# MATH LESSONS

SheLovesMath.com is a free math website that explains math in a simple way, and includes lots of examples, from Counting through Calculus!

Here are the topics that She Loves Math covers, as expanded below:  Basic Math, Pre-Algebra, Beginning Algebra, Intermediate Algebra, Advanced Algebra, Pre-Calculus, Trigonometry, and Calculus (to be completed).

.Note: To report any typos/errors on this math lessons website, please email me at lisa@shelovesmath.com.  Thanks for all your support!  You guys are the best!

# Calculus (Differential)

## 70 thoughts on “MATH LESSONS”

1. This was VERY helpful! I had a test just tomorrow in Pre-Algebra and I was practically clueless. I am beginning to understand the areas of math we are going through, though I currently have a D in my class because the 13/20 on my unit test lowered it from a C+. I was about to give up, but I knew acing my next test would help my grade a lot. I recommend this site to any girls that are struggling with math or just need a little re-learning. Wish me good luck on tomorrows test! I hope I get this one, and I will be happier than ever…I’ve always wanted to show my mom a great math test score… 🙂

• Thanks for the sweet comment, Emme! I’m thinking good math thoughts your way, and I know you can do it!! Thanks for visiting the site and let me know if you ever have any math questions!! Lisa 🙂

2. Really, it is very helpful to explain and understand your way. Keep on 🙂 waiting for the new materials.

3. I just wanted to say thank you! I homeschool my autistic daughter and this site has been very helpful.

• Renee,
Congratulations for home-schooling your precious daughter. Thanks so much for kind words and let me know if there’s anything I can do to make the site better 🙂
Lisa

4. This site is exactly what I’ve been looking for to help my mathematically talented but easily bored son. The examples and explanations are thorough enough to educate but concise enough to hold interest.

• Diane,
Thank you so much – you just made my day! Please let me know if there’s anything I can add to make it better 🙂
Lisa

5. I’m having a really hard time when it comes to radicals 🙁 Too bad, I can’t find something here that will help me. 🙁 maybe you should add them to your site. 🙂 Thank you! Your website is so cool

• Hi!
You might look here or here or here.

Sorry you can’t find them here – can you give me an example of a problem and I can see if I address that? I probably just don’t know label it very well. Thanks! Lisa

• I also have a hard time with that we just started radicals and it’s confusing

6. I’m working on “Laplace Transform”…
I need some real life examples which can be solve by L.T.
can you help me ?

7. I love the site and how much has gotten done since I last looked, but I also see that there isn’t a geometry section. Could you work on this after your other projects?

• Hi Tyler! Thanks for writing!! YES – I hope to do a Geometry section, but it won’t be for awhile ;( Hope all is well! Lisa

8. A cubic polynomial function f(x) has leading coefficient -2 and intercepts the y-axis at 2.If f(1)=1, and f(-2) =-2, find f(-1) and write the complete
function.

• You can solve this using systems. let y = -2x^3 + ax^2 + bx +c. Plug in (0, 2) to see that c = 2. Then we have y = -2x^3 + ax^2 + bx +2. Plug in (1,1) to see that 1 = a + b. Then plug in (-2,-2) to see that -10 = 2a – b. Solve that system to see that a = -3, and b = 4. So we have y = -2x^3 – 3x^2 + 4x +2. So then f(-1) = -3. Hope that helps! Lisa

9. I gave this link to my Elementary Functions/Trigonometry class. I hope they use it 😉

• Thanks so much for writing, and please spread the word about She Loves Math. This makes me want to keep writing 😉 Thanks again, and let me know if I can make it better, Lisa

10. Lisa
I stuck on trying to figure out how to parametrize
y=(-x^2/72)+x so as to get x=24 sqrt[2] t. Once I have the x
I can figure the y which is -16t^2 +24Sqrt[2] t.
(sqrt is abbreviation for square root)
Manuel

• Thanks for writing, Manuel. I think when you go from an equation with x and y to parametric equations with t, t could basically be anything. But if they wanted the y in the form y = -16t^2 + something, we could set -x^2/72 to -16t^2 (the first term of the y), and solve for x to get 24sqrt(2)t. Does that make sense? Lisa

• Thank you very much. I’d like to work this out a bit to be sure I understand and then, if I may, get back to you.
Needless to say you are terrific.
Manuel

11. Lisa
As you said, I could have used any parameter for x.
I was just wondering where the author chose
24Sqrt[2] t, from the rectangular equation y=x-x^2/72 to get
y=-16t^2 +24Sqrt[2] t. I tried all kinds of parameters for
y=x^2/72 and sure enough, while I get different values for y,
Thanks again. Basically, I didn’t understand what was meant by
parametrization.
Manuel

12. Hello Lisa
I was wondering why you haven’t recommended the use of I believe the best calculator available, namely, Wolfram Alpha. It can be downloaded thru your app in your iPad or from the internet for free. It’s better on the iPad. I think it would help to take the drudgery out matrix`manipulations, vectors, quadratic equations, 3D graphing, et al.
As far as I’m concerned you are the best Just thought you may be interested.
You probably know this anyway.
Manuel

• Manuel,
Thanks for writing! I don’t have a lot of experience with Wolfram Alpha and just downloaded the app on my ipad. It looks fantastic! I have used Mathematica in the past; I guess it was based on this.
Thanks so much for the suggestion; that’s a great idea! They don’t use it at the high school, so I haven’t had to help any kids on it, but it’s definitely on my to-do list.

