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Example Problem. #6, Lesson 4.4
 . Given
the function at the left, produce graphs of f that reveal all the
important aspects of the curve. In particular, you should use graphs of
f ' and f '' to estimate
theintervals of increase and decrease, extreme values, intervals of
concavity, and inflection points. |
Given Problem. #6, Lesson 4.4 |
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We calculate the first derivative. |
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We calculate the second derivative. |
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This is the graph of the function f(x). We will use the graphs of f '(x) and f ''(x) below to compare with the results we see here. Take the time to visually check the observations below with the graph of the function at the left. |
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This is the graph of f '(x). Notice several things: 1) Where the this graph crosses the x-axis, we should have relative maxima or minima on the graph of the function. These appear to be at about: .2, 1.1, 2.1, 2.9. 2) Where this graph is above the x-axis,
our original function should be increasing, and where it is negative,
we should have the original function decreasing, 3) Where this graph's tangent line slope is zero, we should have inflection points on the original function. These points appear to be at about x = .8 and x = 2.4. |
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This is the graph of f ''(x). Notice that: 1) The positions where this graph crosses the x-axis should be points of inflection on the orginal function. These appear to be at about x = .8 and x = 2.4. Notice how this is confirmed in observation #3 above. 2) Where this graph is negative, the orginal function
should be "concave down", and where it is positive, the original
function should be "concave up". |
, 
,
, 
, 
, (1.5, 2.4)
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1-21, Every Other Odd
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