Description

I’m working on calculus and need an explanation to help me understand better.

Wladis, BMCC/CUNY

Project 4: Applications of Derivatives

As with other group projects, you can do this project in a group and just submit one copy for the whole group. Be sure to mark the

names of who created each problem, solved and explained, as you work through the project.

To do this project, we need to choose one of the following topics and read and work through the examples in the lecture notes:

1.

2.

3.

differentials

related rates

optimization problems

Select one of these topics. You may want to choose the topic that seems most relevant to your future career or life

goals. Then, for the topic that you have picked:

a) Write three “real world” questions that use the topic that you have chosen to solve a problem in either your

chosen career field, or in an area in which you have particular interest. In creating these problems:

You can use problems that you find in the lecture notes, in the textbook, or online as a basis for creating

your own questions, but the questions you create must be your own.

You will get extra points if you are more creative in the questions that you choose.

Try to pick questions for which the topic you have chosen would be a reasonable approach to solving

them—for example, if you use differentials to approximate something that is easy to calculate directly,

this would be a case where no one would actually use differentials in the real world, and therefore isn’t

a great example (if you end up with this kind of example, try to modify it to make a differentials

approach more reasonable).

Be sure that the questions that you choose can be solved using the methods you have chosen (if they

don’t seem to be solvable using the method you have selected, try modifying the problem).

If you have trouble finding realistic “real world” values and/or equations to represent the topic you are

interested in modeling, you can make up values and/or equations in your problems—just try to make

them realistic. (For example, don’t pick a sine or cosine function to model something like the number of

sales of an item as a result of the selling price, since this relationship is not typically periodic.)

b) For each of the three problems you have selected, solve each problem step‐by‐step, in the role of the “prover”.

c) Then for each of the three problems, explain each step, in the role of the “explainer”.

Wladis, BMCC/CUNY

Name______________________

Math 301 Test 2—Take Home Test

You may use the book, notes, web resources, etc. for this exam and you may discuss the exam questions with others,

but all of your answers should be your own work, and you should understand your answers well enough that if I

asked you to explain any of your answers one‐on‐one, you would be comfortable doing so.

1. Suppose there are two mystery operations, the arrow and the three dots, for example: and

We don’t know anything about what these mystery operations mean—however, we have been told what the

derivative of the arrow function is: If

, then the derivative is

⋅

Use this information about the derivative of the arrow function, along with your knowledge of derivative rules to

find

for the following function:

3

2. For

2

1, find

ln 2

2

using one of the formal limit definitions of the derivative:

a. Show how the derivative should be calculated using the formula.

b. Graph y, and then draw and label the various parts of the derivative definition on that graph.

3. Draw the following graphs:

a. Draw a picture of a graph which has a point where the graph is NOT differentiable. Label that point “A.”

b. Draw a picture of a graph which has a point along a curve where it is differentiable. Label that point “B.”

c. Make up a graph for a function

that has one local maximum, one local minimum, and a vertical

of that graph.

asymptote. Now, on a separate coordinate plane, draw the graph of the derivative

Label both graphs.

d. Make up a graph for a function

that has a cusp and a horizontal asymptote. Now, on a separate

of that graph. Label both graphs.

coordinate plane, draw the graph of the derivative

e. Make up a graph for a function

that has at least one local maximum or minimum. Now, on a

separate coordinate plane, draw one possible graph of

. Label both graphs.

4. Consider the following equations. Using only the techniques of calculus, and SHOWING ALL WORK, sketch the

graph. Be sure to label:

All x and y‐intercepts

All relative extrema (relative maximums and minimums)

All points of inflection

All vertical, horizontal, and slant asymptotes

a.

b.

5.

Draw a section of a curve so that:

a. The original curve is becoming more positive, but the derivative is becoming more negative on the

whole interval.

b. The original curve is becoming more negative, but the derivative is becoming more positive on the

whole interval.

c. The original curve is becoming more positive, and the derivative is becoming more positive on the whole

interval.

d. The original curve is becoming more negative, and the derivative is becoming more negative on the

whole interval.

Wladis, BMCC/CUNY

Name_______________________

Final exam (take home)

1. Answer questions about each of the following functions, given the graph:

a.

b.

c.

d.

lim

(see graph to right)

→

lim

(see graph to right)

lim

(see graph to right)

→ .

→ .

lim

(see graph to right)

→

e. On the graph above, draw another function

equal to

for ANY value of .

f. On the graph above, draw another function

NOT equal to

for ANY value of .

that has the same limit as

as

→

∞, but that is NOT

that has the same limit as

as

→ 0.5 , but that is

2. How many different ways can we have a limit that does not exist? Draw below at least two functions that show

two different function behaviors that would produce a limit that does not exist as → ∞.

(If you can think of more than two, draw extra functions for extra credit.)

3. Consider the following functions:

a. Find lim

2

,

4

,

→

b. Find lim

→

c. Find lim

→

d. Find lim

→

e. Use your answers to a‐e above to help you to explain WHY:

i.

is an indeterminate form

ii.

is NOT an indeterminate form

6

Wladis, BMCC/CUNY

Name_______________________

4. Consider the function in the graph below:

a. Graph the derivative of this function.

b. Over which intervals of the domain on this graph will the second derivative be positive?

5. How many different ways can a function fail to be differentiable at a point? Draw below at least two functions

that show two different function behaviors that would produce a derivative that does not exist at

2.

(If you can think of more than two, draw extra functions for extra credit.)

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