Curve Sketch

Definition (Critical Points) We will call the number a first order critical number if or does not exist and a second-order critical number if or does not exist.

Definition (Concave Up and Concave Down) If the graph of lies above all of its tangents on an interval it is called concave upward on If the graph of lies below all of these tangents, it is called concave downward on

Definition (Infection Point) A point on a curve is called an inflection point of the graph is concave up on one side of and concave down on the other side.

Proposition (Test for Concavity) Suppose is twice differentiable on an interval Then,

    (i) If for all in then the graph of is concave upward on

    (ii) If for all in then the graph of is concave downward on
    

Example (Concavity and Inflection Points) Determine where the curve is concave upward, where it is concave downward, and where the points of inflection are (if any).

    Solution. We use the Concavity Test and find the first and second derivatives:
    

    

    
which allows us to find the second order critical numbers, namely when and We summarize the Concavity Test in the following table:  



Therefore, is concave up on and concave down on The points and are inflection points.

Example (Concavity and Inflection Points) Determine where the curve is concave upward and where it is concave downward. Find all inflection points, local extrema, and sketch the curve.

    Solution. We apply the first derivative test and the concavity test by find the first and second derivatives





and then finding the first and second order critical numbers, namely:





Therefore, the first order critical numbers are and the second order critical numbers are We summarize the First Derivative Test and the Concavity Test in the following table:



Therefore, the function has a local maximum at and a local minimum at This function is increasing on the and decreasing on The point is an inflection point because is concave up on the interval and concave down on

Definition (Critical Points) We will call the number a first order critical number if or does not exist and a second-order critical number if or does not exist.

Proposition (Second Derivative Test) Suppose in continuous on an open interval that contains with Then

    (i) If then has a relative (local) minimum at
    
    (ii) If then has a relative (local) maximum at

    Warning. The second derivative test doe not tell us anything if both and For example, if and both

and

The point is a minimum for but is neither a maximum not a minimum for However, the first derivative test is still useful.

Example (Second Derivative Test) Use the second derivative test to determine whether each critical number of the function corresponds to a relative maximum, a relative minimum, or neither.

    Solution. We find the first order critical numbers by



and so and are the critical numbers. Now to apply the Second Derivative Test we find the second derivative, Since the point is a local maximum and since the point is a local minimum.

Example (Second Derivative Test) Use the second derivative test to determine whether each critical number of the function corresponds to a relative maximum, a relative minimum, or neither.

    Solution. We find the first order critical numbers by



and so are the first order critical numbers of Note that, even though is undefined, so is and so is not a critical number. We compute and since the point is a local minimum and since the point is a local maximum.

Example (Sketching the Graph of a Function)  For the function . Find all first and second order critical numbers. Apply the First Derivative Test, Concavity Test, and the Second Derivative Test. Sketch the graph of the function.

    Solution. The first and second derivatives are:





Solving we find the first order critical numbers to be Solving we find the second order critical numbers to be   Applying the First Derivative Test we find,  



Checking these local extrema by using the Second Derivative Test: determining and so is a local minimum and and so is a local maximum. Applying the Concavity Test:  



A sketch of the graph of follows:

Example (Sketching the Graph of a Function)  For the function . Find all first and second order critical numbers. Apply the First Derivative Test, Concavity Test, and the Second Derivative Test. Sketch the graph of the function.

    Solution. The first and second derivatives are:





Note that is not a critical number because is not defined. Solving we find the first order critical numbers to be Solving we find the second order critical numbers to be   Applying the First Derivative Test we find,  



Checking this local extrema by using the Second Derivative Test: determining



and so

is a local minimum.

Applying the Concavity Test:  



A sketch of the graph of follows:

    There are four possibilities for unbounded behavior of a derivative around a given real number They are:



which are called vertical tangents; and when these limits differ in sign they are:



which are called cusps. Here are the formal definitions of vertical tangents and cusps followed by an example of each.

Definition (Vertical Tangent) Suppose the function is continuous at the point Then the graph of has a vertical tangent at if one of the following holds:

Example (Vertical Tangent) Sketch the graph of and explain why there is a vertical tangent at .

    Solution. Using the product rule we find the derivative as
    








To determine any vertical tangents we consider where is undefined. Notice that at the derivative is undefined but Thus, the point is a candidate for a being a vertical tangent to the graph of We check the following limits to determine if is a vertical tangent.
    
    Since and as   


    
    Since and as   




By the definition of a vertical tangent is a vertical tangent which can be seen from the sketch of the graph of

Definition (Cusp) Suppose the function is continuous at the point Then the graph of has a cusp at if one of the following holds:

Example (Cusp) Sketch the graph of and explain why there is a cusp at .

    Solution. Using the product rule we find the derivative as
    










To determine any cusps we consider where is undefined. Notice that at the derivative is undefined but Thus, the point is a candidate for a being a cusp for the graph of the function . We check the following limits to determine if is a cusp.
    
    Since and as   


    
    Since and as   



By the definition of a cusp is a cusp which can be seen from the sketch of the graph of

Example (Curve Sketching) Sketch the graph of the rational function



showing all special features.

    Solution. The domain is



The and intercepts are both Since the function is even. The curve is symmetric about the -axis.



Therefore, the line is a horizontal asymptote. Since the denominator is 0 when we compute the following limits:



Therefore, the lines and are vertical asymptotes.

Next we find the derivative function.



Since when and when is increasing on and and decreasing on and The only critical number is Since changes sign from positive to negative at is a local maximum by the First Derivative Test. Also,



Since for all we have



and



Thus the curve is concave downward on the intervals and and concave downward on There is no point of inflection since and are not in the domain of Here is a sketch of the graph:


  

Example (Curve Sketching) Sketch the graph of the trigonometric function



showing all special features.

    Solution. The domain is The -intercept is The -intercept occur when



which is precisely when and , because we need only consider since function is periodic via,  



There are no asymptotes. Computing the derivative,







Thus, when or when so in when consider the critical number and and Applying the First Derivative Test we find:



Computing



we find the second order critical numbers as and where and Applying the Concavity Test we find,



Here is a sketch of the graph:

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Curve Sketch
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/curve-sketch.html