Featured 1 month ago

After using

=

=

=

=

=

=

Featured 1 month ago

Equation of tangent at

Observe that

Now, as slope of tangent at any point is given by the value of first derivative

As

and putting

or

i.e.

As slope of tangent is

Featured 1 month ago

Integral is

Let

=

=

Note that when

Now observe the graph of

graph{(x-3)^3 [0.542, 5.542, -1.2, 1.3]}

Note that area of curve in the interval

Featured 1 month ago

Volume

I am assuming that you already know about area under a curve.

In this problem we have the area under

Or

And the summation of these gives us the required volume.

In this problem we are rotating about the line

If you look at the diagram, you can see that we are 9 units below the line

Our required integral will be:

Volume=V

V

V

Plugging in upper and lower bounds:

V

V

Volume of revolution:

I hope you can follow this. It is really quite simple to do, but really difficult to explain it in a simple way.

Featured 4 weeks ago

The limit is

We can rewrite as

#L = lim_(x-> oo) tan(2/x)/(1/x)#

We see that

#L = lim_(x->oo) (-2/x^2 * sec^2(2/x))/(-1/x^2)#

#L = lim_(x-> oo) 2sec^2(2/x)#

The same principal applies with

#L = 2sec^2(0)#

#L = 2(1)#

#L = 2#

A graphical verification confirms.

In the above graph, the red curve is

Hopefully this helps!

Featured 4 weeks ago

=

=

and then

=

Featured 4 weeks ago

Please see below.

Every point on the curve has coordinates

We can minimize the distance by minimizing the radicand:

Differentiate:

Use some technology or approximation method to get

There cannot be a maximum. There is a minimum at this

Using the distance above, we find a minimum distance of approximately

Featured 4 weeks ago

See below.

The critical points are the points where the first derivative equals zero, or the point doesnâ€™t exist.

This is a polynomial so it is continuous for all

First derivative of

Equating this to zero:

Factor:

Putting these values in

Critical Points:

Increasing in

Decreasing in

GRAPH:

Featured 2 weeks ago

A few thoughts...

**Definite vs indefinite integral**

A definite integral includes a specification of the set of values over which the integral should be calculated. As a result, it has a definite value, e.g. the area under a curve in a given interval.

By way of contrast, an indefinite integral does not specify the set of values over which the integral should be calculated. It basically identifies what the antiderivative function looks like, including some constant of integration to be determined. For example:

#int x^2 dx = 1/3 x^3 + C#

**Non-elementary integrals of elementary functions**

Unlike derivatives, the integral of an elementary function is not necessarily elementary. The term "elementary function" denotes functions constructed using basic arithmetic operations,

There are some very useful non-elementary functions expressible as integrals of elementary functions. For example, the Gamma function:

#Gamma(x) = int_0^oo t^(x-1) e^(-t) dt#

The Gamma function extends the definition of factorial to values apart from non-negative integers.

**Poles and Cauchy principal value**

If a function has a singularity such as a simple pole, then its definite integral over a range including that pole is not automatically well defined. A workaround for such cases is provided by the Cauchy principal value.

For example:

#int_(-1)^1 dt/t = lim_(epsilon -> 0+) (int_(-1)^-epsilon dt/t + int_epsilon^1 dt/t) = 0#

**Non measurable sets**

If the set over which you are trying to integrate is non-measurable, then the integral is usually not defined. An exception would be if the value of the function on that set was zero.

To 'construct' a non-measurable set you would typically use the axiom of choice.

For example, you could define an equivalence relation on

#a ~ b <=> (a-b) " is rational"#

This equivalence relation partitions

Use the axiom of choice to choose exactly one element of each equivalence class to create a subset

For any rational number

We can define a non-integrable function by:

#f(t) = { (1 " if " t in S_x " where " x = p/q " in lowest terms and " q " is even"), (0 " otherwise") :}#

This function is not integrable over any interval.

Featured 3 weeks ago

We have

Now, let's try to remember the definition of an integral. This might seem strange, but it will come of great use.

The definite integral

This concept is called a

There are infinitely many rectangles, therefore, if there are

Now, let's take the case where the width is constant between all boxes. If we define

The Riemann Sum aproximates an integral by being the sum of all the rectangles. In order to simplify this further, let's take the particular case where

This is only true if

One way to make

This looks familiar to our original sum,

All we are left to do is to assume they're equal and then solve the integral.

We have to find the integral of this function:

Since

This is because it forms a rectangle with the x-axis, the width and lenght of which is

As a conclusion,