Test II
Math 261, Section 1
Instructor: Sam Watson
March 5, 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
x2 − 2
x ( x − 5)
x2 − 2
x ( x − 5)
x2 − 2
x ( x − 5)
x →5+
lim
x →5−
lim
x →0+
lim
x →0−
lim
x →1
x2 − 2
x ( x − 5)
x2 − 2
x ( x − 5)
2. (10 points) What are the vertical asymptotes of the graph of f ( x ) =
x2 − x − 2
?
x ( x − 2)(12 − x )
3. (10 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).
3
2
1
−2
−1
0
−1
1
2
3
4
5
6
7
−2
−3
−4
−5
−6
x2 − 81
has a removable discontinuity at x = 9. If we
x−9
want to extend the function f so that it is continuous at x = 9, then what value should be
assigned to f (9)?
4. (10 points) The function f ( x ) =
5. (16 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)
| x2 − 1| + cot(2x )
(b)
x + 2x − 3
x3 + 5x2
(c)
x12 − 4x2 + x +
(d)
√
1
x2 −
1
4
1 − 11x
6. (10 points) Using the definition of a derivative, find f 0 ( x ) where f ( x ) = x2 .
7. (10 points) Find the derivative of the following function.
f (x) =
x3 + 2x
x+1
8. (10 points) Find the two values of x0 for which the graph of y = x3 + 6x2 + 9x has a
horizontal tangent at x = x0 .
9. (10 points) Suppose f and g are functions whose domains include 849, and suppose
that f (849) = 100, f 0 (849) = 2/13, g(849) = 39, and g0 (849) = −1/10. If we define the
function h( x ) = f ( x ) g( x ), then what is h0 (849)?
10. (4 points) Calculate
f (1 + h ) g (1 + h ) − f (1) g (1)
,
h
h →0
2
where f ( x ) = x5 + 2x and g( x ) = . (Hint: you do not have to work with the terrible
x
expression you get when you try to plug in).
lim