Download Antiderivatives and Initial Value Problems

Antiderivatives and Initial Value Problems
Warm up
If
d
f (x) = 2x, what is f (x)?
dx
Can you think of another function that f (x) could be?
If
d
f (x) = 3x 2 + 1, what is f (x)?
dx
Can you think of another function that f (x) could be?
Definition
An antiderivative of a function f on an interval I is another
function F such that F 0 (x) = f (x) for all x 2 I .
Examples:
1. An antiderivative of f (x) = 2x is F (x) = x 2 .
2. Another antiderivative of f (x) = 2x is F (x) = x 2 + 1.
3. There are lots of antiderivatives of f (x) = 2x which look like
F (x) = x 2 + C .
Suppose that h is di↵erentiable in an interval I ,
and h0 (x) = 0 for all x in I .
Then h is a constant function!
i.e. h(x) = C for all x 2 I , where C is a constant.
So, if F (x) is one antiderivative of f (x), then any other
antiderivative must be of the form F (x) + C .
Example: All of the antiderivatives of f (x) = 2x look like
F (x) = x 2 + C
for some constant C .
Every function f that has at least one antiderivative F has
infinitely many antiderivatives
F (x) + C .
We refer to F (x) + C as the
general antiderivative or the indefinite integral
and denote it by
F (x) + C =
Example:
Z
Z
f (x)dx.
2x dx = x 2 + C .
Examples
Z
x 2 dx = 13 x 3 + C ,
because
d 1 3
dx ( 3 x
+ C) =
1
3
⇤ 3x 2 = x 2
Z
x 3 dx = 14 x 4 + C ,
because
d 1 4
dx ( 4 x
+ C) =
1
4
⇤ 4x 3 = x 3
Z
x 5 dx =
Z
x
Z
x k dx =
3
dx =
Some important basic integrals
Z
Z
Z
Z
x
1
1
x k+1 + C
k +1
if k 6=
1
dx =
sin(x) dx =
Z
Z
x k dx =
cos(x) dx =
Z
e x dx =
sec2 (x) dx =
p
1
1
x2
dx =
Theorem (Opposite of sum and constant rules)
Suppose the functions f and g both have antiderivatives on the
interval I . Then for any constants a and b, the function af + bg
has an antiderivative on I and
Z
Z
Z
a ⇤ f (x) + b ⇤ g (x) dx = a f (x)dx + b g (x)dx
Di↵erential equations
A di↵erential equation is an equation involving derivatives.
The goal is usually to solve for y .
Just like you could use algebra to solve
y2 + x2 = 1
for y , you can use calculus (and algebra) to solve things like
dy
5y = 0
for y .
dx
A solution to a di↵erential equation is a function you can plug in
that satisfies the equation.
For example, y = e 5x is a solution to the di↵erential equation
above since
d 5x
e = 5e 5x ,
dx
so
dy
5y = (5e 5x ) 5(e 5x ) = 0 X
dx
.
Simplest di↵erential equations: antiderivatives
Finding an antiderivative can also be thought of as solving a
di↵erential equation:
“Solve the di↵erential equation
Answer:
Check:
y=
Z
d
dx y
= x 2 .”
1
x 2 dx = x 3 + C .
3
d 1 3
1
x + C . = ⇤ 3 ⇤ x2 + 0 = x2
dx 3
3
X
Examples
(1) Solve the di↵erential equation
y 0 = 2x + sin(x).
(2) Check that cos(x) + sin(x) is a solution to
d 2y
+ y = 0.
dx 2
Initial value problems
Find a solution to the di↵erential equation
also satisfies y (2) = 8/3.
general solution:
d
dx y
= x 2 + 1 which
y = 13 x 3 + x + C
Each color corresponds to a choice of C .
Red cuve is the particular solution.
Initial value problems
Find a solution to the di↵erential equation
also satisfies y (2) = 8/3.
general solution:
particular solution:
d
dx y
= x 2 + 1 which
y = 13 x 3 + x + C
y = 13 x 3 + x
2
Each color corresponds to a choice of C .
Red cuve is the particular solution.
Definition
An initial-value problem is a di↵erential equation together with
enough additional conditions to specify the constants of
integration that appear in the general solution.
The particular solution of the problem is then a function that
satisfies both the di↵erential equation and also the additional
conditions.
Solve the initial value problem
dy
= 2x + sin(x)
dx
subject to y (0) = 0.
general solution:
y = x2
cos(x) + C
Algebraically: get a particular solution by solving
0 = y(0) = (0)2
C = 1,
cos(0) + C =
so y = x 2
1+C
(for C )
cos(x) + 1.
