The answer is: 12.

First of all: "How much is the sum of a geometric series?".

Let's list all the terms of the series:

or, better:

The first sum is:

The second sum is:

The third sum is:

The fourth sum is:

So:

3 years ago

(see below for solution method)

Start by ignoring the leakage and determine the rate of inflow required to achieve the specified rate of height (depth) of water increase.

Later we'll use the fact that

**Actual inflow rate
= Inflow Rate for Increased Depth + Leakage Rate**

For the given cone the ratio of **r** adius to **h** eight is

so

The formula for the volume of a cone:

We are interested in the change in Volume with respect to time and note that

Using the value we've already calculated for

we get:

or roughly

This is the Inflow Rate Required to Cause Height Increase and

ignores the Rate of Leakage

The Actual Inflow Rate needs to be the sum of these two:

3 years ago

An interesting example related to this one is to consider the piecewise-defined function

Again, let

Certainly, when

But

In other words,

Here's a graph of the function

Pretty amazing, isn't it?!?!

Here a closer view near the origin and also making sure we can still see the blue curve. It's oscillating infinitely often as

For a non-rigorous proof, please see below.

For a positive central angle of

Source:

The geometric idea is that

So we have:

For small positive

So

so

We also have, for these small

so

Since both one sided limits are

**Note**

This proof uses the fact that

That fact can be proved from the fact that

Which can be proved using the squeeze theorem in a argument rather like the one used above.

Furthermore: Using both of those facts we can show that the sine and cosine functions are continuous at every real number.

3 years ago

See the explanation section below.

A point of inflection is a point *on the graph* at which the concavity of the graph changes.

If a function is undefined at some value of

However, concavity *can* change as we pass, left to right across an

**Example**

The concavity changes "at"

But, since

graph{1/x [-10.6, 11.9, -5.985, 5.265]}

**Example 2**

The second derivative is undefined at

But, since

graph{x^(1/3) [-3.735, 5.034, -2.55, 1.835]}

I found:

First you need to find the slope

at

Now we need the

The equation of a line passing through

Graphically:

3 years ago

To evaluate inverse trigonometric functions, a valuable technique is implicit differentiation:

In this case, we also will use the chain rule:

as well as the following derivatives:

Now, let

Through implicit differentiation:

And through the chain rule

So

But we want our final answer entirely in terms of

From this we can see that

Thus we have

From this, we can get our final answer

1) make sure that the degree of your denominator is greater than the degree of your numerator

2) factorize the denominator

3) perform the partial fraction decomposition

4) solve the integral!

Solution:

1) **Check degrees of denominator and numerator**

First of all, you can't immediately start building partial fractions since the degree of your numerator is equal to the degree of your denominator (the strongest power in both expressions is

However, the partial fraction decomposition will only work if the degree of the denominator is greater than the degree of the numerator.

If the expression was more complicated, normally, at this point, a polynomial division is in order. Here we can achieve the goal in an easier way:

So, now we can build the partial fractions decomposition of the latter fraction.

2) **Complete factorization of the denominator**

To do so, let's factorize the denominator completely first:

[You can do so by setting

3) **Partial fraction decomposition**

So, our goal is to find

First, multiply both sides of the equation with

To find the solution of this equation, we need to "collect" the

The solution of this linear equation is

So, our partial fraction decomposition is:

In total:

4) **Solving the integral**

The last thing left to do is solve the integral!

Hope that this helped!

Your chain can be defined as follows:

Thus, to compute the derivative, you need to build the derivatives of

Now, the only thing left to do is to multiply the three derivatives!

2 years ago

The "why" depends on how you've defined

**Define #lnx# first**

One approach is to define

then to define

finally, define

In this case

Differentiating implicitly gets us

So,

**Define #e^x# independently of #lnx#**

**Definition 1**

For positive

(We owe you a proof that this is well-defined.)

Then, using the definition of derivative:

We then define

With this definition we get

# = e^xlim_(hrarr0)((e^h-1)/h) = e^x#

**Definition 2**

(

(We owe you a proof that this is well defined.)

Differentiating term by term (we owe you a proof that this is possible), we get

Which simplifies to