Math 202 – Test 2 Review



Math 202 – Test 2 Review

Chapter 12:

(See review of natural log properties below.)

***Natural log rule review:

• ln(a × b) = ln a + ln b

• ln(a / b) = ln a – ln b

• ln ar = r (ln a)

• ln 1 = 0

• ln e = 1

• e ln x = x

• ln e x = x

• log a b = [pic]

1. Be able to take the following derivatives:

• Derivative of a natural log, page 529; [pic] and [pic]

• Derivative of y = eu, where u = f(x), page 536; . [pic] and [pic]

2. Be able to accomplish an implicit differentiation.

3. Be able to utilize logarithmic differentiation, that is, take the log or natural log of both sides of an equation, then

differentiate. Use properties stated above for natural log if needed.

4. Be able to take higher order derivatives.

• Recognize and be able to use alternate notations that represent higher order derivatives. (See table page 557.)

• Be able to interpret a 2nd degree derivative in a practical situation.

5. Be able to work with such derivatives in practical situations such as those presented in the text.

Chapter 13:

1. Understand and be able to work with the concepts of an increasing or decreasing function.

• Recognize that a function is increasing when its 1st derivative is positive.

• Recognize that a function is decreasing when its 1st derivative is negative.

• Be able to determine whether a function is increasing or decreasing in a specific interval.

2. Understand and be able to work with the concepts of a relative maximum or relative minimum in the evaluation

(and in the graph) of a function.

• Recognize that such a max or min may occur when the 1st derivative of a function is zero.

• Recognize that such a max or min does occur at a specific point in an interval when a function that is continuous in that interval changes from increasing to decreasing (or vice-versa).

• Be able to locate relative maximum and/or minimum points of a given function.

3. Be able to locate the absolute maximum and/or absolute minimum points of a given function over an identified

interval.

4. Understand and be able to work with the interpretation of the 2nd derivative of a function as the rate of change in

the slope of lines tangent to the graph of the function.

• Recognize that an increasing slope curves the function upward (concave up).

• Recognize that a decreasing slope curves the function downward (concave down).

• Be able to evaluate the 2nd derivative of a function in a given interval to determine whether the function is concave up or concave down in that interval.

• Be able to identify maximum or minimum points of a function through consideration of points where the first derivative is 0, followed by consideration of the 2nd derivative at any such point.

o If the second derivative is negative, the graph is concave down at that point and the point is a maximum.

o If the second derivative is positive, the graph is concave up at that point and the point is a minimum.

• Recognize that an inflection point occurs where a function changes from concave up to concave down (or vice versa). Such points may occur when the 2nd derivative of the function is zero.

o Be able to identify inflection points by analyzing concavity in the neighborhood surrounding the point.

5. Be able to graph a function by:

• Identifying x and y intercepts of the function if possible.

• Identifying any symmetry about x axis, y axis or origin.

• Option 1:

o Identifying portions of the graph of the function that are increasing and/or decreasing through use of the 1st derivative.

o Identifying relative maximum and/or minimum points and inflection points through use of 1st and 2nd derivatives and the "increasing/decreasing" analysis above.

• Option 2:

o Identifying maximum/ minimum critical values by finding values that make the first derivative 0.

o Determining if the values represent max/min points by checking those values in the second derivative. If the result is negative, the function is concave down and the point is a maximum. If the result is positive, the function in concave up and the point is a minimum.

• Consideration of the action of the function as x( + ( and as x ( – (.

• "Connecting the dots" once the above information is established.

6. In addition to the graphing techniques listed in #5 above, be able to recognize and include asymptotes in the graphs of

rational functions.

• Recognize that a vertical asymptote occurs in the graph of a rational function at any x value that causes the function to be undefined. Be able to analyze and indicate the action of the graph in the vicinity of an asymptote.

• Recognize that a horizontal asymptote may occur in the graph of a rational function. Such asymptotes are identified through consideration of [pic].

