Here are the solutions to these even-numbered problems.

Calc1-Ch1.1#44-46-48-50.pdf (The PDF has #49 instead of #50. See below for #50)

**Chapter 1.1 #50**

Here are the solutions to these even-numbered problems.

Calc1-Ch1.1#44-46-48-50.pdf (The PDF has #49 instead of #50. See below for #50)

**Chapter 1.1 #50**

What you are looking for is a function like

C = f(x) where C is cost and x is mile

If 0 <= x <= 1, C=2

If 1 < x < =2, C = 2 + 0.2(10(x-1))

Using Wolfram here is what it looks like

If you are looking for details for a transcript for Calculus 1, this should do the trick.

Calculus is about change. The tools of calculus allow us to model change. Once we have equations for something, we can use calculus to see the impact of change on the system.

In Algebra, we learned how to generalized basic arithmetic to show relationships. So, instead of saying 2 + 2 = 4, we might say 2 + x = y, allowing a more general equation that shows a relationship between x and y.

In Geometry (which also includes trigonometry), we studied the relationships of shape.

In Calculus, we study continuous change. For example, our car gets so much gas milage which is impacted by the car’s weight. But the longer you drive, the less gas you have and therefore the less weight. So the weight of the car changes over time, therefore impacting the use of fuel over time.

Calculus is a necessary tool for any science or engineering. It is heavily used in physics and engineering and found in biology, business, social sciences, etc. No matter what we are studying, the only constant thing is change.

Calculus is broken up into two areas – Derivative and Integral Calculus. This course covers derivative calculus, which is equivalent to most Calculus 1 courses at most universities. Calculus 2 covers integral calculus.

Students who complete Calculus are prepared to enter college and should be able to step into a college-level Calculus 1 or Calculus 2 course – depending on how the individual university defines its course. Calculus is not required for the ACT and the SAT.

**Topics Covered in this Course:**

- Limits and limit laws
- Continuity
- Derivatives
- Differentiation rules
- Application of differentiation
- Antiderivatives

Look for the following material or heading in the contents.

- Limits and Derivatives

· Limit of a function

· Calculating limits

· Limit laws

· Continuity - Differentiation Rules

· Rates of change

· Derivative rules (polynomials and exponentials)

· Product rule

· Quotient rule

· Derivatives of trigonometric functions

· The chain rule

· Implicit differentiation

· Derivatives of logarithmic functions - Applications of Differentiation (subtitles may differ widely here)

Notes for use: This course should be equivalent to a Calculus One course at a local university. If you know what school your child plans to attend, find out which textbook they use and go with it. In fact, ask for a syllabus and sample tests. A good deal of information can be gleaned from the UAB website on the screening test and the UAB test bank. (See the resources at the end of the chapter.)

Also, note that these books are often written for Calculus I, II, and III. So your text will be a lot bigger than what you will teach in high school. Also, if you do go with a college textbook you will pay a hefty price. An alternative is to find the previous edition on eBay – much cheaper with probably little change in material.

**Beverly asked:**

Looks like you have a simplification error in the second derivative. You have:

f”(x)=12x^2-4=4(x-1)

But it should be

f”(x)=12x^2-4=4(3x^2-1)

then the intervals for concavity all change.

Am I missing something? Thank you so much for your good product and wonderful support!

You are correct. Thanks for pointing that out.

See the plot of the original equation (a good idea is to look at plots when things are not working out)

From here we can see how it is shaped.

Now if we look at the second derivative, it should be

and look like

which has roots

and looks like

The concavity chart will now show up, down, and up but around the points

**Question from Beverly:**

#3 – on the test copy I have says to use three equal subintervals. But in the handwritten solution, it appears that they used 6. If the student uses 3 subintervals wouldn’t the ∆X be 2 and the right endpoint sum be 0.2, left end point be 39.8, and midpoint be 15.8?