Solve the initial-value problem
y 00 = cos x, y 0 ( ⇡2 ) = 2, y ( ⇡2 ) = 3⇡.
Step 1: Calculate the antiderivative of cos(x) to find the general
solution for y 0 .
Step 2: Plug in the values y 0 ( ⇡2 ) = 2 to calculate C .
Step 3: Write down the particular solution for y 0 .
Step 4: Calculate the antiderivative of your particular solution in Step
3 to find the general solution for y .
Step 5: Plug in the values y ( ⇡2 ) = 3⇡ to solve for the new constant.
Step 6: Write down the particular solution for y .
Word problem:
An object dropped from a cli↵ has acceleration a = 9.8 m/sec 2
under the influence of gravity. What is the function s(t) that
models its height at time t?
Initial value problem:
Solve
d 2s
= 9.8,
dt 2
s(0) = s0 , s 0 (0) = 0.
Word problem:
Suppose that a baseball is thrown upward from the roof of a 100
meter high building. It hits the street below eight seconds later.
What was the initial velocity of the baseball, and how high did it
rise above the street before beginning its descent?
Initial value problem:
Solve
d 2s
= 9.8,
s(0) = 100, s(8) = 0.
dt 2
Use your solution to
(1) calculate s 0 (0), and
(2) solve s 0 (t1 ) = 0 for t1 and calculate s(t1 ).
Extra practice: Basic antiderivatives
Problem A. Indefinite integrals.
Z
1.
x7 dx
2.
3.
4.
5.
6.
Z
Z
Z
Z
x
7
dx
11.
x
1
dx
12.
x5/3 dx
5/4
x
Z p
3
dx
x2 dx
7.
Z
1
p
dx
4
x3
8.
Z
2
dx
x2
9.
Z
10.
(8
x + 2x3
6/x3 + 2x
5
+ 5x
1
Z
Z
Z
(2
5x)(3 + 2x)(1
p
x(ax2 + bx + c) dx
(x2
1/x2 )3 dx
13.
Z
14.
Z ✓
15.
Z
(1 + 2x)3
dx
x4
16.
Z
(1 + x)3
p
dx
x
17.
Z
2x2 + x 2
dx
x 2
) dx
18. If
p
( x
p
x) dx
p
1/ x) dx
1
x+ p
x
df
=x
dx
◆2
dx
1/x2 and f (1) = 1/2 find f (x).
Problem B. Indefinite integrals with trigonometric functions.
Z
◆
Z ✓
1 + 2 sin x
6
2
2
6.
dx
1.
9 sin x 7 cos x
+
+
cot
x
dx
cos2 x
cos2 x sin2 x
2.
3.
4.
5.
Z ✓
Z
Z
Z
cot x
sin x
2
tan x
tan x
2
+
cos x
cos2 x
sec x(sec x + tan x) dx
csc x(csc x
◆
dx
7.
Z
8.
Z
9.
Z
cot x) dx
2
(tan x + cot x) dx
3 cos x + 4
dx
sin2 x
1
dx
1 cos x
[hint: mult and divide by (1 + cos(x)) and
rewrite using csc(x), etc.]
1
dx
1 + cos x
10.
11.
12.
13.
14.
Z
Z
Z
Z
Z
tan x
dx
sec x + tan x
15.
Z
p
csc x
dx
csc x cot x
16.
Z
1
dx
1 + cos 2x
cos x
dx
1 + cos x
17.
Z
1
1
dx
cos 2x
18.
Z
p
1 + sin 2x dx
19.
Z
sin3 x + cos3 x
dx
sin2 x cos2 x
sin x
dx
1 sin x
p
1 + cos 2x dx
1
cos 2x dx
Problem C. Integrals with exponential functions and inverse functions.
Z
Z
x2
1.
2x dx
6.
dx
1 + x2
[hint: add and subtract 1 in the numerator]
Z
Z 2
2.
(6x5 2x 4 7x + 3/x 5 + 4ex + 7x ) dx
x
1
7.
dx
2+1
x
Z
3.
(x/a + a/x + xa + ax + ax) dx
Z 6
x
1
8.
dx
2
x +1
◆
Z ✓
p
p
7
3
4.
x
x4 + p
6ex + 1 dx
Z
3
x4
x2
9.
dx
1 + x2
◆
Z ✓
1
2
5
Z
p
5.
1+
+ p
+ ax dx
2
2
1+x
1 x
x x2 1
10.
cos 1 (sin x) dx