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AP CALCULUS: COMPREHENSIVE REVIEW OF SECTIONS 3.1-3.6

A. Critical Numbers

To find critical numbers of a function  f,

a.       Find the derivative of  f.

b.      Find the  x-coordinates where the derivative is zero or undefined.

c.       The coordinates from (b) are the critical numbers of the function.

Example: Find the critical numbers of 

            a.

            b.

            c. The critical numbers are .

Now you try: Find the critical numbers of .

B. Maxima and Minima on Closed Intervals

To find absolute extrema of a function on a closed interval,

a.       Find the critical numbers of the function.

b.      Evaluate the function at the critical numbers contained within the interval.

c.       Evaluate the function at the endpoints of the interval.

d.      The smallest value from (b) and (c) is the absolute minimum of the function on the closed interval, and the largest value from (b) and (c) is the absolute maximum.

Example: Find the absolute extrema of  on the closed interval .

a.       From (A), we found the critical number of  f  to be .

b.     

c. ;     

            d. Absolute Minimum:                   Absolute Maximum:

Now you try: Find the absolute extrema of  on the closed interval .

C. Rolle’s Theorem

If all three of the following conditions of a function  f  are satisfied,

            a.  is continuous on

            b.  is differentiable on

            c.

Then Rolle’s Theorem states there must be at least one value  c,  for which .

Example: Explain why Rolle’s Theorem does not apply to  on .

Rolle’s Theorem does not apply to  f  on the closed interval since  f  is discontinuous at .

Now you try: Explain why Rolle’s Theorem does not apply to  on .

Example: Verify Rolle’s Theorem for +3 on the closed interval  and find the value  c, , for which .

            a.  is continuous on  since polynomials are continuous everywhere.

            b.  is differentiable on  since polynomials are differentiable everywhere.

            c.

. Setting this equal to zero yields , which is contained within the specified interval.

Now you try: Verify Rolle’s Theorem for  on , and find the value  c, , for which .

D. The Mean Value Theorem

If both of the following conditions of a function   are satisfied,

a.       f  is continuous on

b.      f  is differentiable on

Then the Mean Value Theorem states that there exists at least one value of  c, , for which the derivative of  f  at  c  is equal to the average rate of change of  f on :

Example: Verify that the Mean Value Theorem applies to  on , and find the value of  c, , for which .

a.       f  is continuous on , since polynomials are continuous everywhere.

b.      f  is differentiable on , since polynomials are differentiable everywhere.

. Setting the derivative equal to 8 (x

Now you try: Verify the Mean Value Theorem for  on , and find the value of  c, , for which .

E. Increasing and Decreasing Functions

To find where a function is increasing or decreasing,

a.       Calculate the derivative of the function.

b.      Determine the intervals on which the derivative is positive.

c.       Determine the intervals on which the derivative is negative.

d.      The function is increasing on the intervals from (b), and decreasing on the intervals from (c). In other words, a function is increasing when its derivative is positive and decreasing when its derivative is negative.

Example: Determine the interval(s) of increase and the interval(s) of decrease of the function .

a.       From (A), we know the derivative of  f  is .

b.       is positive on .

c.        is negative on

d.      Increasing , Decreasing

Now you try: Determine the interval(s) of increase and the interval(s) of decrease of the function .

F. Relative Extrema

A function  f  has a relative maximum at   if

            a.  is a critical number of the function  f.

            b.  exists.

            c. The sign of  changes from positive to negative at .

A function  f  has a relative minimum at  if

            a.  is a critical number of the function  f.

            b.  exists.

            c. The sign of  changes from negative to positive at .

First Derivative Test: If a function  f is defined at , andis a critical number of , and the sign of  changes from positive/negative to negative/positive at , then has a relative maximum/minimum at .

Example: Determine all relative extrema of the function .

a.       From (A), we know that  has a critical number at .

b.      From (A), we also know that  exists and that .

c.       From (E), we know that the sign of  changes from negative to positive at , so f  has a relative (and absolute) minimum at .

Now you try: Determine all relative extrema of the function .

G. Concavity

A function is concave up on an open interval if it is above all of its tangent lines on that interval. A function is concave down on an open interval if it is below all of its tangent lines on that interval. Visually, they appear as “smiley” faces and “frownie” faces:

A function is concave up where its second derivative is positive.

A function is concave down where its second derivative is negative. 

Example: Determine the intervals of concavity of .

.

 is positive on  so  f  is concave up on .

 is negative on  so  is concave down on .

Now you try: Determine the intervals of concavity of the function .

H. Points of Inflection

A function  f  has a point of inflection at  if  exists and the sign of  changes at  c. Think of it as where a function changes from smiley face to frownie face or vice versa:

To find points of inflection,

a.       Determine the second derivative of a function.

b.      Determine where the second derivative is zero or undefined.

c.       If the sign of   changes at the values from (b) and  f  exists at the values from (b), then  f  has points of inflection at the values from (b).

Now you try: Determine all points of inflection of .

I. The Second Derivative Test

Let   be a function for which  exists and . The Second Derivative Test states that

            a. If ,  f  has a relative maximum at  c.

            b. If ,  f  has a relative minimum at  c.

            c. If , the Second Derivative Test is inconclusive and the First Derivative Test

            must be used.

Example: Use the Second Derivative Test to determine the extrema of .

                    for                   

;                            

Relative Maximum:                  Relative Minimum:

Now you try: Use the second derivative test to find the relative extrema of .

J. Limits at Infinity

To determine an infinite limit,

a.       Divide both the numerator and the denominator by the highest power in the denominator.

b.      Evaluate the limit.

Example: Evaluate .

.

Shortcut: If   is a rational function,

            a. , if the degree of the numerator is less than the degree of the denominator.

            b. , if the degree of the numerator equals the degree of the denominator.

            c. , if the degree of the numerator exceeds the degree of the denominator.

Now you try: Evaluate:

K. Asymptotes

A function has a horizontal asymptote at   if

              or                                    or                      

A function has a vertical asymptote at  if it increases or decreases without bound as  x approaches  c (the limit is an infinity). If  f  has a vertical asymptote at  c, neither the limit nor the function will exist at  c.

A function will have a slant asymptote if the degree of the numerator exceeds the degree of the denominator by one. Use polynomial long division to find this asymptote.

Now you try: Find all asymptotes of .

Final: Use the information from this review to produce a detailed graph of .

L. Review Table (submitted by Andy Knapp)

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Last modified: 01/09/08