7. Be able to work with application problems such as those presented in the text.

Math 202 – Test 2 Review

Chapter 12:

(See review of natural log properties below.)

***Natural log rule review:

• ln(a × b) = ln a + ln b

• ln(a / b) = ln a – ln b

• ln ar = r (ln a)

• ln 1 = 0

• ln e = 1

• e ln x = x

• ln e x = x

• log a b = [pic]

2. Be able to take the following derivatives:

• Derivative of a natural log, page 529; [pic] and [pic]

• Derivative of y = eu, where u = f(x), page 536; . [pic] and [pic]

2. Be able to accomplish an implicit differentiation.

3. Be able to utilize logarithmic differentiation, that is, take the log or natural log of both sides of an equation, then

differentiate. Use properties stated above for natural log if needed.

4. Be able to take higher order derivatives.

• Recognize and be able to use alternate notations that represent higher order derivatives. (See table page 557.)

• Be able to interpret a 2nd degree derivative in a practical situation.

5. Be able to work with such derivatives in practical situations such as those presented in the text.

Chapter 13:

1. Understand and be able to work with the concepts of an increasing or decreasing function.

• Recognize that a function is increasing when its 1st derivative is positive.

• Recognize that a function is decreasing when its 1st derivative is negative.

• Be able to determine whether a function is increasing or decreasing in a specific interval.

2. Understand and be able to work with the concepts of a relative maximum or relative minimum in the evaluation

(and in the graph) of a function.

• Recognize that such a max or min may occur when the 1st derivative of a function is zero.

• Recognize that such a max or min does occur at a specific point in an interval when a function that is continuous in that interval changes from increasing to decreasing (or vice-versa).

• Be able to locate relative maximum and/or minimum points of a given function.

3. Be able to locate the absolute maximum and/or absolute minimum points of a given function over an identified

interval.

4. Understand and be able to work with the interpretation of the 2nd derivative of a function as the rate of change in

the slope of lines tangent to the graph of the function.

• Recognize that an increasing slope curves the function upward (concave up).

• Recognize that a decreasing slope curves the function downward (concave down).

• Be able to evaluate the 2nd derivative of a function in a given interval to determine whether the function is concave up or concave down in that interval.

• Be able to identify maximum or minimum points of a function through consideration of points where the first derivative is 0, followed by consideration of the 2nd derivative at any such point.

o If the second derivative is negative, the graph is concave down at that point and the point is a maximum.

o If the second derivative is positive, the graph is concave up at that point and the point is a minimum.

• Recognize that an inflection point occurs where a function changes from concave up to concave down (or vice versa). Such points may occur when the 2nd derivative of the function is zero.

o Be able to identify inflection points by analyzing concavity in the neighborhood surrounding the point.

5. Be able to graph a function by:

• Identifying x and y intercepts of the function if possible.

• Identifying any symmetry about x axis, y axis or origin.

• Option 1:

o Identifying portions of the graph of the function that are increasing and/or decreasing through use of the 1st derivative.

o Identifying relative maximum and/or minimum points and inflection points through use of 1st and 2nd derivatives and the "increasing/decreasing" analysis above.

• Option 2:

o Identifying maximum/ minimum critical values by finding values that make the first derivative 0.

o Determining if the values represent max/min points by checking those values in the second derivative. If the result is negative, the function is concave down and the point is a maximum. If the result is positive, the function in concave up and the point is a minimum.

• Consideration of the action of the function as x( + ( and as x ( – (.

• "Connecting the dots" once the above information is established.

6. In addition to the graphing techniques listed in #5 above, be able to recognize and include asymptotes in the graphs of

rational functions.

• Recognize that a vertical asymptote occurs in the graph of a rational function at any x value that causes the function to be undefined. Be able to analyze and indicate the action of the graph in the vicinity of an asymptote.

• Recognize that a horizontal asymptote may occur in the graph of a rational function. Such asymptotes are identified through consideration of [pic].

