· Web viewaxis based upon if the function is increasing or decreasing. Based upon the previous days, define increasing and decreasing of the original function based upon the derivative.

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The Second Derivative Answer key

To the best of your ability, define…

concave up: Variable answers concave down Variable answers point of inflection Variable answers

1. This activity will help us to better state these ideas sing the language of calculus. Open the Sketchpad file. http://www.uhigh.ilstu.edu/math/thompson/sketchpad/apcalc/5.3%20concavity.gsp Page one shows coloring of the x axis based upon if the function is increasing or decreasing. Based upon the previous days, define increasing and decreasing of the original function based upon the derivative.a. A continuous function is increasing on an interval if

the derivative is positive, they could also say that the slope of the tangent line is positiveb. A continuous function is decreasing on an interval if

the derivative is negative they could also say that the slope of the tangent line is negativec. Give a thorough answer: In order to find a local maximum or minimum for a function, I would…

Find where f ' is either zero or undefined. If it is undefined test values on either side to make a determination

d. Explain why the sign of the derivative is connected to the function being increasing or decreasing. Draw a diagram to help you explain.

When a function is increasing we know that a larger x value has a larger y value. If this is so, the slope of a secant line between these points must be positive. This would be true no matter how close the points are thus if the function is increasing, the derivative must then be positive as the derivative is the instantaneous slope.

2. Go to the second page of the Sketchpad document. On this page, the sketch will shade the x axis different

colors based upon if the function, , is concave up or down at that point. Press the animate button.

a. Is it possible for the function to be increasing and concave down? __YES!_________

b. Thinking about the slope of the tangent line that is drawn in Sketchpad, define the concepts below as carefully as possible.

Concave up When the slope of the tangent line is increasing. Students will be prone to write positive here be sure to address this misconception

Concave down When the slope of the tangent line is decreasing. Students will be prone to write negative here be sure to address this misconception

c. Estimate the x-value at which the graph changes concavity._____-3, 0__________

3. Based upon previous knowledge, calculate the derivative of .

4. When you click the button: "Show Functions1" in Sketchpad, it should verify the answer above. Determine where the derivative is increasing and decreasing through analytic methods—solve for critical numbers etc. Write several sentences explaining how you found the answers below. Since we want to

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The Second Derivative Answer key

know if the derivative is increasing or decreasing, we need to find the second derivative and find the critical numbes.

Increasing:Decreasing:Local maximums:Local minimums

5. Do you notice any connection between your answers for the above problem and the answers you provided for the intervals of concave up/ concave down? Explain any connection and why it exists.The derivative has a minimum and a maximum where the original function changes concavity. The derivative changes from increasing to decreasing or vice versa at these points since they are the extrema for the derivative. Thus since concavity is where the slope of tangent line (or derivative) is either increasing or decreasing, concavity must change at these same x values of the original function,

6. Calculate the second derivative of . Graph this function in Sketchpad. Press the button "Show Functions2". This second derivative is most easily connected to the concepts of concavity.

a. Write a new definition to define when a function is concave up or concave down.The original function is concave up when the second derivative is positive. The original function is concave down when the the second derivative is negative.

b. Now connect all graphs to the concepts of concavity of the original function. Describe fully below the connections. Be sure to address intercepts, max/mins, intervals of increasing and decreasing. It will be important that you are clear as to what function you are discussing. Including a graph may be helpful.The original function is concave up when the first derivative is increasing and the second derivative is positive. The original function is concave down when the first derivative is decreasing and the second derivative is negative. The intercepts of the second derivative correspond to the extrema of the first derivative and the points of inflection of the original function.

7. What is significant about where f''(x) = 0? Find a function where f''(x) = 0 yet the concavity does not change at that x value? F(x)=x 4

8. The definition in our text for point of inflection is : A point where the graph of a function has a tangent line and where the concavity changes is the point. For the original function, list the point(s) of inflection if any exist.. X= -3,0

9. If , determine the intervals where this function is/has WITHOUT graphing.

Increasing: Decreasing: Maximum:Minimum:Concave up:Concave down:Points of inflection: