SOLUTIONS TO 18.01 EXERCISES
Unit 1. Dierentiation
1A. Graphing
1A-1,2 a) y = (x
1)2
2
b) y = 3(x2 + 2x) + 2 = 3(x + 1)2
1
2
2
1
1
-2
1b
1a
1A-3 a) f ( x) =
( x)3 3x
=
1 ( x)4
1
-2
-1
2a
x3 3x
=
1 x4
2b
f (x), so it is odd.
b) (sin( x)2 = (sin x)2 , so it
The Function sinc(x)
The unnormalized sinc function is dened to be:
sinc(x) =
sin x
.
x
This function is used in signal processing, a eld which includes sound recording
and radio transmission.
1
Use your understanding of the graphs of sin(x) and x togethe
Operations on Power Series Related to Taylor Series
In this problem, we perform elementary operations on Taylor series term by term di erentiation and integration to obtain new examples of power series for which we know their sum.
Suppose that a function
1. Compute the following integral:
Z
4
p
t ln t dt
1
We apply integration by parts:
v = 2p
t3/2
3
dv = t dt
u = ln t
du = 1 dt
t
Then:
Z
4
p
Z 4
4
2 3/2
2 3/2
t ln t
t t 1 dt
3
3 | cfw_z
1
1
t1/2
4
2 3/2
4 3/2
t ln t
t
3
9
1
16
4 3/2
ln 4
(4
1)
3
9
16
28
Implicit Dierentiation and the Second Derivative
Calculate y 00 using implicit dierentiation; simplify as much as possible.
x2 + 4y 2 = 1
Solution
As with the direct method, we calculate the second derivative by dierentiating
twice. With implicit dierenti
Conrming an Integral Converges
Use limit comparison to show that
Solution
Z
Z
1
1
dx
is nite.
(5x + 2)2
1
dx
1
=
. While this calculation was not
(5x + 2)2
35
1
terribly di cult, many calculations are. Limit comparison provides us with a
useful technique
Limits and Discontinuity
For which of the following should one use a one-sided limit? In each case,
evaluate the one- or two-sided limit.
p
1. lim x
x! 0
2. lim
x! 1
3. lim
x! 1
1
x+1
1
(x
1)4
4. lim | sin x|
x!0
|x|
x! 0 x
5. lim
Solutions
p
1. lim x
x!
Smoothing a Piecewise Polynomial
For each of the following, nd all values of a and b for which f (x) is dierentiable.
ax2 + bx + 6,
x 0;
a) f (x) =
2x5 + 3x4 + 4x2 + 5x + 6, x > 0.
ax2 + bx + 6,
x 1;
b) f (x) =
2x5 + 3x4 + 4x2 + 5x + 6, x > 1.
Solution
a)
Summing the Geometric Series
1 1 1
In lecture we saw a geometric argument that 1 + + + + = 2. By an2 4 8
swering the questions below, we complete an algebraic proof that this is true.
We start by proving by induction that:
N
X 1
2N +1 1
=
.
n
2
2N
n=0
SN
Derivatives of Sine and Cosine
Using the Creating the Derivative mathlet, select the (default) function f (x) =
sin(x) from the pull-down menu in the lower left corner of the screen. Do not
check any of the boxes.
Move the slider or use the > button to di
When does a function equal its Taylor series?
We have computed the Taylor series for a di erentiable function, and earlier in the course, we
explored how to use their partial sums, i.e. Taylor polynomials, to approximate the function. So
we know that the
18.01 EXERCISES
Unit 1. Dierentiation
1A. Graphing
1A-1 By completing the square, use translation and change of scale to sketch
b) y = 3x2 + 6x + 2
a) y = x2 2x 1
1A-2 Sketch, using translation and change of scale
2
a) y = 1 + |x + 2|
b) y =
(x 1)2
1A-3 I
Secants and Tangents
We dened the tangent line as a limit of secant lines. We also know that as
x approaches 0 the secants slope f approaches the slope of the tangent line.
x
How close to 0 does x have to be for f to be close to the slope of the tangent
x
Implicit Dierentiation and the Chain Rule
The chain rule tells us that:
d
(f
dx
g) =
df dg
.
dg dx
While implicitly dierentiating an expression like x + y 2 we use the chain rule
as follows:
d 2
d(y 2 ) dy
(y ) =
= 2yy 0 .
dx
dy dx
Why can we treat y as a
Using the Ratio Test
The ratio test for convergence is another way to tell whether a sum of the form
1
X
an , with an > 0 for all n, converges or diverges. To perform the ratio test
n=n0
we nd the ratio
an+1
and let:
an
L = lim
n!1
an+1
.
an
The test has
1. Compute the following derivatives. (Simplify your answers when possible.)
(a) f 0 (x) where f (x) =
f 0 (x) =
x
1
1(1
x2
x2 ) (x)( 2x)
1 x2 + 2x2
1 + x2
=
=
(1 x2 )2
(1 x2 )2
(1 x2 )2
(b) f 0 (x) where f (x) = ln(cos x)
1 2
sin (x)
2
1
1
( sin x)
2 si
Month
January
February
March
.
.
.
Checking
Balance ( )
175
220
255
.
.
.
ccount Balances
The derivative of a function f (t) describes how the functions output changes
as the value of t changes.
Suppose that a checking account has balance f (t) = 5t + 60t
The Derivative of |x|
The slope of the graph of f (x) = |x| changes abruptly when x = 0. Does this
function have a derivative? If so, what is it? If not, why not?
Solution
At rst glance, this seems like a simple question. To the right of y-axis the graph
Continuous but not Smooth
Find values of the constants a and b for which the following function is continuous but not dierentiable.
ax + b, x > 0;
f (x) =
sin 2x, x 0.
In other words, the graph of the function should have a sharp corner at the pont
(0, f