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The Fundamental Theorem of Calculus
This theorem bridges the antiderivative concept with the area problem. Indeed, let f (x) be a
function defined and continuous on [a, b]. Consider the function
F(x) =
f (t)dt
defined on [a, b]. Then we have
=
Since
f (t)
Calculus 2.2 - Notes
Name_
Objective A: Using a Numeric Approach to find the limit of a function:
Finish the chart using a calculator. f ( x) =
x
F(x)
0.75
.9
.99
x3 1
,x 1
x 1
.999
1
1.001
1.01
1.1
1.25
We say lim f ( x) = ? , the limit as x approaches 1
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Definition of a Polynomial in One Variable- A polynomial of degree n in one variable x is an
expression of the form a0xn An-2x +a,n where the coefficients a0,aa1,a2 an represent real
numbers, a0 is not zero, and n represents a nonnegative integer
Definiti
Order of operation:simplify the expressions inside grouping symbols, such as parentheses,
brarckets, braces, and fraction bars-evaluate all powers, do all multiplications and divisions from
left to right, then all additions and subtraction from left to ri
Negative exponents-for any real number a and any integer n where a does not equal 0 a to the
negative equals one over a to the n and 1 over a to the negative n equals a to the n
Multiplying powers-For any real number a and integers m and n, a to the m tim
Partial derivatives
In two dimensions, when we have a function y(x), we can readily define dy/dx as the
slope of the curve y(x). Here we'll use two concrete examples to illustrate partial
derivatives: first we'll look at a curve y(x) that is also a functi
Analytical derivatives
But what if we 'know' the formula for the function x(t)? I have put 'know' in quotation
marks, because for anything in physics, the only things that we know are the
measurements. There are only a finite number of these, so we just
cpcalc day39
Ch4.1 (I) Extrema of a
function on the given interval
cp calculus
block 39
HW: p209,x3s
#3-#45
2/19/2013
cpcalc day39
extrema of a function
on an Interval
1
Madeline Valentine
2/19/2013
cpcalc day39
2
Ch4 Objective
How to use calculus
to anal
Calculus 2.2 - Notes
Name_
Objective A: Using a Numeric Approach to find the limit of a function:
Finish the chart using a calculator. f ( x) =
x
F(x)
0.75
.9
.99
x3 1
,x 1
x 1
.999
1
1.001
1.01
1.1
1.25
We say lim f ( x) = ? , the limit as x approaches 1
Trigonometric functions
Sine and cos functions are important, especially in circular motion, simple harmonic
motion, components of forces and other cases involving components of vectors.
Fortunately, the derivatives here are simple. Let's work them out, u
Partial derivatives
In two dimensions, when we have a function y(x), we can readily define dy/dx as the
slope of the curve y(x). Here we'll use two concrete examples to illustrate partial
derivatives: first we'll look at a curve y(x) that is also a functi
Analytical derivatives
But what if we 'know' the formula for the function x(t)? I have put 'know' in quotation
marks, because for anything in physics, the only things that we know are the
measurements. There are only a finite number of these, so we just
Integration: How do the results of a variable rate add up?
Let's leave displacement time graphs for a moment, because my favourite example
of an integrator is a bucket. A bucket integrates the flow of water from a tap above
it. In our example, someone is
The Fundamental Theorem of Calculus
This theorem bridges the antiderivative concept with the area problem. Indeed, let f (x) be a
function defined and continuous on [a, b]. Consider the function
F(x) =
f (t)dt
defined on [a, b]. Then we have
=
Since
f (t)
Trigonometric functions
Sine and cos functions are important, especially in circular motion, simple harmonic
motion, components of forces and other cases involving components of vectors.
Fortunately, the derivatives here are simple. Let's work them out, u
Calculus Notes
Definition of an anti-derivative-A function f is an antiderivative of f on an interval I if F(x) =f(x)
for all x in I.
Theorem 5.1 Representation of Antiderivatives- If F is an antiderivative of f on an interval I then
G is an antiderivativ