Name______
Part 1: Given the picture below answer the following questions:
1)The graph above is for the function. If you did not have the picture to look at but wanted to sketch it what points on the graph would be valuable points to know the locations of?
Local minima, maxima, x-intercepts, and y-intercepts are key locations that need to be determined and graphed.
2)What information about the shape of the graph or behavior of the curve in relationship to the x-axis would be important to have to be sure your sketch was as accurate as possible?
As mentioned in question 1, the x-intercepts are key locations to an accurate sketch, but also the magnitude of the slope as the curve hits those locations.
Part 2: Now that we know what kinds of information we are looking for we need to discover how to find it. For each question below set the function equal to zero and solve for x. Then, plug those values back into f(x) and locate where the resulting point(s) exists on the graph at the top of page one. For questions 2 and 3 see if you can find a way to use the point(s) you found to break the graph into intervals that will help you fill out the tables. Also, using either or both of your textbook and lecture notes find the first and second derivative tests and determine how they can help you.
1)What information about its graph can we determine from the function itself? Hint: Think back to our discussion of limits. Use the function above to find this information.
Solving for x = 0 I can determine that the y-intercept is 0, so f(x) = 0 when y = 0. This function intersects the origin (0,0).
2)What information about its graph can we determine from the 1st derivative of the function? The first derivative of our function is .
Where the first derivative equals 0 is where the slope is 0, or a local maxima or minima. By picking x values on the two sides of these points we can determine the sign of the slope, and make a conclusion about whether the points where the derivative equals zero is a minima or maxima.
=-1.045=6.379
Interval / / -1.045< /X-value in interval / -2 / 2 / 8
Sign of f’(x) at chosen value / + / - / +
Conclusion about increasing/decreasing behavior / Increasing / Decreasing / Increasing
3)What information about its graph can we determine from the 2nd derivative of the function? The second derivative of our function is .
The x-coordinate where the second derivative equals 0 is the x-coordinate where concavity shifts in the graph of the original function.
Interval / /x-value in interval / -4 / 4
Sign of f”(x) at chosen value / - / +
Conclusion about concavity. / Concave down / Concave up
4)What is the 1st derivative test and how can it help you answer part of question 2? In other words, how are we using the increasing/decreasing behavior of a function to locate the minima and maxima of that function?
By picking x values on the two sides of these points we can determine the sign of the slope, and make a conclusion about whether the points where the derivative equals zero is a minima or maxima. If both sides of the point are increasing, the point is a minima, but if both sides are decreasing, then the point is a maxima.
5)What is the 2nd derivative test and how can it help you answer part of question 2? In other words, how are we using the concavity of the function to locate the minima and maxima of the function?
If the sign of an x-value for the second derivative in the interval is positive, then the original function is concave up, if the sign is negative then the original function is concave down. A first derivative x-coordinate, where the first derivative evaluates to 0 at the x-coordinate, is a maxima for a concave down area, and a minima for a concave up area.
Part 3: Using what you have learned above find all the important information for the function below and then sketch it.
The function passes through the origin (0,0).
Interval / / / / /X-value in interval / -4 / -1 / 0.5 / 1 / 8
Sign of f’(x) at chosen value / + / - / + / - / +
Conclusion about increasing/decreasing behavior / Increasing / Decreasing / Increasing / Decreasing / Increasing
Based on the chart above:
(-2, 0) maxima
(-0.894, -9.159) minima
(0.894, 9.159) maxima
(2, 0) minima
Interval / / / /x-value in interval / -4 / -1 / 1 / 4
Sign of f”(x) at chosen value / - / + / - / +
Conclusion about concavity. / Concave down / Concave up / Concave down / Concave up
Concavity shifts at (-1.549, -3.966), (0, 0), and (1.549, 3.966).
Part 4: Optimization
Directions: A rectangular garden will be fenced off with 220 feet of available material. What is the largest area that can be fenced off? Do the following to answer this question.
1)Draw a picture of the garden and label the key dimensions.
2)List all the known and unknown quantities related to the garden and assign each a variable.
The unknown objects are the two lengths (x and y) of the fenced off area (A).
3)Write two or more equations relating the variables listed above. Be sure that one of them involves the quantity that is being optimized. Label this one the primary equation.
4)Does the primary equation have two or more independent variables? If so, how can we rewrite it so that it contains only one independent variable? Hint: Can you make use of the other equation(s) you wrote in question 3.
The dependent variable is the area (A), which I seek to maximize by finding the optimal lengths (x and y), which are independent variables.
5)We have previously seen that extrema, optimal values, occur when the derivative of a function equals zero. So, take the derivative of the primary equation from question 4, set it equal to zero and solve for the unknown variable. What information do you now have? Does this answer the initial question asked? If not, how can we use this value to get to the answer we need?
I now know that were x equals 55 feet is the optimal value for the area of the rectangular area. I can now use the other equation, , to solve for y, and plug y back into the original equation to get the maximum area. However, I do not need to know y to solve for the maximum area, I can just use x in the equation to find the maximum area that can be fenced off.
6)What is the answer to the question asked?
Now, try these and see if you can solve them.
1)A prison is building a new exercise yard. One wall will be the side of the existing prison complex. The other three walls will be built from electrified barb wired fencing. Prison officials want to maximize the area of the exercise yard. Budget allows them 450 feet of fencing. What will the dimension of the new yard be?
Dimensions of the yard will the 112.5 ft by 112.5 ft, for a maximized area of 12656.25 ft2.
2)A cardboard box manufacturing company is going to start making a new box. They start with a flat piece of square cardboard 20 feet per side. Then, they cut smaller squares out of each corner and fold up the sides to create the box. What will the dimensions of the resulting box be if the company wants to maximize the volume?
Start with width and depth: