Station #1:

1.  Extrema on a closed interval: Find the absolute extrema of f(x) = x(x2 – 12) on the interval [0, 3].

2.  Global extrema: Use the first derivative test to find any local extrema of h(t) = ¼t4 – 8t.

3.  What is the difference between absolute extrema and local extrema?

Station #2:

Given f(x) = x2(x-3), find the following:

a)  f ‘(x) [product rule!]

b)  Extrema (classify as minima or maxima; absolute or local)

c)  Where is f(x) decreasing?

d)  f “(x)

e)  Points of inflection

f)  Where is f(x) concave up?

g)  Use this information to sketch the graph of f(x).

Station #3:

1.  Find the critical numbers of fx=3x2/3-2x.

2.  Find the absolute extrema for y=x2(x-3) on the closed interval [0, 4].

3.  Use the first derivative test to find and classify extrema of gx=x+1x. Determine the open intervals on which the function is increasing and decreasing.

Station #4 (optional):

1.  For what values of x is the function fx=-x+4-1 not differentiable? Why?

2.  Differentiate each of the following:

y=x-43x+2 fx=x(2x-5)2 y=sin2(4x)

3.  Write the equation of the tangent line to y=2x+1 at (0, 2).

Station #5:

1.  The derivative of a function f is given for all x by f ‘(x) = x2(x + 1)3(x – 4)2. The set of x values for which f has a relative maximum is:

a)  {0, -1, 4} b) {-1} c) {0, 4} d) {1} e) none of these

2.  The value of c for which fx=x+cx has a local minimum at x =3 is

a)  -9 b) -6 c) -3 d) 6 e) 9

3.  At how many points on the interval [0,π] does fx=2sinx+sin4x satisfy the Mean Value Theorem? (calculator active)

a)  None b) 1 c) 2 d) 3 e) 4

4.  The polynomial function f has selected values of its second derivative f’’ given in the table below. Which of the following statements must be true?

x / 0 / 1 / 2 / 3
f’’(x) / 5 / 0 / -7 / 4

a)  f is increasing on the interval (0, 2)

b)  f is decreasing on the interval (0, 2)

c)  f has a local maximum at x = 1

d)  The graph of f has a point of inflection at x =1

e)  The graph of f changes concavity in the interval (0, 2)

5.  The function f is continuous for -2≤x≤2 and f-2=f2=0. If there is no c, where -2<c<2, for which f'c=0, which of the following statements must be true?

a)  For -2<k<2, f'k>0

b)  For -2<k<2, f'k0

c)  For -2<k<2, f'k exists

d)  For -2<k<2, f'k exists, but f' is not continuous

e)  For some k, where -2<k<2, f'(k) does not exist