Tangents, velocities, other rates of change given a curve and a point on the graph, the slope tangent line to the curve at is given by lim given that the position of an object moving in a line at time is given by then the average velocity from to is given
The Mean Value Theorem Consider: Let be a function that is: 1. continuous on the closed interval 2. differentiable on the open interval 3.
Rolle's Theorem Let be a function that is: 1. continuous on the closed interval 2. differentiable on the open interv
antiderivatives
Determining antiderivatives or indefinite integrals is essentially "undoing differentiation." When we are to determine the antiderivative of , we are to find the functions such that . The basic integration rules on the top half of referenc
Absolute and local minimum and maximum values Defn: A function has an absolute maximum at if domain of . The number is called the maximum value of A function has an absolute minimum at if . The number is called the minimum value of The maximum and minimum
We know that, given a differentiable function , that the line tangent to the curve at
, and a point on the graph has slope and equation
If is "sufficiently close" to , then we can use points on the tangent line as approximations to . In this sense, the ta
4.
If
and
, determine
differentiate implicitly wrt time
when at and at
then , we have , we have
, so
8. If a snowball melts so that its surface area decreases at a rate of cm /min, determine the rate at which the diameter decreases when the diameter is 10
Since
ln
log
we can differentiate
wrt ln
to obtain
more generally, we have log because log
ln
ln
ln
ln
ln
ln
The chain rule version for ln example: ln or we can rewrite given ln ln what is as ln
ln is
ln ?
recall that
so that then
ln ln
ln ln
so that
ln
c
Implicit differentation
Consider the equation The graph is below. This equation is not a function.
A point on the curve is . Does the curve have slope there? Does it have a tangent line there?
If we just look at the graph "near"
, it is a function - see b
The Chain Rule
The Chain Rule says that if , then
Sometimes we say: Given and , then
So, in Here we decompose the function into two functions and , which are
then, with we have
and and
so that
2.
decompose some function, where some function ,
so that then
Derivatives of the Trig Functions Consider lim sin
There are several ways to determine the value of this indeterminate limit One way is to graph both sin and on the same plane: as , how do the graphs compare? a zoomed in graph is below
Another way is to g
Rates of Change Recall that And then and the average rate of change of And the instantaneous rate of change of (of ) with respect to with respect to lim 208-2 A particle moves according to a. Determine the velocity at time Since b. the rate of change of p
The product rule If Proof:
lim
and
are differentiable functions, then lim
lim lim lim
lim lim lim
lim lim
lim lim
lim lim
The Quotient rule If and are differentiable functions, then
MTH 173 ection 3.2 notes page1
4.
differentiate
Note that:
product rule
8
Derivatives of Polynomial & Exponential Functions
Given a constant function lim Given the function lim Given the function Since lim lim Given the function Since , lim lim lim lim , its derivative . lim lim , its derivative lim . lim .
, its derivative lim
The Derivative as a Function Recall that lim gives the derivative of a function at the point But we saw that, given , that . This is evaluated to give the derivative's value for any number replacing . But it could also be considered a function (of ). Now
Definition A function 0 is increasing on an interval if for any two numbers B" and B# in the interval, B" B# implies 0 aB" b 0 aB# b. A function 0 is decreasing on an interval if for any two numbers B" and B# in the interval, B" B# implies 0 aB" b 0 aB# b
MTH 174, section 4.4 Recall that, given , the tangent line to the graph at has equation
^ Thm L'Hopital's rule Let and be functions that are differentiable on an open interval containing , except possibly at itself. Assume that for all near except possibl
Limits at Infinity; horizontal asymptotes By a limit at infinity, we mean "is there a number the function begins to act like as gets large without bound?" denotes "large without bound," while without bound." Consider the function denotes "negatively large
Continuity Continuity of a function means the graph has no breaks, jumps, holes, or gaps. Continuity at a point means the function exists there, that the limit exists there, and that these values are the same. Specifically, Definition: A function is conti
Calculating Limits using the limit laws Limit Laws Suppose that is a constant and the limits lim exist. Then 1. 2. 3. 4. 5. lim lim lim lim lim lim lim lim lim lim lim lim if lim lim lim and lim
112-2.
lim lim d. a. lim lim
lim lim
lim
MTH 173, section 2.
The Limit of a Function Consider the function Domain ? What happens to the function near ? Both the numerator and denominator gets closer and closer to What happens to the graph near ? It exists, seems to be near As the values of become ever closer to , w
Section 7.7 This section deals with techniques for approximating the value of definite integrals. We know we can determine the value of a definite integral by determining an antiderivative of . However, not all functions have antiderivatives, and also som
Average value of a function Defn: The average value 0ave of a function 0 on the interval c+ ,d is 0ave
, " oe ( 0 aBb .B ,+ +
2.
Determine the average value of 0 aBb oe
%
" in c" 0 B
0ave
% " " " " ln % oe .B oe ln B oe aln % ln "b oe ( %" " B $ $ $ "
The
Section 6.1 This section deals with the area of regions bounded by two (or more) curves. In setting up the integral, I illustrate the Riemann sum "typical" rectangle with a thick black line segment. The width of the segment represents either ?B or ?C. The
This section extends your antiderivative finding skills dramatically with a technique called substitution. Essentially, if a function's antiderivative can be found by "backing through" the chain rule, then this technique is useful. To have sucess here, yo
14.
sin sin
sin cos sin
cos
sin
22
40.
sin
sin
sin
cos
cos
44.
area of region by writing
as a function of
and then integrating wrt .
y=1
y=x1/4
48. A honeybee population starts with 100 bees and increases at a rate of per week. What does represent? The nu
Section 5.3 FTC 1
the fundamental theorem of calculus If is continuous on , then the function defined by
is continuous on
, differentiable on
and
Proof: We need to show that lim
So
Animate Point Move B -> D
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
-2
-1 -0. 5
1
2
3
The Definite Integral in this section we define the definite integral and learn some ways to evaluate it using the summation formulas below and using areas of geometric figures. Summation Formulas 1. 2.
3.
4.
Example:
Example: lim
lim
lim lim Example:
lim
Areas & Distances
The area problem:
What's the area of the shaded region:
We can approximate the area with a set of rectangles. The more rectangles, the better the approximation. These four give a very rough approximation:
These eight give a better approx
8.
cardboard
ft maximize volume
base
side
x
cut out squares feet on a side. by by
x
dimensions of box are so volume is
Since all dimensions must be positive, we know that . Thus we are to maximize in this interval. At the end points, we have we determine