AP Calculus
AP Calculus
Name:____________________________________
Lesson- Antiderivatives and Indefinite Integrals
Date:_____________________________________
Anti-derivative = Anti-differentiation = Working backwards
Example:x2 is an antiderivative of 2xThe general form of the
antiderivative is given as F(x) + C, where C is any constant,
called the constant of integration. Any 2 antiderivatives of f will
differ only by a constant.
Notation:
Let y = x2 + C. Then 2x
. This is called a differential equation, an equation that
involves
x, y, and a derivative of y.
We can re-write this equation as 2x . This is called the
differential form of the equation.
In general,
Let and
Then
f(x)
is a differential equation and
f(x) dx
is the differential form of the equation.
We write this using integral notation:
is the indefinite integral sign; it means the antiderivative of
f with respect to x.
f(x) is the integrand, the function to be
antidifferentiated.
x = the variable of integration
C = the constant of integration
Rules of Integration
=
Steps for Integrating:
(1) Re-write the original integral into a form in the table.
(2) Integrate (take the antiderivative).
(3) Simplify.
Examples
1.
2.
3.
4.Find the general solution for differential equation
5.
6.
7.
8.
AP Calculus
Name:____________________________________
Lesson- Rolles Theorem, MVT, AVT
Date:_____________________________________
Objective:To learn about and apply Rolles Theorem and the Mean
Value Theorem
Rolles Theorem:Given an interval (a, b) if f(a)=f(b) then there
is a location in the interval where the first
derivative = 0.
MVT:
On the closed interval [a,b]:
{used for finding average velocity)
The idea:
The average slope is the slope of a line drawn from endpoint to
endpoint. No matter how much the function may change between
endpoints, we say that its average change is simply the ratio of
changes in y to changes in x.
For average slope, use the slope formula.
The MVT says that there has to be a point somewhere between the
endpoints where the instantaneous rate-of-change (derivative) is
the same as the average slope.
The formula
where c is an x-value between a and b.
Find the values of x that satisfy the MVT for each of the
following:
1. y = x2 on [1,5]
2. f(x) = sin (x) on [0, (]
Determine if Rolles Theorem can be applied. If so, apply it. If
not, explain why.
3.
4.
Determine if Mean Value Theorem can be applied. If so, apply it.
If not, explain why.
5.
6.
AVT:
The average value of a function is a number that represents the
central tendency of the output. The calculation of this value is
straightforward; it is simply the integral of the function over an
interval divided by the interval.
Average Value =
Practice: 7.Find the average value of y on [2, 6] for y = 2x +
3.8.Find the average value of y on [0, (] for y = cos(x) + x9.Find
the value of c that makes the average value of x2 equal to 9 on
[0,c].10.Design a function of your own whose average value is 0 on
[-2, 2].
AP Calculus
Name:____________________________________
Lesson- Antiderivatives and Initial Conditions
and Particular Solutions
Date:_____________________________________
Objective- To learn how to find anti-derivatives involving
particular values and initial conditions.The general solution to
the differential equation is
If we are given the value of y for one value of x this is called
an initial condition then we can get a particular solution; that
is, we can find a particular constant C and write a specific
antiderivative as our answer.
Examples
1.Given and Find the particular solution.
F(x) =
2.Find g(x) if and g(0) = -1.
3.A ball is thrown upward with an initial velocity of 64 ft/sec
from an initial height of 80 ft. Using the fact that the
acceleration due to gravity is 32 ft/sec,
(a) find the position function giving the height s as a function
of time t.
(b) When does the ball hit the ground?
4.The rate of growth dP/dt of a population of bacteria is
proportional to the square root of t, where P is the population
size and t is the time in days . That is, . The initial size of the
population is 500. After 1 day the population has grown to 600.
Estimate the population after 7 days.
5.The Grand Canyon is 1800 m deep at its deepest point. A rock
is dropped from the rim above this point. Write the height of the
rock as a function of the time t in seconds. How long will it take
the rock to hit the canyon floor?
