Featured 1 week ago

To find the value for

We know from the hint that

Now we need to make

Featured 1 week ago

See below.

This differential equation is linear so it can be represented as the sum

The solution for

now supposing that

we get

Featured 6 days ago

See below.

Since the derivative of a function is a function for finding the gradient at a given point, we find this function in a similar way as we would find the gradient of a line. i.e.

We start with a point

From example we have:

Expanding:

Simplifying:

Cancelling:

We now take the limit:

Featured 6 days ago

#"let "y=(sinx-cosx)/(sinx+cosx)#

#"differentiate using the "color(blue)"quotient rule"#

#"given "y=(g(x))/(h(x))" then"#

#dy/dx=(h(x)g'(x)-g(x)h'(x))/(h(x))^2larrcolor(blue)"quotient rule"#

#g(x)=sinx-cosxrArrg'(x)=cosx+sinx#

#h(x)=sinx+cosxrArrh'(x)=cosx-sinx#

#dy/dx=((sinx+cosx)(sinx+cosx)-(sinx-cosx)-1(sinx-cosx))/(sinx+cosx)^2#

#=((sinx+cosx)^2+(sinx-cosx)^2)/(sinx+cosx)^2#

#=(sinx+cosx)^2/(sinx+cosx)^2+(sinx-cosx)^2/(sinx+cosx)^2#

#=1+y^2#

Featured 6 days ago

**Solution 1**

Use the formula for an arithmetic series (the sum of an arithmetic sequence).

Now solve

**Solution 2**

The sum of the first

The sum of the first

So we have

Featured 6 days ago

See below.

Exponential decays typically start with a differential equation of the form:

That is, the rate at which a population of something decays is directly proportional to the negative of the current population at time

We will now solve the equation to find a function of

This is the general form of the exponential decay formula and will typically have graphs that look like this:

graph{e^-x [-1.465, 3.9, -0.902, 1.782]}

**Perhaps an example might help?**

Consider a lump of plutonium 239 which initially has

We are told the lump has

Now at 1 million years:

Rearrange to get:

So

For the next part:

Rearrange to get

Now for the last part, the decay rate is already defined a way back at the very start, simply evaluate it at the given time:

The idea is to start with differential equation above, which gives the decay rate, and solve it to get the population at any given time.

Featured yesterday

To evaluate

So,

where

For

Featured yesterday

See below.

1)

and

and for small values of

2)

Making

now

Featured yesterday

Please see below.

The standard (basic) definition of convergence of an improper integral tells us that

If

There is another definition, called the Cauchy Principle Value. For the integral you are working on,

the Cauchy Principle Value is

The standard (or Basic) definition requires us to take the two limit separately. The Cauchy principal value takes the limit of the sum.

As for you specific question, I am not confident about any definition of convergence. that would allow us to answer the question, "Does

I can say that in the extended real numbers this evaluates to

Featured yesterday

Yes, we do multiply by the conjugate, but it is easy to be mistaken about what the conjugate is.

The conjugate of

So if either of

The analog for cube roots is based on the fact that

So the conjugate of

# = 1/((x+h)^(2/3)+(x(x+h))^(1/3) + x^(2/3))#

And, now we can evaluate the limit