Math 180 sample problems for Hour Exam Two
1. Differentiate the following functions:
(a) x2ln x,
(b) sin(a+bx),
(c) arctan(3x).
2. Differentiate the following functions:
(a) e2−x2,
(b) xcos(x),
(c) arcsin(x/2).
3. Let y=f(x) be the function defined implicitly by y3−y+x= 0 and f(6) = −2.
(a) Find dy
dx at the point (6,−2).
(b) Find the equation of the tangent line at (6,−2).
4. Use the information in the table about fand gto find:
(a) h0(0), where h(x) = f(g(x)).
(b) k0(2), where k(x) = f(x)g(x).
x f(x)f0(x)g(x)g0(x)
0 1 −1 2 5
1−1 2 4 3
2 7 3 1 4
5. Find the critical points of the function f(x) = x3+ 3x2−9x−11 and find the global minimum of f(x)
on the interval −4≤x≤3.
6. Find the x- and y-coordinates of all local maxima, local minima, and inflection points of f(x) = x3−
3x+ 2.
7. Find lim
x→0
1−ex
x−x2.
8. You wish to enclose a 400 square-foot rectangular garden with shrubs costing $40 per foot on three sides
and a wall costing $20 per foot on the fourth side. Find the dimensions that minimize the total cost.
9. Find the largest interval on which f(x) = (x2+ 1)e−xis concave down.
10. The graph below is of the derivative f0(x) on the interval (−0.5,5.5). Determine the intervals on which
the original function fis:
(a) increasing,
(b) decreasing,
(c) concave up,
(d) concave down.
(e) Give one value of xat which fhas a local maximum.
012345
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y=f0(x)