AP Calculus
Name:____________________________________
Lesson- Sigma Notation
Date:_____________________________________
Summation NotationSigma Notation
, read sigma, is the uppercase Greek letter S. It is used to
denote a sum.
Let a1 be the first term of a sum,
a2 be the second term of a sum,
a3 be the third term of a sum,
.
.
.
an be the nth term of a sum.
Then we write the sum a1 + a2 + a3 + + an in sigma notation
as
where i is called the index of summation.
Example 1
Here ai = i.
Example 2
Here ai = i + 1.
Example 3Write the sum 42 + 52 + 62 + 72 + 82 + 92 + 102 in
sigma notation.
Example 4Write the sum represented by
Example 5Expand .
Properties of Summation
(1)
Example:(2)
, by the commutative property of addition
and
Summation Formulas
(1)
(2)Sum of 1st n positive integers =
(3)Sum of squares of 1st n positive integers =
(4)Sum of cubes of 1st n positive integers =
Example 6 Use the properties of summation and summation formulas
to evaluate the sums:
(a)
(b) Evaluate the sum in part (a) when n = 10.
Evaluate Each:
7.
8.
9.
Write in Sigma Notation
10.
11.
12.
How to do Summations on your Calculator
(1) In MODE menu, set 4th line to SEQ.
(2) In LIST menu (2nd STAT) select MATH menu #5, get display:
sum(
(3) In LIST (2nd STAT) menu, select OPS menu #5, press ENTER,
get display: sum(seq(
(4) Enter, in turn: formula, n, lower bound of summation, upper
bound of summation))(5) Hit ENTER. This will return the sum of the
indicated terms in your list.
Try this with Example 7, 8, and 9 above.
AP Calculus
Name:____________________________________
Lesson- Area of a Plane Region, Upper and Lower Sums
Date:_____________________________________
Given a continuous, non-negative function f(x) find the area
under the graph of y = f(x) between the vertical lines x = a and x
= b.
Pictures:
Method:
Step 1:
Break the interval [a, b] into n equal parts.
Each little sub-interval so formed will have length
Step 2:
Label the endpoints of the sub-intervals as follows:
Step 3:
Draw vertical lines at each of these endpoints to inscribe
rectangles inside the region
Since f is continuous, f has a minimum value on each of these
sub-intervals.
Call the x value in the interval where the minimum value of f
occurs .
Step 4:
Compute the area of these inscribed rectangles. This is called a
lower sum.s(n) =
Area under the curve of f
Step 5:
Repeat Step 3, only this time draw circumscribed rectangles.
Since f is continuous, f has a maximum value on each of these
sub-intervals.
Call the x value in the interval where the maximum value of f
occurs .
Step 6:
Repeat Step 4, only this time use circumscribed rectangles.
Compute the area of these circumscribed rectangles. This is
called an upper sum.Area under the curve of f S(n) =
Example 1Find the upper and lower sums for the region bounded by
the graph ofand the x-axis between x = 0 and x = 2 for n = 4.
Sketch a graph.
Find
Lower Sum
Find for each subinterval.
Find for each subinterval.
Find areas and add.
Find areas and add.
Actual area A is between theses two values.
Example 2Find the upper and lower sums for the region bounded by
the graph ofand the x-axis between x = 0 and x = 2 for n = 8.
=
=
=
s(8) =
S(8)=
In a similar manner we, can compute s(16) =
and S(16) =
,
So
Example 3Find the upper and lower sums for the region bounded by
the graph of and the
x-axis between x = 0 and x = 2 for general n.
=
=
=
Theorem: If f is a continuous, nonnegative function on the
interval [a, b], then the limits as of both the lower and upper
sums exist and are equal to each other. In symbols,
where , the minimum value of f on the subinterval, and
the maximum value of f on the subinterval.
Definition: Let f be a continuous, non-negative function on
[a,b]. The area of the region bounded by the graph of f, the
x-axis, and the vertical lines x = a and x = b is
Area = , where .
