Math 1310
Homework 1: Solutions
Section 1.1 Problem 29) Evaluate the difference quotient for the given
function. Simplify your answer.
Solution: Looking at this problem we should notice a few things.
1) We have a function and are asked to evaluate the given difference quotient, 2) the difference quotient asks us to evaluate the
function at certain points, namely x and a, 3) we need to simplify.
This is the information given to us, so what do we need and how
does this information relate? The concept here is how to evaluate function and how can we simplify these evaluations, related
to this problem is the definition of the difference quotient, in particular version 1, and there are really no underlying theorems or
results to use. We just look at the problem and do it.
We are given that f (x) = x1 and we need to evaluate
if you can follow the computation.
f (x)−f (a)
,
x−a
see
1
− a1
f (x) − f (a)
x
=
x−a
x−a
a
− x
= ax ax
x−a
=
a−x
ax
x−a
a−x
=
(ax)(x − a)
−(x − a)
=
(ax)(x − a)
−1
=
.
ax
And we’re done!
Q.E.D.
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Section 1.2 Problem 16) (Since problems 13 and 16 from this section are
similar I am going to give a solution to 16, 13 is basically the same, but less
involved.) The manager of a furniture factory finds that is costs $2200 to
manufacture 100 chairs in one day and $4800 to produce 300 chairs in one
day.
(a) Express the cost function of the number of chairs produced, assuming
that it is linear. Then sketch the graph.
(b) What is the slope of the graph and what does it represent?
(c) What is the y- intercept of the graph and what does it represent?
Solution: This problem is full of concepts! We need to know
about slopes, interecepts, and how to make linear functions.
Luckily we can deal with all of this, we know that (y − y0 ) =
m(x − x0 ) gives us the equation of a line, we can find slope
by the difference quotient and we know that the y- intercept
is the value of the function when x = 0. So let’s begin.
Notice that we have the following ordered pairs, (100, 2200)
and (300, 4800), these ordered pairs are a part of our function
since they relate the number of chairs to the cost of production.
Given two points we can find slope. By means of the first
= 13.
version of the difference quotient then m = 4800−2200
300−100
Well now we have a slope, and we also have some points, to
find the equation of the line it seems logical to use point slope
form. I will use the point (100, 2200), so y−2200 = 13(x−100),
which yeilds the equation y = 13x + 900. This finishes part (a)
(I omit the graphs because you can all sketch graphs).
From our result in part (a) we can figure out parts (b) and (c).
The slope of our function is 13, this means that in one day for
every 1 chair more made we need 13 more dollars to produce
it, and part (b) is done. For part (c) we need to know that yintercept, so when x = 0 then y = 900, this is the cost when
no chairs are produced, in other words this is the amount of
money we need per day just to open the factory, it is a fixed
cost.
Q.E.D.
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Section 1.3 Problem 57) The Heaviside function H is defined by
(
0 if t < 0
H(t) =
.
1 if t ≥ 0
(a) Sketch the graph of the Heaviside function (you can do this, I am not
going to).
(b) Sketch the graph of the voltage V (t) in a circuit of the swtich is turned
on at time t = 0 and 120 volts are applied instantaneously to the circuit
(as you can guess I am not providing graphs). Write a formula for V (t)
in terms of H(t)
(c) Write a formula for a switch being turned on at t = 5 and 240 volts is
applied.
Solution: This is not a bad problem at all, all we need to do is
figure out which translation to use, thats really all the coneptual
stuff we need. In part (b) at time t = 0 we apply 120 volts,
so before there was nothing, now there is 120 volts, okay so we
are multiplying the Heaviside function by 120. Hence, V(t) =
120H(t). For part (c), we are applying 240 volts, but turn the
switch on after 5 seconds, so we multiply the Heaviside function
by 240 and are shiting it to the right by 5 seconds, hence V(t) =
240H(t - 5). Just basic translation stuff, do not let the piecwise
function get the better of you.
Q.E.D.
Section 1.6 Problem 62) When a camera flash goes off, the batteries immediately begin to recharge the flash’s capacitor, which stores electric charge
given by Q(t) = Q0 (1 − e−t/a ).
(a) Find the inverse of this function and explain its meaning.
(b) How long does it take to recharge the capacitor to 90% of capacity if
a = 2.
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Solution: This problem is really only making sure that you know
how to solve for values of exponentials. We have all of these
properties for exponential and this special (inverse) relationship
between exponentials and logarithms, everything else is just arithmetic. Let’s watch.
Q = Q0 (1 − e−t/a ) ⇒
Q
− 1 = −e−t/a ⇒
Q0
Q
1−
= e−t/a ⇒
Q0
Q
−t
ln(1 −
)=
⇒
Q0
a
t = −a ln(1 −
Q
)
Q0
This is the inverse function and it is interpreted as the amount
of time t needed to recharge to some percetage of Q0 . Using (a),
we can do (b) easily. We want Q to be at 90% of total capactiy,
but Q0 is total capacity, so we want for Q = .9Q0 . To find the
0
) = −2 ln(1 − .9),
time it takes to do this we let t = −2 ln(1 − .9Q
Q0
and we get about 4.6 seconds. The interesting thing about math
is that we do not need to know physics to do physics
Q.E.D.
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