AP Calculus Test #2: Graphs and Limits
Name: ________________
Section A: Fill in the Blank (no working required in questions worth 1 mark)
8
3x 2  2x  1
x2  4
a) has vertical asymptotes x= ________
1. A function with equation f (x) 
(1)
b) has an horizontal asymptote whose equation is y= __________ (1)
x 2  4x  3
=
x3 x 3  3x 2
2. Find: lim
(2)
3. Consider the graph shown below. A possible equation is y= ______________
(3)
25
20
15
10
5
-10
-5
5
10
15
20
-5
-10
-15
-20
-25
-30
-35
4. On the graph shown above, estimate the slope of the tangent at x=0. (1)
dy
 ______________
dx
Section B- Two Questions- Full solution Required- #2 on next page
16
x3
. Include any intercepts, asymptotes, local
x2  5
max/min values, providing algebraic justification.
(8)
1. Graph the function y 
2. The function p(x) is continuous and differentiable. It is given by the
equation:
16a
, where x  4
x
p(x)  bx 2  2x  56, where x  4
a) Prove that a  and b 
(4)
b) Use the values a  and b 
. Find the value(s) of x in the interval
[0,16] at which the instantaneous slope equals the average slope.
(4)
p(x) 
Section B – Multiple Choice- Only write solutions if worth >1 mark)
12
1. If f (x)  | x |, which of the following statements is true: (1)
i) f (x) is an even function ii) it is continuous at x  0 iii) it is differentiable at x  0
a) i only
b) ii only
c) i,ii only
d) ii, iii only
e) i,ii, and iii
2x  k
a
(x  1)2
has a horizontal tangent at the point (0,6), then the PRODUCT of the values of k
and a is: a) 7
b) - 7
c) 0
d) -1
e) 1
(4)
2. Determine the value of k and a such that the function g(x) 
ax 2  b
and it is know that f (x) has
xc
an x intercept at x=1, a verticle asymptote of x= -2 and a y intercept at (0,3). The
equation of the slant asymptote is:
(3)
3. Given that a function is defined by f (x) 
a) y= -x
b) y = - 6x
c) y= -x + 6
d) y = -6x +12
e) y= 6
4. At which of the points on the graph shown below of y  f (x) is f (x)  f (x) ?
a) A only
b) B only
c) C only
d) A and B
e) A and C
(1)
5. Let f (x) be a continuous and differentiable function. If g(x) 
the value of g (1) , using the table shown below.
Take Home Portion:
1. Consider the graphs of two functions shown below:
f(x)
g(x)
1
, find
f (x)
(3)
10
8
4
6
2
4
-10
-5
5
10
2
-2
-10
-5
5
10
-4
-2
-6
-4
-8
-6
-10
-8
-12
-10
f (x)
g(h)  g(0)
f (h)g(h)  f (0)g(0)
b) lim
c) lim
h0
h0
g(x)
h
h
f (h) f (0)

g(x)
g( f (h))  g( f (0))
g(h) g(0)
d) lim
e) lim
f) lim
h0
x 3 f (x)
h0
h
h
f (x)
g) State the value(s) of x for which the function y 
is discontinuous in the
g(x)
interval [-4,10]
h) Find the closest integer value of x on the graph of g(x) which satisfies the mean
value theorem in the interval [0,9]
a) lim
x
g(x)
as accurately as possible, where
f (x)
f (x), g(x) are the functions defined in question #1. Assume that the domain
required is all real numbers. Include any asymptotes. You need not find the
equation of any oblique asymptotes.
2. Draw a sketch of the graph of y 
k
, where k is a constant.
x
If B is the y intercept of the tangent drawn at any point on the curve and A is the
3. a) Consider any tangent line drawn to the curve y 
x intercept of the same tangent, prove that the area of triangle ABO, where O is
the origin is always the same area.
b) Find the y value of the point of tangency to the curve if B is the point (0, 10),
where B is, as before, the y intercept of the tangent line.