Flashcards in Math 150 Chapter 4.1 - 5.3 Deck (49)

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1

## Define absolute maximum

###
A function f has an absolute maximum at c if

f(c) ≥ f(x) for all x ∈ D, the domain of f. The number f(c) is called the maximum value of f on D.

2

## Define absolute minimum

###
A function f has an absolute minimum at c if

f(c) ≤ f(x) for all x ∈ D, the domain of f. The number f(c) is called the minimum value of f on D.

3

## Define extreme values.

### Absolute minimum and absolute maximum.

4

## Define local maximum

###
A function f has a local maximum at c if

f(c) ≥ f(x) for all x in an open interval in the domain, containing c.

5

## Define local minimum

###
A function f has a local minimum at c if

f(c) ≤ f(x) for all x in an open interval in the domain containing c.

6

## Define 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,d ∈ [a,b].

7

## Define fermat's theorem

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

8

## Define 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.

9

## Define the closed interval method (3 steps)

###
To find the absolute maximum and absolute minimum values of a continuous function f on a closed interval [a,b]:

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

B) Find the values of f at the endpoints of the interval

C) The largest of the values from step (a) and (b) is the absolute maximum value, the smallest of these values is the absolute minimum value

10

## Define Rolle's theorem (3 steps)

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

A) f is continuous on the closed interval [a,b]

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

C) f(a) = f(b)

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

11

## Define the mean value theorem (2 steps)

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

A) f is continuous on the closed interval [a,b].

B) 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) or equivalently

f(b)-f(a) = f'(c)[b-a]

12

## If f'(x) = 0 for all x in an interval (a,b), then what?

### If f'(x) = 0 for all x in an interval (a,b), then f is constant on (a,b).

13

## Prove that "If f'(x) = 0 for all x in an interval (a,b), then f is constant on (a,b)."

###
Suppose f'(x) = 0 for all x ∈ (a,b). Let x1 and x2 ∈ (a,b).

We have f is continuous on [x1,x2] and f is differentiable on (x1, x2). By the mean value theorem, there is a c ∈ (x1, x2) such that f(x2) - f(x1) = f'(c)[x2-x1] → f(x2) = f(x1). therefore f is constant.

14

## What does arcsin[(x-1)÷(x+1)] equal to?

### arcsin[(x-1)÷(x+1)] = 2arctan(√x) - (π/2)

15

## Prove that arcsin[(x-1)÷(x+1)] = 2arctan(√x) - (π/2)

###
f'(x) = g'(x) then f(x) = g(x) + c

f'(x) = 1÷[√(1-((x-1)/(x+1))^2)] × (d/dx(x-1)/(x+1)) = 1÷[√(1-((x-1)/(x+1))^2)] × (x+1-(x-1)) ÷ (x+1)^2 = 2÷√([(x+1)^2-(x-1)^2]÷(x+1)^2) × 1/(x+1)^2 = 1÷[(√x)(x+1)]

g'(x) = 2÷(1+(√x)^2) × 1/(2(√x)) = 1÷[(√x)(x+1)]

To find c plug in x=0

f(0) = - (π/2)

g(0) = 2arctan(0) = 0

f(0) = g(0) +c

- (π/2) = 0 + c

c= - (π/2)

16

## Define increasing/decreasing test (2 steps)

###
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

17

## If f'(x) = g'(x) for all x in an interval (a,b), then what?

### If f'(x) = g'(x) for all x in an interval (a,b), then f - g is constant on (a,b) that is f(x) = g(x) + c where c is a constant.

18

## Apply f(x) = g(x) + c to "h(x) = f(x) - g(x)

### h'(x) = f'(x) - g'(x) = 0 → h is constant → f(x) = g(x) + c

19

## Define the first derivative test (3 steps)

###
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' does not change at c, then f has no local minimum or maximum at c.

20

## Define concave upward

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

21

## Define concave downward

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

22

## Define concavity test. (2 steps)

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

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

23

## Define inflection point

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

24

## Define the second derivative test (2 steps)

###
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.

25

## Define L'Hospital's rule

###
Suppose that f and g are differentiable and g'(x) ≠ 0 near a (except possibly at a.) suppose that

lim┬(x→a)〖f(x) 〗= 0 and lim┬(x→a)〖g(x) 〗= 0

OR

lim┬(x→a)〖f(x) 〗= ±∞ and lim┬(x→a)〖g(x) 〗= ±∞

Then, lim┬(x→a)〖f(x)/g(x) 〗= lim┬(x→a)〖f'(x)/g'(x) 〗

if the limit on the right side exists ( or is ∞ or -∞)

26

## Define parametric curve

### The set of points C = {(f(t), g(t)) : t ∈ I}

27

## Define parameter

### The variable t from C = {(f(t), g(t)) : t ∈ I}

28

## Define parametric equations

### We say that the curve is defined by x = f(t), y = g(t)

29

## Define parametrization of C

### x = f(t) and y = g(t) is a parametrization of C

30