Test II
Math 261, Section 1
Instructor: Sam Watson
February 25, 2009
Name:
Please solve the following problems in the space provided. You may use a scientific calculator, but you will not need one. You must show work to support your answer.
1. (10 points) Calculate the following limits.
(a)
(b)
(c)
(d)
(e)
lim
x−2
x 4 (3 − x )
x−2
4
x (3 − x )
x−2
4
x (3 − x )
x →3+
lim
x →3−
lim
x →0+
lim
x−2
4
x (3 − x )
x−2
4
x (3 − x )
x →0−
lim
x →−1
2. (10 points) What are the vertical asymptotes of the graph of f ( x ) =
x2 − 9
?
x ( x + 1)( x − 3)
3. (12 points) Find all values x0 in the interval [−2, ∞) for which the function f ( x )
sketched below is not continuous at x0 . For each x0 you list, write the type of discontinuity (e.g. jump, infinite, oscillatory, or removable).
6
5
4
3
2
1
−2
−1
0
1
√
2
3
4
5
6
7
x−5
has a removable discontinuity at x = 25. If we
x − 25
want to extend the function f so that it is continuous at x = 25, then what value should
be assigned to f (25)?
4. (5 points) The function f ( x ) =
5. (15 points) For what values of x are the following functions continuous? (You may
express your answer in the form “all real numbers except...” when it is convenient).
(a)
tan(πx )
x
(b)
x+1
x2 − 2x
(c)
| x + 2| + sin(5x )
(d)
x+
(e)
√
1
x
4x + 1
6.
√ (12 points) (a) Using the definition of a derivative, calculate the derivative of f ( x ) =
2x.
(b) Using
your answer to part (a), find the equation of a line tangent to the graph of
√
y = 2x at (32, 8).
7. (16 points) Find the first and second derivatives of the following functions
(a)
y = 8x3 − 6x + 1
(b)
y=
5
+x
x10
(c)
y=
x−3
x+5
(d)
4x2 −
y=
x
√
x
x
has a horizontal
8. (8 points) Find the two values of x0 for which the graph of y = 2
x +1
tangent at x = x0 .
9. (8 points) Suppose f and g are functions whose domains include 0, and suppose that
f (0) = 2, f 0 (0) = −1, g(0) = −4, and g0 (0) = 1/2. If we define the function h( x ) =
f ( x ) g( x ), then what is h0 (0)?
10. (4 points) Using the definition of a derivative, find the derivative of the function sin( x )
at x = 0.