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MATH 105
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EXAM 1 FEBRUARY 12, 2009
NAME: ~
YOUR GRADE IS BASED ON CORRECTNESS, COMPLETENESS, AND CLARITY ON EACH EXERCISE. You MAY USE A CALCULATOR, BUT NO NOTES, BOOKS, OR OTHER STUDENTS. GOOD LUCK!
1.) (10 pts.) Is y = x + 1 a solution of the differential equation xy' + (y')2 - Y = 07 Justify
your answer..
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2.) (15 pts.) Use the graph of l' below to answer the following questions about f. [Note:
the graph of f is not shown.]
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a.) (5 pts.) For which intervals of x-values is f increasing?
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b.) (5 pts.) At which x-values does f have stationary points?
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c.) (5 pts.) For which intervals of x-values is f concave down?
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4.) (10 pts.) Use the (SAME!) graph of l' below to complete the following. [Note: the graph
of I is still not shown.] f'(x) 2 ~- - - - - -:- - - - - -. - - - - - -.- - - - - - ~ - - - - - -'- - - - ~ - - - - - -, - - - - -
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-2f1- - - - -:- - - - - -; - - - - - -'- - - - - - - -'- - - - - - -'- - - - - -'- - - - - - -'- a.) (5 pts.) Use the First Derivative Test to explain how this graph shows that I has a
local maximum at x = 6.
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b.) (5 pts.) Sketch a possiblegraph of If! on the axes below.
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5.) (20 pts.) Pediatricians frequently compare young children to standard growth charts as part of assessing health and progress. The table below comes from one such growth chart, compiled by the Centers for Disease Control and Prevention. The table shows median heights of girls from birth to two years of age. Age (months) Height (inches) a. ) (5 pts.) Is this a function? Explain.
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~fre l S^ ~ "i-.t>.. c.k l <r () /\ e 0 u.~~ \A.-t b.) (5 pts.) Zoom in numerically to find two different approximate growth rates (in inches/month): the rate for a 3-month-old girl and the rate for a 21-month-old girl.
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(v l D S]~ I;'\O'\e.~ "-'. ;n , I MO~ MO~~ c.) (5 pts.) At which age (from 0 to 24 months) do girls' growth rates appear to be fastest? What is the overall trend of growth rates as girls age from 0 to 24 months? Justify your answer using the values in the table. ~e. ā¬J>.rt\dt- ~ru ~V'f ~e k&'~st ~rou~) D-..I'\J 3fow1'h ~&.5 It.> _s-_ DI..U a..S ~\ rlr OJLT c\t)Jtr. ''b 2 t.t MVl the ~ro"" vaJ)u.ā¬J '1/\ t'tble', cJ...().J~.s _11_ ~ o.j'f oJL~) :3> MClI\W.,..., ~v.:- c..~J ;.1\ 'he', "D~+- a. f(', 5Mo-M ~ ~1we.E:f\ slAt.Le.fsi ve.. (0..,' d o-f. d.) (5 pts.) Based on your answer from part (c.), If we were to graph the data from thef^ aI Ai-} table, with height on the vertical aXis . and age on the horizontal axis, what kind Of
concavity should the graph have? Explain. -:If TtJo)e S~)WS , l.-..J/ r-~s of ~ro~ eve ~'(.lJ.I
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7.) (15 pts.) Suppose f(x) - # - 7X2+ 3.
a.) (5 pts.) Compute F(x), an antiderivative of f(x).
b.) (5 pts.) Compute f'(x).
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c.) (5 pts.) Compute f"(x).
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