BC 1-2
Problem Set #3
Spring 2013
Name
Due Date: Tuesday, 22 Feb. (at beginning of class)
Please show appropriate work – no big calculator leaps – except as indicated. Work should be shown
clearly, using correct mathematical notation. Please show enough work on all problems (unless
specified otherwise) so that others could follow your work and do a similar problem without help.
Collaboration is encouraged, but in the end, the work should be your own.
1.
Differentiate the following function. Don’t even think of simplifying or using the TI-89!
 e
f ( x)  cosh e
2.
x

   sin( x)ln
sec x 2
x  
2
 
sin 4 x3
x3

Suppose h  x   f x 2  2 x , f   0   3, f   2   3, f   4   5, f   6   7, f  8   2 . Determine the
following value, if possible, showing your work.
h  4  
3.
Suppose (1) = 0 and that  is the
function graphed at the right. If
G  x      x   , is G concave up
4
or concave down at x = –4?
Justify your answer.
Graph of 
BC 1-2
Problem Set #3
Spring 2013
4. The function f :

Name
Due Date: Tuesday, 22 Feb. (at beginning of class)
satisfies f  x 2  f ( x)  f ( x) f   x 2  for all x 
.
  f (1)
 .
If f (1)  1 and f (1) 8, find f (1)
5. Let h  x     g  x   . Use the graphs of f and g below to estimate the values of x where the graph of
h has stationary points. (That is, estimate all values of x such that h( x)  0 ).
y = f(x)
y = g(x)
BC 1-2
Problem Set #3
Spring 2013
Name
Due Date: Tuesday, 22 Feb. (at beginning of class)
6. Suppose that the function f :  has a first and second derivative for all x  . (We say f is
twice differentiable).
a. Give an example of such a function f such that f (a )  0 , but f has neither a local maximum nor
a local minimum at x  a . Give the function f and the value of a.
b. Give an example of such a function f such that f (a)  0 , but f does not have a point of
inflection at x  a . Give the function f and the value of a.
7.
Sketch a graph of the function g(x) over the closed interval [–5, 5] given that g satisfies the
following conditions. Clearly identify all local maximum points, local minimum points, and
inflection points on your graph. Neatness and accuracy count.

g(–5) = 4 and g(5) = –6

g(x) > 0 for –2 < x < 1 only

g(x) < 0 f for –5 ≤ x < –2, 1 < x < 3, and 3 < x ≤ 5 only

g(x) > 0 for –5 ≤ x < 0 and 2 < x < 3 only

g(x) < 0 for 0 < x < 2 and 3 < x ≤ 5 only
BC 1-2
Problem Set #3
Spring 2013
8.
Name
Due Date: Tuesday, 22 Feb. (at beginning of class)
The cost of fuel for a luxury liner is proportional to the square of the speed, v. At a speed of 10
km/hr, fuel costs $3000 per hour. Other fixed costs (e.g., labor) amount to $12,000 per hour.
Assuming that the ship is to make a trip of total distance d, find the speed that minimizes the cost
of the trip. Does this speed depend on the distance?
9.
a) If u, and v are twice differentiable functions of x, find the formula for the second derivative of
their product, Dx2 (uv )
b) Generalize your work in part (a) by guessing the formula for Dxn (uv ) — the nth derivative of
the product of (n – times) differentiable functions. Then prove your formula by mathematical
induction.