Math 251, Winter 2009
Practice second midterm.
(1) Calculate the following limits. If the limits do not exist, state if they converge to โ
or to โโ. Otherwise if they do not exist, state that they do not exist.
(a)
lim
xโโ
x2
ex.
(b)
lim
xโ0+xln(x).
(c)
lim
xโ0
sin(3x)
tan(5x)
(d)
lim
xโ0
x
cos x+x
(2) Let f(t) = 4t
1+t2. Use derivatives to show where this function is increasing and where
it is decreasing. Also show where it is concave up and concave down. List all critical
points. List all points of inflection. Sketch the graph of the function from this data.
At what tis there a global maximum (if there is one) and at what tis there a global
minimum? Indicate clearly on your graph any horizontal or vertical asymptotes.
(3) Let f(x) = exโ3x. Find the minimum value of f(x) on the interval [โ1,5]. Either
give a precise answer, or if you give a decimal approximation, give six decimal places.
(4) The best size medium box for UPS shipping has to have dimensions subject to the
following restriction
l+ 2w+ 2h= 130
where lis the length, wis the width, and his the height. Assume h=w, and find
the land the wwhich give the largest volume subject to this restriction.
(5) Consider the ellipse x2
4+y2
9= 1. Use implicit differentiation to find the tangent line
to the ellipse when x=โ20/3. (Using the positive value of y.)
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