Example 4Find the area under the graph of y = 1 x2 over the
interval [-1, 1].
AP Calculus
Name:____________________________________
Lesson- Riemann Sums and Definite Integrals
Date:_____________________________________
Consider the sum whose limit is defined to be the area of the
region under the graph of (continuous, nonnegative function) f(x)
between x = a and x = b:
,
where is the same for every sub-interval.
If we generalize this sum to allow for varying widths in each
subinterval, we have the definition of a Riemann sum.
Definition:Let f be defined on the closed interval [a, b]
and
let be a partition of the interval [a, b] given by
where is the width of the ith subinterval.
If ci is any point in the ith subinterval, then the sum
is called a Riemann sum for the partition .
The upper and lower sums we computed in Examples 1 through 4 are
Riemann sums.
Notation/Vocabulary:The width of the largest subinterval of a
partition is called the norm of the
partition and is designated by .
A partition where every subinterval is of equal width is called
a regular partition.
Definition of a Definite Integral: If f is defined on the closed
interval [a, b] and the limit
exists, then f is said to be integrable on [a,b] and the limit
is denoted by:
=
We call this limit the definite integral of f from a to b.
a is called the lower limit of integration;
b is called the upper limit of integration.
Note: A definite integral is a NUMBER. An indefinite integral is
a FAMILY OF FUNCTIONS.
Example 5Write these limits as definite integrals:
on the interval [0,4]
on the interval [1,3]
Example 6 Use the limit definition to evaluate the definite
integral:
Important Theorem: The Definite Integral as the Area of a
RegionIf f is continuous and nonnegative on the closed interval [a,
b], then the area of the region bounded by the graph of f, the
x-axis, and the vertical lines x = a and x = b is given by:
This theorem simply states that the area under the graph of f (f
continuous and nonnegative) is the limit of the Riemann sum of any
partition of f.
Example 7Set up the definite integrals that give the area of the
regions shown.
Example 8Sketch the region whose area is described by the
definite integral ; then use a geometric formula to evaluate the
integral.
Important Properties of Definite Integrals
(1) If f is defined at x = a, then
(2) If f is integrable on [a,b], then
(Reversing the limits of integration reverses the sign.)
(3) Additive Interval Property: If f is integrable on the closed
intervals determined by a, b, and c, with
,then:
(4) Constant Multiple Property: If f is integrable on [a, b] and
k is a constant, then:
(5) Additive Function Property: If f and g are integrable on [a,
b], then
(6) Preservation of Sign Property: If f is integrable and
nonnegative on [a, b], then
(7) Preservation of Inequality Property: If f and g are
integrable on [a, b], and for every x in
[a, b], then
Examples of Use:
Example 9: Given evaluate
Example 10: Given evaluate
a.
b.
c.
d.
Properties of Definite Integrals Practice
Given that the shaded region A has an area of 1.5, and evaluate
the following integrals.
(a)
(b)
(c)
(d)
(e) The average value of f over the interval [0,6] is
AP Calculus
Name:____________________________________
Lesson- The Fundamental Theorem of Calculus & MVT
Date:_____________________________________
This is the important theorem that links the 2 branches of
calculus--differential and integral.
Fundamental Theorem of Calculus
If the function f is continuous on [a, b] and F is an
antiderivative of f on the interval [a, b], then
This says that taking a definite integral, the limit of a
Riemann sum, the area under a curve, is the same as taking an
antiderivative.
Look Ma, No long sum!
Notation:
Example 1Use the Fundamental Theorem of Calculus to evaluate the
following definite integrals:
a.
b.
c.
d.
e.
Example 2 Find the area of the region bounded by the graphs of ,
the x-axis, x=1 and x=2.
Example 3Find the area under the graph of from 0 to
0
Example 4Evaluate
Mean Value Theorem for Integrals
If f is continuous on [a, b], then there exists a number c in
[a, b] such that
Example 5Find the value of c guaranteed by the Mean Value
Theorem for Integrals for on the
interval [1,3].