**Answer from Dr Callahan** – That is correct – in the solution we used an interval of 1 and it should have been 2.

#8b – The handwritten solution has dV= √t and V = t^1.5. Shouldn’t V = 2/3*t^1.5? With that, our final solution came out to be 3.39.

**Answer from Dr Callahan** – That is also correct – we have an error in our solution.

#9a – Instead of factoring the cos^5x into (cos^2x)(cos^2x)cosx, my student used the Form 74 from the Table of Integrals followed by Form 68. But he didn’t get the same answer as he would have solving it as we learned in 5.7 (p. 403) like the written solution did. Why don’t the two methods yield the same answer? I’m thinking it has to do with the limits of the integral, but it’s been 25 years since I had calculus, and I’m struggling with explaining the why on this one.

**Answer from Dr Callahan** – Yes you should get the same answer – but I noticed an error in our solution!!!! When you integrate we added a 4. Should be as below

#9b – In the handwritten solution it looks like it evaluates 3*ln 3 as 9.89, but isn’t it 3.3? I believe the answer would be -0.17 if 3.3 is correct.

**Answer from Dr Callahan** – Yes it should be 3.3

Extra Credit – The solution just substituted into the Form #95. Did you not expect them to continue to solve the remaining integral or just stop with that first substitution?

**Answer from Dr Callahan** – Yes we just stopped there because that was the challenging part!

**Question from Carl:**

The question on the test has one slight variation from the book:

f (x) = x if -1< x<1the text book showsf (x) = x if -1< x<1

Please clarify if the teacher syllabus is incorrect. We’re finding it is not possible to be both (0,1) and (1,1)…thus only (0,1) is solid and (1,1) is open on the graph.

**Answer from Dr. Callahan:**

Good catch – you are correct – cannot be both. The question should say

f (x) = x if -1< x<1

The answers are not given in the book since this is even, so here they are:

a) 3

b) 4

c) 2

d) does not exist

e) 3

Here is the solution to this even numbered problem:

**Question from Alyanya:**

Hello,

I am looking for a math course for my daughter who just finished her sophomore year of high school. She received a 97 in Algebra 2 Pre-AP and was planning on taking Pre-Calculus Pre-AP next year. However we have decided to home school next year because of our travel schedule, so I’m looking for a course for her. I noticed that you don’t have a separate Pre-Calculus class. In your opinion would it be ok for her to go straight to the Calculus class? She’s a very strong student, but I wouldn’t want her to take something that she is not prepared for.

Any input would be appreciated.

Answer from Dr. Callahan:

My guess is she would be OK as long as she has had some trig.

I would first have her go to out online textbookhttp://www.mhhe.com/math/precalc/barnettcat7/student_index.mhtml

and take the chapter quizzes of each chapter, 1-8.

This will show any weak spots. If she has any, a quick study of those sections should help her get ready.

Hope this helps.

**Question from Donna:**

But I’m wondering if calculus will be too difficult for me to teach. Trig was a bit of a challenge for me, just wondering what your ideas were. Thanks

**Answer from Cassidy Cash:**

Calculus is designed to be mostly self-taught on the part of the student. The parent might come in and assist in grading, or provide accountability for homework, etc, but students at this level should be able to not only teach themselves using the dvds and the textbook, but they are more than capable of finding the answers to problems they don’t understand.

Also, not only do they have Homework Help as a resource, but we are actually expecting Calculus 1 students to be proactive at this level about asking questions and seeking help when they need it. (some students even do internet and library searches to find answers–which is great. Considering the next step for most students after this course is to take a college course, this is definitely the time to be learning how to be proactive about their own education. Being able to find answers outside the textbook is an excellent place to be)

To sum up, I would say not to worry about it. If your student ever arrives at a problem or concept that they cannot figure out from the materials they have available to them, we are here to help them. And we are here to help you as well. If you have questions about anything related to teaching this course, we will assist you as well. 🙂 I think you will do just fine 🙂

And we are here to help whenever you need.