BC 1-2
Quiz #3
Show all appropriate work clearly for full credit.
Name:_________________
NO CALCULATORS
Skills:
1.
For each function below, find its first derivative. You need not Simplify the Result.
a.
  x   tan 1 (sin x)
(x) =
b.
g  x   x cos( x 2  1)
g(x) =
2.
3.
Suppose q  x  
f ( x)
. If f (1)  2, g (1)  1, f (1)  4, and g (1)  3 , what is the value of q 1 ?
g ( x)
Suppose r ( x)  g (2 x)  f  x  ln( x)  .
If g (1)  1, g (2)  3, g (1)  2, g (2)  3, f (1)  4, f (2)  3, f (1)  5, and f (2)  0 , find the value
of r  1 .
BC 1-2
4.
5.
Quiz #3
Name:_________________
Consider the curve defined by the equation 2 x 2  xy  y 2  14 .
dy
.
dx
a.
Find and simplify
b.
Find all point(s) on the curve where the tangent line to the curve is vertical.
Suppose g ( x)  f (sin 1 ( x)) . Given the information about  , , and f´´ provided in the table.
x
0
π/6
π/4
π/3
f  x
5
–1
14
35
f  x
–2
8
4
12
 2
1
Find the values of g   and g  
 . Show work/thinking.
2
 2 
f ( x)
-5
6
16
3
BC 1-2
Quiz #3
6. Suppose  is the function shown at the right.
a.
Name:_________________
Graph of f
 
If p  x   x  f x 2 , is p  x  increasing or
decreasing at x  1 ? Justify your answer
clearly and completely.
b.
 
If p  x   x  f x 2 , is p  x  concave up or concave down at x  1 ? Justify your answer clearly
and completely. You may assume that the graph of f has points of inflection at x  0.8 and x  1.5 .
 e x  e4 
7. Evaluate the limit lim 
 by thinking of it as the derivative of some function f at some fixed x
x2
 x  2 
value, x  a .
2
BC 1-2
Quiz #3
Name:_________________
Concepts:
8. We know that
the domain of
that f ( x ) 
d
1
(ln( x))  . Note that the domain of the natural log is all positive real numbers, but
dx
x
1
is all real numbers except 0? Is there a function f with domain (,0)  (0, ) such
x
1
for all x  (,0)  (0, ) ? Explain.
x
9. Suppose that h is defined on all the reals, and that g ( x)  h( x)cos( x) . If g has a terrace point at


x  , what can you conclude about the behavior of h at x  .
2
2
BC 1-2
Quiz #3
Name:_________________
10. Let f ( x)  sin( x), where x  [0, 2 ] . Describe the set of points (a, b) in the plane with the property
that there is exactly one tangent line to f that goes through (a, b) .