7. Be able to work with application problems such as those presented in the text.

Math 202 – Test 2 Review

Chapter 12:

(See review of natural log properties below.)

***Natural log rule review:

• ln(a × b) = ln a + ln b

• ln(a / b) = ln a – ln b

• ln ar = r (ln a)

• ln 1 = 0

• ln e = 1

• e ln x = x

• ln e x = x

• log a b = [pic]

3. Be able to take the following derivatives:

• Derivative of a natural log, page 529; [pic] and [pic]

• Derivative of y = eu, where u = f(x), page 536; . [pic] and [pic]

2. Be able to accomplish an implicit differentiation.

3. Be able to utilize logarithmic differentiation, that is, take the log or natural log of both sides of an equation, then

differentiate. Use properties stated above for natural log if needed.

4. Be able to take higher order derivatives.

• Recognize and be able to use alternate notations that represent higher order derivatives. (See table page 557.)

• Be able to interpret a 2nd degree derivative in a practical situation.

5. Be able to work with such derivatives in practical situations such as those presented in the text.

Chapter 13:

1. Understand and be able to work with the concepts of an increasing or decreasing function.

• Recognize that a function is increasing when its 1st derivative is positive.

• Recognize that a function is decreasing when its 1st derivative is negative.

• Be able to determine whether a function is increasing or decreasing in a specific interval.

2. Understand and be able to work with the concepts of a relative maximum or relative minimum in the evaluation

(and in the graph) of a function.

• Recognize that such a max or min may occur when the 1st derivative of a function is zero.

• Recognize that such a max or min does occur at a specific point in an interval when a function that is continuous in that interval changes from increasing to decreasing (or vice-versa).

• Be able to locate relative maximum and/or minimum points of a given function.

3. Be able to locate the absolute maximum and/or absolute minimum points of a given function over an identified

interval.

4. Understand and be able to work with the interpretation of the 2nd derivative of a function as the rate of change in

the slope of lines tangent to the graph of the function.

• Recognize that an increasing slope curves the function upward (concave up).

• Recognize that a decreasing slope curves the function downward (concave down).

• Be able to evaluate the 2nd derivative of a function in a given interval to determine whether the function is concave up or concave down in that interval.

• Be able to identify maximum or minimum points of a function through consideration of points where the first derivative is 0, followed by consideration of the 2nd derivative at any such point.

o If the second derivative is negative, the graph is concave down at that point and the point is a maximum.

o If the second derivative is positive, the graph is concave up at that point and the point is a minimum.

• Recognize that an inflection point occurs where a function changes from concave up to concave down (or vice versa). Such points may occur when the 2nd derivative of the function is zero.

o Be able to identify inflection points by analyzing concavity in the neighborhood surrounding the point.

5. Be able to graph a function by:

• Identifying x and y intercepts of the function if possible.

• Identifying any symmetry about x axis, y axis or origin.

• Option 1:

o Identifying portions of the graph of the function that are increasing and/or decreasing through use of the 1st derivative.

o Identifying relative maximum and/or minimum points and inflection points through use of 1st and 2nd derivatives and the "increasing/decreasing" analysis above.

• Option 2:

o Identifying maximum/ minimum critical values by finding values that make the first derivative 0.

o Determining if the values represent max/min points by checking those values in the second derivative. If the result is negative, the function is concave down and the point is a maximum. If the result is positive, the function in concave up and the point is a minimum.

• Consideration of the action of the function as x( + ( and as x ( – (.

• "Connecting the dots" once the above information is established.

6. In addition to the graphing techniques listed in #5 above, be able to recognize and include asymptotes in the graphs of

rational functions.

• Recognize that a vertical asymptote occurs in the graph of a rational function at any x value that causes the function to be undefined. Be able to analyze and indicate the action of the graph in the vicinity of an asymptote.

• Recognize that a horizontal asymptote may occur in the graph of a rational function. Such asymptotes are identified through consideration of [pic].

7. Be able to work with application problems such as those presented in the text.

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