Integration by Advanced Guessing
xdx
1 + x2
If we use the method of substitution, we start by setting u equal to the
ugliest part of our integral:
Example:
u = 1 + x2
The calculation looks like:
xdx
1 + x2
and du = 2xdx.
1
2 du
u 2
du
2
=
=
u
1
1
=
=
=
=
Newton's Method: What Could Go Wrong?
Newton's method works (very) well if |f | is not too small, |f | is not too big, and x0 starts near the solution x. We're not going to discuss these conditions in detail, but let's see why they're there. If f is too l
The Mean Value Theorem and Linear Approximation
Whats the dierence between the mean value theorem and the linear approxi
mation?
The linear approximation to f (x) near a has the formula:
f (x) f (a) + f (a)(x a) x near a.
If we let x = x a, we get:
f (x)
The Mean Value Theorem and Inequalities
The mean value theorem tells us that if f and f are continuous on [a, b] then:
f (b) f (a)
= f (c)
ba
for some value c between a and b. Since f is continuous, f (c) must lie between
the minimum and maximum values of
Dierentials and Linear Approximation
Linear approximation allows us to estimate the value of f (x + x) based on the
values of f (x) and f ' (x). We replace the change in horizontal position x by
the dierential dx. Similarly, we replace the change in heigh
Mean Value Theorem: Consequences
The rst thing we apply the MVT to is graphing, but well see later that this
is signicant in all the rest of calculus.
If f > 0 then f is increasing.
If f < 0 then f is decreasing.
If f = 0 then f is constant.
We told yo
Ring on a String
Were going to do one more max/min problem.
Consider a ring on a string held xed at two ends at (0, 0) and (a, b) (see
Fig. 1). The ring is free to slide to any point. Find the position (x, y ) that the
ring slides to.
Note that if b = 0,
Dierentials
Today we move on from dierentiation to integration. For this well need a new
notation for quantities called dierentials.
Given a function y = f (x), the dierential of y is
dy = f (x)dx
Because y = f (x) we sometimes call this the dierential of
Introduction to Related Rates
Were continuing with a related rates problem from last class.
Example: Police are 30 feet from the side of the road. Their radar sees
your car approaching at 80 feet per second when your car is 50 feet away from
the radar gun
Denition of the Denite Integral
b
The denite integral a f (x)dx describes the area under the graph of f (x) on
the interval a < x < b.
a
b
Figure 1: Area under a curve
Abstractly, the way we compute this area is to divide it up into rectangles
then take a
Dierential Equations and Slope, Part 1
Suppose the tangent line to a curve at each point (x, y ) on the curve is twice as
steep as the ray from the origin to that point. Find a general equation for this
curve. (See Fig. 1.)
(x,y)
Figure 1: The slope of th
Related Rates, A Conical Tank
Example: Consider a conical tank whose radius at the top is 4 feet and whose
depth is 10 feet. Its being lled with water at the rate of 2 cubic feet per
minute. How fast is the water level rising when it is at depth 5 feet?
A
Newtons Method
Today well discuss the accuracy of Newtons Method.
Recall how Newtons method works: to nd the point at which a graph
crosses the x-axis you make an initial guess x0 at the x-coordinate of that
crossing. You then nd the tangent line to the g
Areas Between Curves
Suppose you have two curves, y = f (x) above and y = g (x) below. You want
to nd the area between the two curves bounded on the left by x = a and on
the right by x = b.
11111111111111111111111111111111111
00000000000000000000000000000
Max/Min Example 2
This is an example of a minimization problem with a constraint.
Example: Find the box (without a top) with least surface area for a xed
volume.
Again, we start by drawing a diagram and choosing variables. (This time
well have four variab
Introduction to Related Rates
Next well look at the subject of related rates, which will give us another op
portunity to practice working with several dierent variables and equations.
Example: Police are 30 feet from the side of the road. Their radar sees
Summation Notation
Youll have noticed working with sums like 12 + 22 + 32 + + (n 1)2 + n2 is
extremely cumbersome; its really too large for us to deal with. Mathematicians
have a shorthand for calculations like this which doesnt make the arithmetic
any ea
Introduction to Curve Sketching
Goal: To draw the graph of f using information about whether f and f are
positive or negative. We want the graph to be qualitatively correct, but not
necessarily to scale.
WARNING: Dont abandon your precalculus skills and c
Implicit Differentiation and Min/Max
Example: Find the box (without a top) with least surface area for a fixed volume. Another way to solve this problem is by using implicit differentiation. As before, this method has some advantages and some disadvantage
Newtons Method
Newtons method is a powerful tool for solving equations of the form f (x) = 0.
Example: Solve x2 = 5.
Were going to use Newtons method to nd a numerical approximation for
5. Any equation that you understand can be solved this way. In order
Substitution:
x3 (x4 + 2)5 dx
We want to compute x3 (x4 + 2)5 dx.
We already have a formula for xn dx, so we could expand (x4 + 2)5 and
integrate the polynomial. That would be messy. Instead well use the method
of substitution.
Finding the exact integral
Introduction to Antiderivatives
This is a new notation and also a new concept. G(x) = tiderivative of g. Other ways of saying this are: G (x) = g(x) or, dG = g(x)dx g(x)dx is the an
There are a few things to notice about this definition. It includes a di
Introduction to Maxima and Minima
Suppose you have a function like the one in Figure 1. Find the maximum value
of the function. Then nd the minimum.
To nd the maximum value the function could output, we look at the graph
and nd the highest point. To nd th
1
Antiderivatives of sec2 x and
1 x2
Example:
sec2 x dx
Searching for antiderivatives will help you remember the specic formulas
d
for derivatives. In this case, you need to remember that dx tan x = sec2 x.
sec2 x dx = tan x + c
1
dx
1 x2
An alternate wa
Maximum Area of Two Squares
Consider a wire of length 1, cut into two pieces. Bend each piece into a square.
We want to gure out where to cut the wire in order to enclose as much area in
the two squares as possible.
In all of these problems you start with
Example: f (x) = x2
Professor Jerison only does one simple example of computing the denite in
tegral as a limit because computing integrals this way involves a lot of hard
work.
In this example well use the rst interesting curve, f (x) = x2 , with startin
Easy Denite Integrals
Well do two more (much easier) examples so that we can see the pattern in
these calculations.
Example: f (x) = x
b
Compute 0 f (x) dx.
b
b
Figure 1: The area under the curve is
b
0
xdx
Looking at Figure 1 we see that the area under t
Review for Test 2
Exam 2 is usually the hardest test of the course. The topics covered will be:
1. Linear and/or quadratic approximations
2. Sketches of y = f (x)
3. Max/min problems.
4. Related rates.
5. Antiderivatives. Separation of variables.
6. Mean
Riemann Sums
We havent yet nished with approximating the area under a curve using sums
of areas of rectangles, but we wont use any more elaborate geometric arguments
to compute those sums.
y=f(x)
a
ci
b
Figure 1: Area under a curve
The general procedure f
Example:
dy
= f (x)
dx
Well solve our rst, very simple, example using the method of separation of
variables. We start by multiplying both sides by dx:
dy = f (x) dx.
Then integrate both sides:
dy =
f (x) dx
The antiderivative of dy is just y , so we get: