MATH 1100: QUANTITATIVE ANALYSIS
TEST #2 (VERSION A)
1. Find the derivative with respect to x of
1/3
y = x2 + 1
.
Solution:
dy
2
x
=
.
dx
3 (x2 + 1)2/3
2. Find the tangent line to the curve
y = x5 − x3
at x = 1.
Solution:
At x = 1,
y =1−1=0
and
dy
= 5x4 − 3x2 = 5 − 3 = 2.
dx
Therefore the tangent line is
y = 2(x − 1) = 2x − 2.
3. The response y of the body to an amount x of adrenalin is given by
x
y=
a + bx
where a and b are constants determined by experiment.
(a) Where is this function continuous?
(b) Suppose that for any x ≥ 0 the value of y can not be negative. What
can you say about a and b? Explain your answer. (For example, are
they both positive, both negative, or of opposite signs?)
Solution:
(a) The function is a ratio of polynomials, so it is continuous everywhere
except where the denominator ax + b vanishes. So either (i) a = b = 0
and the function is not defined anywhere or (ii) a = 0 and b 6= 0 and
the function is continuous everywhere or (iii) a 6= 0 and the function
is continuous except at x = −b/a.
(b) To have y defined and nonnegative for nonnegative x we must have
ax + b > 0. Taking x large positive, and dividing by x, we see that
a ≥ 0, and taking x = 0 we see that b > 0. Summing up: a ≥ 0 and
b > 0.
Date: September 24, 2001.
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MATH 1100: QUANTITATIVE ANALYSIS
TEST #2 (VERSION A)
4. A product has sales S (in millions of dollars) given as a function of time t
(in months) by
1
S =1+
.
(t + 1)5
(a) Find the rate of change of sales at time t = 1.
(b) Find the second derivative of sales at t = 1, and use it to explain how
the rate of sales is changing at t = 1.
Solution:
(a)
5
dS
=−
dt
(t + 1)6
so at t = 1
dS
5
=− .
dt
64
(b)
d2 S
30
=
dt2
(t + 1)7
so at t = 1
d2 S
15
=
.
dt2
64
Therefore the rate of sales is increasing at a rate of 15/64 at t = 1.