Pre-Calculus - High School Math

This question is asking for the equation's slant asymptote. To find the slant asymptote, divide the numerator by the denominator. Long division gives us the following:

However, because we are considering as it approaches infinity, the effect that the last term has on the overall linear equation quickly becomes negligible (tends to zero). Thus, it can be ignored.

The domain is defined as the set of possible values for the x variable. In order to find the impossible values of x, we should:

a) Set the equation under the radical equal to zero and look for probable x values that make the expression inside the radical negative:

There is no real value for x that will fit this equation, because any real value square is a positive number i.e. cannot be a negative number.

b) Set the denominator of the fractional function equal to zero and look for probable x values:

Now we can solve the equation for x:

There is no real value for x that will fit this equation.

The radical is always positive and denominator is never equal to zero, so the f(x) is defined for all real values of x. That means the set of all real numbers is the domain of the f(x) and the correct answer is .

Alternative solution for the second part of the solution:

After figuring out that the expression under the radical is always positive (part a), we can solve the radical and therefore denominator for the least possible value (minimum value). Setting the x value equal to zero will give the minimum possible value for the denominator.

That means the denominator will always be a positive value greater than 1/2; thus it cannot be equal to zero by setting any real value for x. Therefore the set of all real numbers is the domain of the f(x).

The domain is defined as the set of all values of x for which the function is defined i.e. has a real result. The square root of a negative number isn't defined, so we should find the intervals where that occurs:

The square of any number is positive, so we can't eliminate any x-values yet.

If the denominator is zero, the expression will also be undefined.

Find the x-values which would make the denominator 0:

Therefore, the domain is .

The domain is defined as the set of all possible values of the independent variable . First, must be greater than or equal to zero, because if , thenwill be undefined. In addition, the total expression under the radical, i.e. must be greater than or equal to zero:

That means that the expression under the radical is always positive and therefore is defined. Hence, the domain of the function is .

will approach when approaches , so will be of type as shown below:

So, we can apply the L’ Hospital's Rule:

since:

hence:

The function given is a polynomial with a term , such that is greater than 1.

Whenever this is the case, we can say that the whole function diverges (approaches infinity) in the limit as approaches infinity.

This tells us that the given function is not a very realistic description of a car's speed for large !

This can be rewritten as follows:

We can substitute , noting that as , :

, which is the correct choice.

Notice that on the interval , the term is always less than or equal to . So the function is largest at the points when . This occurs at and .

Plugging in either 1 or 0 into the original function yields the correct answer of 0.

We have a point at which . We know from the second derivative test that if the second derivative is negative, the function has a maximum at that point. If the second derivative is positive, the function has a minimum at that point. If the second derivative is zero, the function has an inflection point at that point.

Plug in 0 into the second derivative to obtain

So the point is an inflection point.

This is a graph of . We know that is undefined; therefore, there is no value for . But as we take a look at the graph, we can see that as approaches 0 from the left, approaches negative infinity.

This can be illustrated by thinking of small negative numbers.

NOTE: Pay attention to one-sided limit specifications, as it is easy to pick the wrong answer choice if you're not careful.

is actually infinity, not negative infinity.

The formula for the sum of an arithmetic series is

,

where is the number of terms in the series, is the first term, and is the last term.

The formula for the sum of an arithmetic series is

,

where is the number of terms in the series, is the first term, and is the last term.

We know that there are terms in the series. The first term is and the last term is . Our formula becomes:

The sum of even integers represents an arithmetic series.

The formula for the partial sum of an arithmetic series is

,

where is the first value in the series, is the number of terms, and is the difference between sequential terms.

Plugging in our values, we get:

To best understand, let's write out the series. So

We can see this is an infinite geometric series with each successive term being multiplied by .

A definition you may wish to remember is

where stands for the common ratio between the numbers, which in this case is or . So we get

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

This is a geometric series whose first term is and whose common ratio is . The sum of this series is:

A function's symmetry is related to its classification as even, odd, or neither.

Even functions obey the following rule:

Because of this, even functions are symmetric about the y-axis.

Odd functions obey the following rule:

Because of this, odd functions are symmetric about the origin.

If a function does not obey either rule, it is neither odd nor even. (A graph that is symmetric about the x-axis is not a function, because it does not pass the vertical line test.)

To test for symmetry, simply substitute into the original equation.

Thus, this equation is even and therefore symmetric about the y-axis.

The sequence can be described by the equation , where is the term in the sequence.

For the 15th term, .