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Transcript
AP Calculus Section 6.2
The Indefinite Integral
Homework: Day 1 Page 363 #9 – 35 odd, 41, 43, 47, 49, 53, 54, 55, 56
Day 2
Day 3
Objective: SWBAT find the indefinite integral and solve a basic differential equation.
They will also be able to create a slope field for a particular differential equation.
1. The Indefinite Integral
The integral of f ( x) with respect to x is equal to F ( x) plus a constant.
 f ( x)dx  F ( x)  C
The differentiation symbol dx identifies the independent variable. That is the variable
that we will take the integral with respect to. This becomes real important when we us
separation of variables to solve differential equations.
To integrate a power of x (other than -1) add one (1) to the exponent and divide by the
new exponent. Think of the integral as the opposite operation of differentiation.
2.
a.
 x dx
b.
x
5
c.

xdx
2
1
dx
3. In the expression  (3x 2  2 x  5)dx

is the integration symbol
3x 2  2 x  5 is called the integrand
dx identifies the independent variable, the variable we will take the integral with
the respect to
C is the constant of integration
4. Find the integral of the following:
a.
  sec x dx
b.

3
2
c.

1
x
2
xdx
dx
5. Some basic concepts about integrals. They seem obvious but they are important.
a.
 cf ( x)dx  c  f ( x)dx
b.
  f ( x)  g ( x)dx   f ( x)dx   g ( x)dx
c.
  f ( x)  g ( x)dx   f ( x)dx   g ( x)dx
6. Find the integrals
 4 cos xdx
 (x  x
 (3x
6
2
)dx
 2 x 2  7 x  1)dx
cos x
dx
2
x
 sin
t 2  2t 4
 t 4 dt
7. Find
8.
d 
x 2  3x  5 dx 




dx
d 
f ( x)dx   f ( x)

dx  
Connecting the concepts of integration and differentiation
9. Suppose that a point moves along some unknown curve y  f ( x) in the xy plane in
such a way that at each point  x, y  on the curve, the tangent line has slope x 2 . Find an
equation for the curve given that it passes through the point  2,1 .
10. Differential equations - an equation that involves a derivative of an unknown
function. This is different then the equations you have been working with in mathematics
up to this point because the unknown is no longer a number, it is a function.
** Another Important Topic Seen On Every AP Test ***
The initial value problem
11. One type of differential equation is called an initial value problem. It is written:
dy
 f ( x) and y  x0   y0 is called the initial condition.
dx
12. To solve all of the differential equations in this course, you MUST separate the
differentials and then integrate both sides. Once this is done, use the given
condition to solve for the dependent variable.
13. Solve the initial value problem with the given initial condition.
dy
 cos x
dx
y (0)  1
14. Use separation of variables to solve the differential equation. Write your result
as y  f ( x) .
dy
1

dx  2 x 3
y 1  0
15. Use separation of variables to solve the differential equation. You will not be
able to solve this as y  f ( x) .
dy
6 x2

dx 2 y  cos y
y 1  
16. Solve the differential equation. Since there is no condition given your answer will
have a constant in it.
y '  x2 y
17. Suppose we are interested in a quantity y (population, radioactivity, money, etc.)
that increases or decreases at a rate proportional to the amount present. If we
also know the amount present at time t = 0, say y0 , we can find y as a function of t
by solving the following initial value problem.
Differential Equation
Initial Condition
dy
 ky
dx
y  y0 when t  0
If y is positive and increasing, then k is positive and the rate of growth is proportional
to what has already been accumulated. If y is positive and decreasing, then k is
negative and the rate of decay is proportional to the amount still left.
18. Solve the growth/decay problem and discuss the constant in the answer.
dy
 ky
dt
Initial Condition
y  y0 when t  0
19. Find the half-life of a radioactive substance with decay equation
initial conditions y  y0 when t  0 . What do you notice?
dy
  ky with
dt
20. Suppose that the cholera bacterium is a colony that grows unchecked according to
dy
 ky . The colony starts with one bacterium and doubles
the differential equation
dt
in number every half hour. How many bacteria will the colony contain after 24
hours?
21. The decay for radon-222 gas is known to follow the differential
equation
dy
 .18 y , with t in days. About how long will it take the amount of radon in
dt
a sealed sample of air to decay to 90% of its original value?
22. Real Life Diffy Q
When a murder is committed, the body, originally at 37 C , cools according to Newton’s
Law of Cooling. Suppose that after two hours the temperature is 35 C , and that the
dy
 k  y  20  .
temperature of the surrounding air is a constant 20 C . Assume that
dt
a. Find the temperature, y, of the body as a function of time, the time in hours since
the murder was committed.
b. Sketch a graph of temperature versus time.
c. What happens to the temperature in the long run? Show this on the graph and
algebraically.
d. If the body is found at 4 P.M. at a temperature of 30 C , when was the murder
committed?