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4) Find the critical value(s) of

By Mario Grant,2014-06-22 08:50
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4) Find the critical value(s) of

    Survey of Calculus: Shumaker Review #3: 5.1 5.2, 5.4 5.6 The problems with *** are problems from 5.2 5.4

    411. *** Find the critical value(s) of . f(x)2xx

    262. *** Find the second derivative of . f(x)(3x4)

    3. *** Given the graph of , state the intervals where is concave downward. f ''(x)f

    4. *** Following is the graph of . State the intervals where the graph of . f(x)f '(x)0

     f(x)

5. *** Following is the graph of . Which one of the next four graphs is the correct matching graph f(x)

     of? f '(x)

     A) B)

    C) D)

    426. Use the Second Derivative Test to determine whether the local extrema of are f(x)x18x50

     maximums or minimums. State both the and coordinates of these extrema. If this test fails for yx

     a particular critical value, just state that fact; there is no need to revert back to the First Derivative Test

     for that critical value.

    32, , and the critical values are . x3, x0, x3f '(x)4x36xf ''(x)12x36

    4327. *** Using a sign chart, find all inflection points for . State both x- and y -f(x)x10x24x3x5

     coordinates.

8. *** Use a sign chart to solve this problem.

    A company estimates that it will sell units of a product after spending $ thousand on N(x)x

    32 advertising, as given by. N(x)2x90x750x2000 5x25

     When is the rate of change of sales decreasing? State your answer in interval notation.

    23x79. For the given function, f(x), algebraically find and state the following: 4x5

    a. State where the function is continuous in interval notation (state the domain).

     b. State the - intercept(s) as point(s). c. State the - intercept(s) as point(s). yx

     d. State the vertical asymptote(s). e. State the horizontal asymptote(s).

10. *** is continuous on . Use the given information to sketch the graph of . f(x)(?~,~)f

     x-3 -2 -1 0 2 3

    0 3 2 0 -3 0 f(x)

     + + + + +0 - - - - - 0 - - - - ND + + + + + f'(x)

     x

     -2 0 2

     - - - - - - - - - 0 + + 0 - - - - ND - - - - - - f"(x)

     x

     -1 0 2

    311. Find the absolute maximum for on the interval . (;3, 5f(x)x48x12

    16f(x)54x12. Find and state both the - and - coordinates of the absolute extrema of for . yxx0x

     If an absolute extrema exists, is this extrema an absolute maximum or an absolute minimum?

    2164x1632 and . f'(x)4f''(x)223xxx

13. *** Use the given information to sketch the graph of . Sketch the graph below. f

    , except ; Domain: All real xx2

     Behavior at point of discontinuity: ; ; lim f(x)~lim f(x)?~ x? 2x? 2

    End behavior: lim f(x)?~lim f(x)1x? ~x? ~

     - 4 0 4 6 x

    0 0 3 2 f(x)

     + + + + ND + + + 0 - - - - - - - - - - - - - - - f'(x)

     x

     -2 4

     + + + + ND - - - - - - - - 0 + + + + + + + f"(x)

     -2 6

    426x9xlim14. Find the indicated limit; this is used for end behavior. (Review from 3.2) 3x? ~2x7

15. *** Use the given graph of to find the intervals where f is increasing. yf '(x)

     f '(x)

     x

    31316. *** Find the second derivative for the following function. f(x)x24x

    6417. Find all local extrema using the second-derivative test. . Describe these f(x)x6x2

     extrema as either maximum or minimum, and state both the and coordinates of these yx

     extrema. If this test fails for a particular critical value, just state this fact; there is no need to revert back

    5342 to the first-derivative test for that critical value. f '(x)6x24xf ''(x)30x72x

     at ,:,:f(x)0x2.4379, 2.4379, 0.7804, 0.7804f '(x)0 at x2, 0, 2

     ,:f ''(x)0 at x1.5492, 0, 1.5492

    18. *** Given the graph of f (x), find the following information:

     f (x)

     x

    a. Identify intervals over which f (x) > 0.

    b. Identify the x-intercept(s).

    c. Identify the y-intercept(s).

    d. Identify the vertical asymptote(s).

    e. Identify the horizontal asymptote(s).

    f. Identify intervals over which the graph of f (x) is decreasing.

    g. Identify intervals over which f (x) > 0.

    h. Identify x coordinates of horizontal tangent lines.

    i. Identify x coordinates of where f ‘ (x) does not exist.

    j. Identify x and y coordinates of where f (x) has local minima and local maxima.

    k. Identify intervals over which the graph of f (x) is concave upward.

    l. Identify intervals over which f ‘‘ (x) < 0.

    m. Identify intervals over which the graph of f (x) would be decreasing.

    n. Identify x and y coordinates of where f (x) has inflection points.

    o. Identify x and y coordinates of where f (x) would have local extrema.

ANSWERS

    45?)221. x = {8} 2. 3. ;;;;;;;;;;;;;;;;~,3?1,6?6,~f ''36x53x46x3x436??(

    4. 5. B 6. local minimums at and ;;;;;;;;~,1?1,33,313,31

     local maximum at ;;0,50

    7. inflection points at and 8. 9. a. ;;;;;;;;;;1,234,1715,25~,54?54,~

    7?)?)?)7579. b. and c. d. e. no h.a. 0,,0,0x??????343((5(

10.

    11. absolute maximum: 12. absolute minimum: ;;;;3, 1292, 21

13.

    165314. 15. 16. ''6~;;;;3,1?1,4fxx3

17. local minimum: and ; test fails at x = 0 ;;;;2,302,30

    18. a. b. c. d. ;;;;;;;;;;;;;;;;1.5,1?1,3?9,~1.5,0; 1,0; 3,0; 9,00,1x4

    e. f. g. ;;;;;;;;;;;;;;;;y35,2?0,1?2,4~,5?2,0?1,2?4,7?7,~

    h. i. j. min at ; max at , , ,:,:;;;;;;;;x1,7x5,2,0,2,41,05,10,12,1

    k. l. m. ;;;;;;;;;;;;;;;;;;~,2?0,2?7,~2,0?2,4?4,72,0?2,4?4,7

n. o. ;;;;;;;;0, 1, 2, 1, 7, 2.57,2.5

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