Reference no: EM131052540

Pre Calculus Assessment Project

Part 1: Box Optimization Lab

You are commissioned with maximizing the volume of an open top box that is built by cutting from the corners of a rectangular sheet. (Specific I know, but let's just say this happened...) So...here's your cardboard!

Activity: You and your classmates will be creating open topped boxes from your cardboard. Your instructor will assign you a value of "x" that will represent the height of the box assigned to your group.

1. Approximate the original dimensions of the rectangular piece of cardboard used to make the open-topped box to the nearest eighth of an inch:

2. To make an open-topped box (with a consistent height), you will need to cut some pieces out of your cardboard box. Below, draw what your cardboard will look like after you cut out these pieces. Let the length that represents the height of the box be "x".

Once you have gotten your diagram checked, you can go ahead and start cutting!

3. Height of the box assigned to your group:      

4. Volume of the box you built: 

Write the two values in 3 and 4 on your box along with the last names of all members of your group. When your class has finished plotting the boxes, plot the class points on the grid below.

x (height of box)	y (volume of box)

Now, you have a pretty good idea of where the maximum of your function will be. You also have noticed a very particular pattern that we find when we plot volumes of boxes.

5. Write the equation that models the volume of all the possible open-topped boxes that can be created from the original rectangle of cardboard as a function of x, the height of the box.

6. Determine the domain restrictions for your function from #5.

7. We know we could use the first derivative of our cubic volume function to determine the x-value that corresponds to the maximum volume of the box. However, we don't know how to find the derivative of a cubic! So let's quickly investigate:

a. If f (x) =ax² + bx + c, then what is f '(x)?

If f (x) = ax³ + bx² + cx + d, we can follow a similar pattern! f '(x) = 3ax² +2bx +c!

b. So, if g(x)=3x³ - 4x² +1 , what is g'(x) ?

c. Use this pattern to find the derivative of your cubic function from #5.

8. Use the derivative to find the maximum volume that can be made in this way from your size of cardboard. (Show work!)

9. Why are you sure that the value you found above will give you a maximum?

Problem Set 1: Rational Equations

Earlier in the semester, we learned that the derivative of a quadratic function is a linear function. Through our box optimization activity in class, we recognized that the derivative of a cubic function is a quadratic function. In fact, this pattern is true of all polynomials!! When we take the derivative of a polynomial of degree "n", we get a polynomial of degree "n - 1". Even more patterns emerge if we look more closely.

Derivative of a quadratic: If f (x) = ax² + bx + c, then f'(x) = 2ax + b

Derivative of a cubic: If g(x) = ax³ + bx² + cx + d, then g'(x) = 3ax² + 2bx + c

As you may have noticed in your work with quadratics, there is a really clear pattern here! The exponent of each term of the polynomial is multiplied by the coefficient of the term, and the degree of each term is reduced by 1 when we take the derivative.

In fact, we can extend this to the derivative of any term in a polynomial:

If f (x) = axⁿ, then f'(x) = nax^n-1

But wait, this section is supposed to be about rational equations! Well, at its very simplest, a rational equation could look like this: f (x) = 1/x. Well, this is the same as f (x) = x^-1! Well, does the formula above work when "n" is a negative number? Let's find out.

1. If the formula above works for negative values of n, what is the derivative of 1/x?

2. Consider some relationships.

a. Fill in the blank: If a function is discontinuous at x = c, then f'(c) =        

b. Does the above statement support the conjecture that your derivative from #1 is correct? Why or why not?

c. Fill in the blank: When a function is increasing, the derivative of the function is __________. When the function is decreasing, the derivative is _________.

d. Does the above statement support the conjecture that your derivative from #1 is correct? Why or why not?

3. Recall how we defined the derivative using a limit. Use this definition to algebraically determine whether your derivative from #1 is correct.          

4. Optimization Problem

(Last Name A-H) Margins of a paper: A theater is making a poster to advertise their new show. A consulting marketing firm told the theater that the optimal poster has a 200in2 area and has 1 inch margins on the sides, 1.5 inch margin on the bottom, and 2 inch margin on the top. However, they didn't tell you what dimensions of the poster you should use. Your job is to determine the dimensions that maximize the printed area of the poster.

(Last Name J-S) Surface area of a can: You are working for a manufacturer of a new energy drink. The company wants the drink to be packaged in 20 ounce cans. Find the dimensions of the can in centimeters that would minimize the surface area in order to decrease manufacturing costs. (1 fluid oz: 29.57cm³)

a. Draw and label a diagram and write a function that models the situation.

b. Write the derivative of that function. (Hint: First rewrite your rational function so that it looks like a polynomial, but may have some negative exponents. Then use the formula above for each term.)

c. Use the derivative to find possible extrema. Show that your extrema is indeed a maximum/minimum in order to answer the optimization problem.

d. In part c, you found the derivative. Find the SECOND derivative (the derivative OF the derivative).

e. Consider the graph of the second derivative.

a. If the second derivative is positive, then the first derivative is __________.  

b. If the second derivative is negative, then the first derivative is              __________.

c. We know a continuous function has a maximum if the derivative changes from __________ to __________.

d. Explain why your second derivative supports the idea that the value you found in part c is a maximum.

Problem Set 2: Sine and Cosine

We haven't talked about the derivative of trig functions yet, but we can certainly think about them.

1. Below, draw the graph of sine. (f(x) = sinx)

We are going to think about the derivative of our sine function, f(x), above.

2. Fill out the chart below. For many of the points, you can observe the exact derivatives using what you know about maximums and minimums. If you cannot determine the derivative by looking at the graph, you should use your calculator. Once you have the chart, plot the points on the graph so that the y-axis represents f'(x).

X	-(3π/2)	-π	-(π/2)	0	π/2	π	3π/2	2π
f'(x)

The graph of the derivative of sine should look familiar to you.

3. Finish the question: If f (x) = sinx, then f'(x) =               

4. Let f (x) = sinx. Find the following:

a. f'(0)

b. f'(π/2)

c. f'(5π/6)

5. Write the equation of the tangent line to the curve y = sinx through the point (11π/6, 1/2)

The following problem will be discussed and "demoed" in class during the project work days. You are more than capable to go ahead and start on it before we demo, just know that we will revisit it in class.

6. Optimization Problem: The most commonly used canal cross section used in irrigation and drainage is one with a trapezoidal cross section. You have been commissioned to help determine the dimensions of the trapezoid to use based on how much water the canal can hold. The company you are working for says that the length of the trough (x + x + x) should be 12 feet. You must find the angle (θ) to use to maximize the amount of water the canal can hold.

a. Write a function that models the situation.

b. Is the function continuous? Why or why not.

c. The derivative of the function is A'(θ) = 16 (cos θ + cos²θ - sin²θ). Algebraically determine when the derivative is zero or does not exist. (Hint: First show A'(θ) =  16(2cosθ - 1)(cos θ + 1).

d. Use the values from c to solve the original problem. Show how you determined your final answer is a maximum by commenting on the sign of the derivative.