Written Homework 1
Math& 151 Online
Write clear, organized, complete, and mathematically precise solutions for the following problems and
submit them according to the Directions for Submitting Written Assignments and Writing Mathematics,
which can be found in the Orientation to Math& 151 Online module.
1. Sketch the graph of a function f that satisfies all of the following conditions.
The domain is ( −4, ) ,
lim f ( x) = −1 ,
x →3
lim f ( x) = 0 ,
x → 0−
f (3) = −2 ,
lim f ( x) = 2 ,
f (0) = 0 ,
lim f ( x) = 3
lim f ( x) = 4
x → 0+
x →− 4+
x →
2. The point P(−2, ln 4) lies on the curve y = ln(2 − x) .
a) If Q is the point ( x, ln(2 − x) ) , use your calculator to find the slope of the secant line PQ , to six
decimal places, for the specified values of x . (Remember - to earn credit, you must show work!)
x
Slope of the secant
line PQ
-1.9
-1.99
-1.999
-2.1
-2.01
-2.001
b) Using the results of part (a), determine the value of the slope of the tangent line to the curve at
P(−2, ln 4) .
3. The graph of a function f is given. Sketch a graph of f ' in the same set of axes. (You will need to
retrace or cut-and-paste the graph of the given function onto your notebook paper.)
4. Evaluate each limit, if it exists. Use any appropriate method from Chapter 2 except for estimates and
tables. You must show complete work and completely justify each answer. (For this problem, you would
get zero points if you use the estimate/table method.)
a) lim
x →−2
7 x 2 + 14 x
7 x 2 + 13 x − 2
b) lim
x →
7 x 2 + 14 x
7 x 2 + 13 x − 2
3e x
x →0 2 x − 13
c) lim
x2
. Use the definition of the derivative to find f '( x) . Show complete work,
3 − 2x
including detailed algebra and limit evaluation steps.
5. Let f ( x) =
(You would get zero points for using any other method. The process here is what counts! Expect to lose
points for omitting key steps or combining too many at once.)