The value f(c) guaranteed by the MVT for Integrals is called the
average value of f on [a, b].
Definition
If f is integrable on [a, b] then the average value of f on the
interval [a, b] is
Example 6Find the average value of the function f(x) = cos x on
the interval [0, ].
Example 7This graph shows the velocity, in ft/sec, of a car
accelerating from rest. Use the graph to estimate the distance the
car travels in 8 seconds.
Second Fundamental Theorem of Calculus
If f is continuous on an open interval I containing a, then, for
every x in the interval,
The function is called an accumulation function.
It accumulates the area under the curve y = f(t).
Example 8Let f(x) = cos x. Consider the values for the function
on the interval [0, ].
Note: F(x) is a strictly increasing function.
The First Fundamental Theorem of Calculus says that integration
undoes differentiation.
The Second Fundamental Theorem of Calculus says that
differentiation undoes integration.
Taken together, they complete the notion that differentiation
and integration are inverse processes.
Examples of Use:
Example 9Find F as a function of x, then evaluate it at x = 2,
5, and 8.
Example 10Integrate to find F(x), then differentiate to show the
2nd Fundamental Theorem holds.
Example 11Use the 2nd Fundamental Theorem to find
2nd Fundamental Theorem and the Chain Rule
What happens when the upper limit of an accumulation function is
not x, but a function of x???
Make a u-substitution, then use the chain rule: Example 12Find
if
Example 13Find if
AP Calculus
Name:____________________________________
Lesson- Integration by Substitution
Date:_____________________________________
This is a technique that allows us to integrate a great many
functions by putting them in the form of some function we have in
our table of integrals.
By the Chain Rule, , so,
by the Fundamental Theorem, .
If we consider u = g(x), some function of x, then and
Steps for Integrating by SubstitutionIndefinite Integrals
1. Choose a substitution u = g(x), such as the inner part of a
composite function.
2.Compute .
3.Re-write the integral in terms of u and du.
4.Find the resulting integral in terms of u.
5.Substitute g(x) back in for u, yielding a function in terms of
x only.
6. Check by differentiating.
Examples
1.
2.
3.
4.
5.
6.Solve the differential equation:
Integration by Substitution Indefinite Integrals with Initial
Conditions
Use the method above to find a general solution, then use the
initial condition to find C and a particular solution.
7.Find a function f that satisfies
and whose graph passes through the point
.8.Find f if
and f(2) = 7.
Integration by Substitution Definite IntegralsChange of
Variables for Definite Integrals:
If the function u = g(x) has a continuous derivative on [a, b]
and f is continuous on the range of g, then
Steps for Integrating by SubstitutionDefinite Integrals
1. Choose a substitution u = g(x), such as the inner part of a
composite function.
2.Compute . Compute new u-limits of integration g(a) and
g(b).
3.Re-write the integral in terms of u and du, with the u-limits
of integration.
4.Find the resulting integral in terms of u.
5.Evaluate using the u-limits. No need to switch back to xs!
9.
10.
11.
12.
13.
AP Calculus
Name:____________________________________
Lesson- Areas and Volumes
Date:_____________________________________
Area Problems1. The area of the region bounded between and .
2. The area of the region bounded by, the x and y axes, and
a. With respect to x.
b. With respect to y.3. The area of the region in quadrant I
bounded by, , and
a. With respect to x.
b. With respect to y.Volume Problems
4. Find the volume of the solid whose base is the region in
quadrant I that is bounded by, , and
. All cross-sections perpendicular to the x-axis are
a. Rectangles with height twice the width.b. Semi-circles.
5. The volume of the solid generated by rotating the region
bounded by , , between and
around the x-axis
6. The volume of the solid generated by rotating the region in
quadrant I bounded byand
around the x-axis.
7. The volume of the solid generated by rotating the region
bounded by , , and around the y-axis.
8. The volume of the solid generated by rotating the region in
quadrant I bounded by,
and around
a. The line.
b. The line.
c. The line.
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