Unit Three: Maxs and Mins Flashcards

1
Q

Definition of extrema

A

Let f be defined on an interval I containing c

1) F(c) is a minimum of f in I if f(c) is less than or equal to f(x) for all x in I
2) F(c) is a maximum of f in I if f(c) is greater than or equal to f(x) for all x in I

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2
Q

The extreme value theroem

A

If f is continuous on [a,b] then f has both a max and a min on that interval (they can be the same number)

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3
Q

Critical numbers

A

Let f be defined at some x-value c. If f’(c) is zero, or if f is not differentiable at c, then c is a critical number

(Can help you find relative/absolute max’s and mins)

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4
Q

Rolle’s Theorem (MEMORIZE)

A

Let f be continuous on [a,b] and differentiable on the open interval (a,b). If f(a) equals f(b), then there is at least one number c in that open interval (a,b) such that f’(c) equals zero

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5
Q

The Mean Value Theorem (MEMORIZE)

A

If f in continuous on [a,b] and differentiable on (a,b), then there exists a c in (a,b) such that f’(c) equals (f(b)-f(a))/(b-a)

(Or the average slope)

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6
Q

How do you find the extrema on an interval?

A

Plug in the end points, then the critical numbers, use the biggest and the smallest as the max/min

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7
Q

How do you know when f is increasing, decreasing, or constant on an interval [a,b]?

A

If f’(x) is greater than zero (For all x in (a,b)), it is increasing.

If f’(x) is less than zero (For all x in (a,b)), it is decreasing.

If f’(x) is zero (For all x in (a,b)), it is constant.

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8
Q

How can you find the relative max’s and min’s for f?

A

If f’(x) changes from pos to neg at point c, (c,f(c)) is a relative max

If f’(x) changes from neg to pos at point c, (c,f(c)) is a relative min

If f’(x) changes from pos to pos or neg to neg at point c, (c,f(c)) is not a relative max or min

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9
Q

What is the meaning of concave up?

A

F is decreasing, then increasing

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10
Q

What is the meaning of concave down?

A

F is increasing, then decreasing

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11
Q

How do you determine if the graph of f is concave up or down?

A

It is up if f’ is increasing (f’’ is greater than zero), it is down if f’ is decreasing (f’’ is less than zero)

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12
Q

What is an inflection point?

A

If (c, f(c)) is a point of inflection of f, then either f’‘(c) is equal to zero or it does not exist at c. An inflection point is a “point” where f changes concavity.

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13
Q

What is the second derivative test? (will not have to use if you don’t want to)

A

Let f be a function such that f’(c) equals zero, and the second derivative of f exists on an interval containing v.

If f’‘(c) is greater than zero, f has a relative min at (c,f(c)).
If f’‘(c) is less than zero, f has a relative max at (c,f(c)).
If f’‘(c) is zero, f has neither.

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14
Q

How do you find y-intercepts?

A

X=0

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15
Q

How do you find x-intercepts?

A

Y=0, only pay attention to numerator

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16
Q

How do you find symmetry?

A

Y-axis is if f(-x)=f(x)

Origin is if f(-x)=-f(x)

17
Q

What is domain?

A

Possible x’s

18
Q

What is range?

A

Possible y’s

19
Q

How do you find continuity (and removable/non)?

A

Removable=hole (cancels out 0/0)

Non-removable=VA (does not cancel out, #/0)

20
Q

How do you find vertical asymptotes?

A

(#/0), set the denominator equal to zero

21
Q

How do you find horizontal asymptotes?

A

0 (degree of n less than degree of d)

DNE (degree of n greater than degree of d)

A number (ratio of leading coefficients if degree of n=degree of d)

22
Q

How do you find slant asymptotes and how do you know if there will be one?

A

If the degree of the numerator is one more than the degree of the denominator, you can find a slant asymptote by dividing the numerator by the denominator where the remainder does not count. (You will get Y=answer)

23
Q

What is newton’s method for approximating the zeros of a function?

A

Make an initial estimate (Xn), determine a new approximation using Xn+1=Xn-(f(Xn)/f’(Xn)), if |Xn-Xn+1| is within the desired accuracy you are done if not keep going with step two

24
Q

What is the equation for differentials?

A

dy=f’(x)*dx

25
Q

What is the change in y equation? (Differentials)

A

Change in y is f(x + change in x) - f(x)

26
Q

What equation do you use to approximate function values?

A

F(x+changeinx)=f(x)+f’(x)*dx