 

[image]

Definition

The rate of change of y is proportional to y itself.

 

[image]

 

Separate variables, integrate, and plug in the initial condition.

Term

Where is increasing?

[image]

Definition

(-,c)  U  (e, ∞)


 

A function is increasing wherever its derivative is positive.

Term

Where is f concave up?

[image]

Definition

(a, b)  U  (d, )

 

A function is concave up wherever

its derivative is increasing.

Term

Does  f   have relative extrema?

[image]

Definition

f  has a relative maximum at x = c

and

a relative minimum at x = e.

 

A max occurs where f ' switches from + to -

and

a min occurs where f ' switches from - to +.

Term

Where is f concave up?

[image]

 

Definition

(-∞, c)  U  (e, )


f  is concave up where its second derivative is positive (because it means its first derivative is increasoing)

Term

Where is f increasing?

[image]

Definition

Not enough information.

 

The only thing that can be determined is whether

f '  is increasing or decreasing.  This says nothing about the values of   f , only how fast it is changing.

Term

 

A function is linear if its derivative is _____.

Definition

 

constant.

 

 

A constant derivative means the

function has a constant slope.

Term

Suppose v(t) is a velocity function of a particle.  What is

[image]   ?

 

Definition

 

The displacement (or change in position) of the particle from t = a to t = b.  It is the net signed area between v(t) to the t-axis.

Term

Suppose v(t) is a velocity function of a particle.  What is

[image]   ?

Definition

 

The total distance traveled by the particle from

t = a  to  t = bIt is the total area between v(t) and the t-axis.  It does not tell the final position of the particle- only the distance it went.

Term

 

What are the hypotheses of the

Mean Value Theorem?

Definition

 

The interval must be closed and the

function must be differentiable everywhere between the endpoints.

Term

True or False?

 

If f '(a) = 0, then   f  has a relative extremum

at x = a.

Definition

False.

 

If f '(a) = 0, then   f  has a horizontal tangent

at x = a; not necessarily an extremum.

Term

 

If  f '(a) = 0  and  f ''(a) > 0, then   f

has a  _________ at x = a.

Definition

Relative minimum.

f '(a) = 0  means f  has a horizontal tangent at

x = a, and  f ''(a) > 0 means f is concave up at

x = a.  Hence the relative minimum.  This is the Second Derivative Test.

Term

 

If f '(a) = 0  and  f ''(a) < 0, then f

has a ____________ at x = a.

Definition

Relative maximum.

f '(a) = 0  means f  has a horizontal tangent at

x = a, and  f ''(a) < 0 means f  is concave down at

x = a.  Hence the relative maximum.  This is the Second Derivative Test.

Term

 

If  f '(a) = 0  and  f ''(a) = 0, then f

has a ____________ at x = a.

Definition

Cannot be determined.

f   has a horizontal tangent at x = a, but since

f ''(a)=0, its concavity at that point cannot be determined.  Hence the Second Derivative Test cannot be used to determine what f  looks like at

x = a.

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