BC 1-2
Quiz #2
Name:_________________
No Calculator
Skills:
1.
Find each limit. No work required. Be Careful..
a.
lim
x3
lim
x5
x5
x5
c.
e.
lim
x
lim  2 
 1
5x  6
3x  2
Find each limit algebraically. Clearly show how you arrived at your answers. An answer without
any algebraic justification will receive little or no credit.
lim
a.
b.
3.
b.
 sin( x  2) 
lim 
x2  2 x  4 
d.
2.
x4
x 3
z3  z 2  2 z
z 1
z 2  5z  6
lim
x3 2
x 1
x1
Find the derivative of each function.
a.
b.
f ( x)  3x 2  2  x 2  2 
g ( x)  x  e x
2
BC 1-2
4.
Quiz #2
Name:_________________
15
for x  [1, 7] and y(1) = 9. Using a step size of 2, construct the graph of the
x
piecewise-defined function for yapprox using Euler’s method making sure to mark and label your
Suppose y  7 
endpoints clearly.
y
6
3
2
4
-3
-6
5.
Find the derivative of f ( x) 
1
using the limit definition of the derivative.
x x
2
6
x
BC 1-2
6.
Quiz #2
Name:_________________
The graph of f is shown at right. Evaluate the following limits (estimate as closely as possible
when necessary). Write DNE if the limit does not exist.
f ( x))
a. Lim(

x 0
y = f(x)
b. Lim( f ( x))
x 0
For c. and d. give a brief
explanation for your answer:
 f (4  h)  f (4) 
c. Lim 

h 0
h


 f ( x)  4 
d. Lim 

x 3
 x3 
Concepts:
7.
Find all points P on the graph of f ( x)  x3  3x 2 such that the tangent line to f at P goes through
the point (0,1).
BC 1-2
Quiz #2
Name:_________________
8.
Is there a value c for which the function y  sin x  c, x  [0, 2 ] is tangent to y  ln x . If so, how
1
many different values? You may use the fact that the derivative of y  ln x is y  and the
x

derivative of y  sin x is y  cos x . Note: two functions f and g are tangent at x  a if and only if
f (a)  g (a) and f (a)  g (a) .
9.
Let f :[0,1) 
be a function defined as follows: for x  [0,1) , let

an
 10
n 1
n
 .a1a2 a3
be the
decimal expansion for x without repeating 9’s. For example, write .3479 as .348. Now define f by

a
1
f ( x)   n2 n . Find f    .
3
n 1 10
BC 1-2
5.
Quiz #2
Determine a value of the constant k so that the following limit exists.
lim
x
24 x  6
2kx  2
Name:_________________