# Calculus 1 : Graphing Functions

## Example Questions

### Example Question #5 : How To Graph Functions Of Area

Find the area bounded by the curve  in the first quadrant.

Explanation:

The curve is in quadrant one over the interval , which are the bounds of integration.  To see this, note that the x-intercepts are 0 and 2 and the parabola opens downward.

The definite integral below is solved by taking the antiderivative of each term of the given polynomial function, evaluating this antiderivative at the bounds of integration, and subtracting the values.

For this particular integral use the rule,  to solve.

### Example Question #6 : How To Graph Functions Of Area

Find the area under the curve between the following bounderies of the following function.

in between the boundaries of  and

Explanation:

We can find the area under the curve by taking the anti-derivative of the function and using the two boundaries as x values.  The anti-derivative of  is .  If we use the two boundaries, we end up with our answer, .

### Example Question #1 : How To Graph Functions Of Area

Evalute the following Definite Integral:

Explanation:

For this integral one may be tempted to directly integrate; however, this is no rule of integrals which allows us to do so.

We must apply a complex method of integration, here u-subtitution works best.

We notice that Cos(x) is the derivative of Sin(x) so it may be best to let .

Now that we have found a proper u and du, we may directly substiute into our original integral

However, our limits are in terms of x so we must substitute our u=Sin(x) back in before evaluating

### Example Question #1 : How To Describe Points By Graphing Functions

Suppose a point on the curve given above has the property that

Based solely on the graph above, which of the following is most likely the  value of the point in question?

Explanation:

If then the graph must be concave up at the point. Based on the picture, we know that the curve is concave up on  at best. The only value that falls on this interval is , which is . Since , this definitely falls on the interval given and we can be sure it is concave up based on the picture.

### Example Question #2 : How To Describe Points By Graphing Functions

What is the critical point for ?

Explanation:

To find the critical point, you must find the derivative first. To do that, multiply the exponent by the coefficient in front of the  and then subtract the exponent by . Therefore, the derivative is: . Then, to find the critical point, set the derivative equal to .

.

### Example Question #1 : Points Of Inflection

Find the inflection point(s) of .

Explanation:

Inflection points can only occur when the second derivative is zero or undefined. Here we have

Therefore possible inflection points occur at  and . However, to have an inflection point we must check that the sign of the second derivative is different on each side of the point. Here we have

Hence, both are inflection points

### Example Question #2 : How To Graph Functions Of Points Of Inflection

Below is the graph of . How many inflection points does  have?

Not enough information

Explanation:

Possible inflection points occur when  . This occurs at three values, . However, to be an inflection point the sign of  must be different on either side of the critical value. Hence, only  are critical points.

### Example Question #3 : How To Graph Functions Of Points Of Inflection

Find the point(s) of inflection for the function .

and

There are no points of inflection.

and

Explanation:

A point of inflection is found where the graph (or image) of a function changes concavity. To find this algebraically, we want to find where the second derivative of the function changes sign, from negative to positive, or vice-versa. So, we find the second derivative of the given function

The first derivative using the power rule

is,

and the seconds derivative is

We then find where this second derivative equals  when .

We then look to see if the second derivative changes signs at this point. Both graphically and algebraically, we can see that the function  does indeed change sign at, and only at, , so this is our inflection point.

### Example Question #4 : How To Graph Functions Of Points Of Inflection

Find all the points of inflection of

.

There are no inflection points.

Explanation:

In order to find the points of inflection, we need to find  using the power rule, .

Now we set , and solve for .

To verify this is a true inflection point we need to plug in a value that is less than it and a value that is greater than it into the second derivative. If there is a sign change around the  point than it is a true inflection point.

Let

Now let

Since the sign changes from a positive to a negative around the point , we can conclude it is an inflection point.

### Example Question #1 : Points Of Inflection

Find all the points of inflection of

There are no points of inflection.

Explanation:

In order to find the points of inflection, we need to find  using the power rule .

Now to find the points of inflection, we need to set .

.

Now we can use the quadratic equation.

Recall that the quadratic equation is

,

where a,b,c refer to the coefficients of the equation  .

In this case, a=12, b=0, c=-4.

Thus the possible points of infection are

.

Now to check if or which are inflection points we need to plug in a value higher and lower than each point. If there is a sign change then the point is an inflection point.

To check  lets plug in .

Therefore  is an inflection point.

Now lets check  with .

Therefore  is also an inflection point.