Math 131 Exam 1 Solution Fall 1999

**1**. (25 points)
a)

**b)** Support your result in part (a) both by a table and by a graph
of the function in a window
containing the point . Notice that
.

**2**.
a) The function
is continuous at the point *x*=2 if and only if
the two one sided limits
and
are equal.
We determine the value of *k* for which the function is continuous by

i) calculating each of the one sided limits in terms of *k*.

ii) set them equal to each other

and solve for *k* to get *k*=2.

**b)**
The function is *not* differentiable even when *k*=2 because we have
a corner at the point (2,6) on the graph.
This can be seen by comparing the *derivatives* at the point *x*=2 of
the two functions 2*x*+2 and (which are not equal). More precisely,
we calculate the left hand and right hand derivative limits:

Since they are not equal, the derivative limit does not exist.

**3**.
a) A line *x*=*a* is a vertical asymptotes of the function

if and only if a one sided limit is *infinite*

The candidates for vertical asymptotes are the roots *x*=1 and *x*=-4 of the
denominator.
We need, however, to check that a one sided limit is indeed infinite
(In question 1, for example, 2 is a root of the denominator, but *x*=2
is *not* a vertical asymptote).

In our case, *x*=1 is a vertical asymptote
because the numerator does not vanish at *x*=1.
(If you stopped here you got the full credit). Similarly,
*x*=-4 is a vertical asymptote because the numerator does not vanish at *x*=-4.
We can show the vertical asymptotes by a graph:
A precise answer would include a calculation of the one sided limits
algebraically:

Similarly,

**b)**
The line *y*=*k* is a horizontal asymptote of the function
because

Thus, for the line *y*=3 to be a horizontal asymptote, we choose *k*=3.

**4**.
We calculate the derivative of using the definition of the derivative as
follows:

**5**. Let
Using the derivative formula for a quotient,

we compute:

**6**.

**a)** The derivative of
is

**b)**
The equation of the tangent line through the point
(1,1+*e*) is

Using part a) we get that . So, the tangent line is

**7**.

**a)** Using the product rule we see that, if , then

**b)**
If *g*'(0)=5, then

so *f*(0)=5.

Fri Oct 22 08:46:04 EDT 1999