Differentials

Problem 1: Differential Approximations

Consider the plots of two continuous and differentiable functions, f and g, shown in Figure 1. Use these charts to approximate, where possible, each of the values below:

Problem 2: Graphing Curves

Consider the function:

Draw the graph of this function in a system of Cartesian axes (by hand), for which you must complete the following basic steps:

1. Determine the domain.

2. Determine the points of intersection with the axes.

3. Find all vertical and horizontal asymptotes.

4. Find the derivative of f and use it to determine the critical points of the function and intervals of increase and decrease.

5. Determine the relative extrema.

6. Find the second derivative of f and Use it to determine the intervals of concavity of the

role and turning points.

7. Calculate the image by f of some points in the domain of the function, in addition to the points

critical and inflection, in order to clarify the graph.

8. Finish graphing the function with the information collected.

9. Use Matlab to verify that your chart is correct.

Problem 3: Optimization

A hiker is at a point E in a forest, three miles of a long straight road. He wants to walk to his cabin located at a point C, twelve kilometers into the forest and three kilometers of the straight road, as shown in Figure 3. The hiker can walk by a straight road at a constant speed of five miles per hour, but can only walk at a constant speed of three miles per hour within the forest. The hiker has only two travel options to get to his cabin (see Figure 3): the first is to walk only by the forest to his cabin, touring the EC segment; The second option is to walk first through the forest to the straight road, then along it, and finally through the forest to the hut, along the polyline EBDC. Refering again to Figure 3, we want to know what should be the value of variable x to minimize the time required to reach the hiker’s cabin, assuming you chose the second option of travel. Then, based on this result, we want to know which of the two options is best for the hiker.

To give answers to these questions, carry out the following tasks:

1. Write a function that expresses the total time spent by the hiker to carry out the path on the polygonal line EBDC in terms of the variable x.

2. Determine the domain (closed interval) of the function found in item 1. Explain choice.

3. graph the function found in item 1 using Matlab. Print out the chart and attach it to the report.

4. Determine the value of x belonging to the domain determined in item 2, for which the time

Total employed by the hiker to reach his cabin following the EBDC path is minimal.

5. Calculate the total distance the hiker walk through the forest

...