CHAPTER FOUR MEMORY ITEMS
Short Answer
1. Name the three types of critical points.
(1)
(2)
(3)
2. If f '(c) = 0 then (c,f(c)) is called a ________________________________ point.
3. If f ' (c) is undefined but f(c) = a, then (c,a) is called a _________________________point.
4. What two properties must f '(c) have if f(c) is an extrema?
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5. What two properties must f '' (x) have in order for (c,f(c)) to be an inflection point?
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6. If f ' (x) > 0 then f(x) is ______________________________________.
7. If f ' (x) < 0 then f(x) is ______________________________________.
8. If f '' (x) > 0 then f(x) is ______________________________________.
9. If f '' (x) < 0 then f(x) is ______________________________________.
10. What is the difference between a relative (or local) maximum and an absolute (or global) maximum?
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11. What is the difference between a relative (or local) minimum and an absolute (or global) minimum?
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12. What is the First Derivative Test?
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13. What is the Second Derivative Test ?
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14. What occurs on the graph of f(x) at x=c if both f ' (c) and f '' (c) equal zero?
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15. Besides points where f ' (c) = 0 where else may extrema occur?
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16. profit = __________________________________________
17. What is the difference between p(x) and P(x) ?
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18. cost = C(x) = _____________________________+ _________________________* _____
19. In cost/profit problems, what does x stand for?
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20. revenue = R(x) = x * _______________________________________
21. When do we suspect a vertical asymptote?
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22. How do we verify that there is a vertical asymptote at x=a ?
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23. How do we verify that there is a horizontal asymptote at y=b ?
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24. What's the quickest way of evaluating limits when x
infinity ?
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25. What are the two requirement for f(x) in order for the Mean Value Theorem to apply ?
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26. What is the conclusion of the Mean Value Theorem to apply ?
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27. How do we compute the average rate of change in f(x) (Also known as the average value of the rate of
change) ?
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28. If f ' (x) = g ' (x) for all x, what is the relationship between f(x) and g(x) ?
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CHAPTER FOUR MEMORY ITEMS
Answer Section
SHORT ANSWER
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endpts, stationary pts, singular pts
stationary
singular
zero or undefined AND sign change or endpoint
zero (or undefined) and change signs at x=c
increasing
decreasing
concave up
concave down
relative (local) is only higher than neighboring points wheras absolute (global) is the highest of all
points
relative (local) is only lower than neighboring points wheras absolute (global) is the lowest of all points
If f ' changes from + to - at x=c then f(c) is a max
If f ' changes from - to + at x=c then f(c) is a min
Given f ' (c) = 0 if f '' (c) > 0 then f(c) is a MIN, if f '' (c) < 0 then f(c) is a MAX, if
f '' (c) = 0 then the test fails to determine anything
Could be either a min, max, or an inflection point.
at endpoints OR wenever f ' (c) is undefined but f(c) is defined
revenue - cost
p(x) is the price, P(x) is the profit
fixed cost + price per item * x
the number of items bought or sold
price of one item
Where denominator = zero
Show that there is an infinite limit as x
a
Show that there is a numerical limit of b as x
infinity
Compare leading terms.
f is continuous on [a,b] and differentiable on (a,b)
f ' (c) = (f(b) - f(a)) / (b-a) for at least one c value between a and b.
f(b) - f(a)
----------b-a
f(x) = g(x) + c or f(x) - g(x) = c