Flashcards in Chapter 4 Deck (21)

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1

## What is the definition of a absolute maximum(minimum)?

### Let c be a number in the domain D of a function f. Then f(c) is the absolute maximum(minimum) value of f on D if f(c) >/= (=) f(x) for all x in D.

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## What is the definition of a local maximum(minimum)?

### The number f(c) is a local maximum/minimum value of f if f(c) >/= (=) f(x) when x is near c.

3

## What is the Extreme Value Theorem?

### If f is continuous on a closed interval (a,b), then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in (a,b)

4

## What is Fermat's Theorem?

### If f has a local maximum or minimum at c, and if f'(c) exists, then f'(c)=0

5

## What is the definition of a critical number?

### A critical number of a function f is a number c in the domain of f such that either f'(c)=0 or f'(c) does not exist. If f has a local maximum or minimum at c, then c is a critical number of f

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##
What is the closed interval method for finding the absolute maximum and minimum values of a continuous function f on a closed interval (a,b)?

###
1.Find the values of f at the critical numbers of f in (a,b)

2.Find the values of f at the endpoints of the interval.

3.The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.

7

## What is Rolle's Theorem?

###
Let f be a function that satisfies the following three hypotheses:

1.f is continuous on the closed interval (a,b)

2.f is differentiable on the open interval (a,b)

3.f(a) =f(b)

Then there is a number c in (a,b) such that f'(c)=0

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## What is The Mean Value Theorem?

###
Let f be a function that satisfies the following hypotheses:

1.f is continuous on the closed interval (a,b)

2.f is differentiable on the open interval (a,b)

Then there is a number c in (a,b) such that f'(c)=(f(b)-f(a))/(b-a)

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## What is the Increasing/Decreasing Test?

###
(a) If f'(x)>0 on an interval, then f is increasing on that interval.

(b) If f'(x)<0 on an interval, then f is decreasing on that interval

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## What is the First Derivative Test?

###
Suppose that c is a critical number of a continuous function f.

(a) If f' changes from positive to negative at c, then f has a local maximum at c.

(b) If f'changes from negative to positive at c, then f has a local minimum at c.

(c) If f' is positive to the left and right of c, or negative to the left and right of c, then f has no local maximum or minimum at c.

11

## What is the definition of concave upward and concave downward?

### If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on I, it is called concave downward on I.

12

## What is the Concavity Test?

###
(a) If f"(x)>0 for all x in I, then the graph of f is concave upward on I.

(b) If f"(x)<0 for all x in I, then the graph of f is concave downward on I.

13

## What is the definition of an inflection point?

### A point P on a curve y=f(x) is called an inflection point if f is continous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P.

14

## What is the Second Derivative Test?

###
Suppose f" is continuous near c.

(a) If f'(c)=0 and f"(c)>0, then f has a local minimum at c.

(b) If f'(c)=0 and f"(c)<0, then f has a local maximum at c.

15

## What is L'Hopsitals's Rule?

###
Suppose f and g are differentiable and g'(x) /=/ 0 on an open interval I that contains a (except possibly at a). Suppose that

limf(x) as x->a =0, and limg(x) as x->a =0

OR

limf(x) as x->a =+/-infinity, and limg(x) as x->a =+/-infinity

(In other words, we have an indeterminate form of type 0/0 or infinity/infinity)

Then lim f(x)/g(x) as x->a =limf'(x)/g'(x) as x->a

16

## How do you find a Vertical and Horizontal Asymptote?

###
A horizontal asymptote is a limit where as x approaches +/- infinity, f(x) gets closer to L.

A vertical Asymptote is a limit where as x approaches a number, f(x) = +/- infinity

17

## What is the circumference and area of a circle?

###
Circumference=2pi*r

Area=pi*r^2

18

## How do you define the linearization of f(x) at x=a?

### L(x)=f(a)+f'(a)(x-a)

19

## What is Newton's Method for approximating a zero?

###
X(n+1)=X(n)-f(Xn)/f'(Xn)

For Newtons Method, must create/be a function that =0 and solve for x

20

## How do we find the differential dy of an equation y(t)=...?

### You find the derivative and answer derivative(dt)

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