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. Do Not Simplify the Result
a.
 
  x   sin 6 ( x)  cos x6  ln

2x 1

(x) =
b.
 
g  x   1  cosh e x
g(x) =
3
c.
ex
h  x 
sec x
h(x) =
2.
Suppose q  x     g  x  . If (1) = –3, (1) = 4, (2) = –2, (2) = 1, g(1) = 2, g(1) = –4,
g(2) = 1, and g(2) = –3, what is the value of q(2)?
BC 1-2
Quiz #3
Graph of f
3.
Suppose  is the function shown at
the right.
a.

Is g  x   2 x  f x 2  3
12

8
increasing or decreasing at
x  1 ? Justify your answer
clearly and completely.
Note that the graph of f has:
A zero at x  3.4;
Local maxs and mins at
x  2, x  1, x  5.
b.
Name:_________________
4
2
1
1
2
3
4
5
4
8


Continuing with the same graph of y  f  x  from above, if g  x   2 x  f x 2  3 , is g  x 
concave up or concave down at x = 1?
Justify your answer clearly and completely.
4.
Suppose that f is an increasing function (i.e., f 1 exists) and that g ( x)  f 1 ( x3 ) . Find g ( x) in
terms of f , g , and f  .
BC 1-2
5.
Quiz #3
Name:_________________
Consider the curve defined by the equation 6 y 2  2 x 2 y  x 2  0 .
dy
.
dx
a.
Find and simplify
b.
Find all point(s) on the curve where the tangent line to the curve is vertical.
BC 1-2
Quiz #3
Name:_________________
Concepts:
1
Let k ( x)  tan 1  x   tan 1  
 x
6.
a. Differentiate k ( x) and simplify completely.
b. Sketch the graph of y  k ( x ) . [Accurately!]
7.
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
Quiz #3
Name:_________________
8. Suppose that f is a function which is defined on all the reals and is differentiable everywhere.
a.
If possible, evaluate lim
h 0
f ( x  h )  f ( x)
.
h
b. Suppose that g is a continuous function with lim  g (h)  0 . What conditions on g might
h0
f ( x  g (h))  f ( x)
?
guarantee that i. lim
h 0
h
[You need not prove your assertions, but justify any conjectures as fully as you are able.]
2
2
9. Hypocycloid. Show that every tangent line to the curve x 3  y 3  1 in first quadrant has the property
that that portion of the line in the first quadrant has length 1.