Lisa

13. I am having trouble with this section, if you could get back to me asap it would be greatly appreciated.
1)In the nineteenth century, the Austrian monk Gregor Mendel noticed while crossbreeding plants (peas in particular) that often a characteristic of the plants would disappear in the first-generation offspring but reappear in the second generation. He theorized that the first-generation plants
contained a hidden factor (which we now call a gene) that was somehow transmitted to the
second generation to enable the characteristic to reappear.
As an example, suppose we denote the gene that produces the yellow seed by Y and the gene that
produces the green seed by g. The uppercase Y indicates that yellow is the dominant gene and
the lowercase g indicates that green is recessive. The table below shows the possible theoretical
outcomes that can occur when we cross two first-generation plants.
First Generation Plant
Y g
First Generation
Plant
Y YY Yg
g gY gg
Notice from the table that of the 4 possible outcomes, 3 of the plants will be yellow (since 3 have
the dominant Y gene) while 1 of the plants (since only 1 has two recessive genes) will be green.
From this we can say that the theoretical probability of the second-generation plant being green
is ¼ or 0.25.
There is of course a difference between theoretical and experimental. The following table lists
some of the actual results that Mendel obtained in his experiments in crossbreeding peas.
Experimental Results
Characteristics That Were
Crossbred First-Generation Plants Second-Generation Plants
Tall versus Short All tall 787 tall
277 short
Smooth versus Wrinkled
Seeds All smooth seeds 5,474 smooth
1,850 wrinkled
Yellow versus Green Seeds All yellow seeds 6,022 yellow
2,001 green
Notice from the results that tall, smooth, and yellow are all dominant genes. Also notice that based on the experimental results, the probability of a second-generation plant being green is 0.2494. This agrees fairly well with the theoretical probability.
From the tables of possible theoretical outcomes and experimental results, answer the following questions.
1) Assume that we are crossbreeding genetically pure tall plants with genetically pure short plants. Create a table (like the one for yellow and green plants) indicating the possible combinations of tall and short peas.
2) What is the theoretical probability that a plant will be short?
3) What is the experimental probability that a plant will be short?
4) How do theoretical and experimental probabilities compare?
5) Assume that we are crossbreeding genetically pure smooth-seed plants with genetically pure wrinkled-seed plants. Create a table (like the one for yellow and green plants) indicating the possible combinations of smooth-seed plants and wrinkle-seed plants.
6) What is the theoretical probability that a plant will have smooth seeds?
7) What is the experimental probability that a plant will have smooth seeds?
8) How do theoretical and experimental probabilities compare?

• Thanks for writing! I’m sorry – I’m having trouble making sense of the tables; let me see if I can find this problem online somewhere. Sorry about that. Lisa

14. I THINK I understand the problem statement and questions. I have re-stated the problem and answered the questions in a file on my website. I hope it helps. The document is available on http://www.PaulkUSA.com. Click on the “Math Docs” tab and look for the Algebra 2 file named “Statistics Example 01.”

• Solution was updated 4/15/15 at 7:00p

15. Hi dear just wanna say thank you for the tutorial they are so useful and as a girl I love Maths too!

16. How long will the calculus site in construction?? coz i hope it will help me in my calculus project assignment. i hope it will settle faster, pleaseeeee

• Thanks for writing! I’m SO sorry about not completing the Calculus portion yet – I hope to by the fall! Keep checking back! Lisa

17. A great resource, you cover almost everything and I see how much work you have put in. Thank you very much and I will be coming back here again

18. this problem makes me confused, can you help solving this?
a receptacle contains x ounces of acid. A second receptacle contains x ounces of water. From the acid y ounces are removed, placed in the water, and the contents thoroughly mixed.Then, from the second receptacle, y ounces of the mixture are removed and placed in the acid. find (a)the concentration of of acid in the second receptacle, and (b)the concentration of water in the first receptacle

• Here’s how I’d do this one – I got y/(x+y) or y/(x+y)*100 % for concentration of acid in the 2nd receptacle. Then I get (y * (x/(x+y))/(y+x-y) = y/(x+y), or y/(x+y)*100 % for the concentration of water in the first receptacle (amount of water over total amount). You can try it with real numbers. Does that make sense? Lisa

19. I’m having trouble at this problem… hope you reply..
Two pipes running simultaneously can fill a swimming pool in 6 hours. If both pipes run for 3 hours and the first is then shut off, it requires 4 hours more for the second to fill the pool. How long does it take each pipe running separately to fill the pool

there’s a problem that I can’t seem to figure out:

log (12x) + log (X-1) > 2

would the domain restriction be X>1 or would it be the domain restriction of the two log combined – log (12x^2 – 12x?

21. Hello Lisa,

I love your website. It would really be helpful if you could cover non-differential Calculus for us as well!

• Thanks so much for your kind words – comments like these make me want to keep writing! I will get to non-differential Calculus eventually – hopefully this summer! Please spread the word about shelovesmath.com 🙂 Lisa

22. This website is amazing! Even though I am a boy (and 42 years old), I just want to say I really appreciate all the hard work you have put into it. I wanted to brush up my skills a bit for a standardized test I have to take soon for nursing school, and this is by far the most comprehensive, complete, logical review I’ve seen. Exactly what I needed! Thank you.

• David,
Thanks so much for writing!! And kudos to you for being a nurse – such a wonderful field. I’ve thought about making the site more generic – not “she loves math” but something else – so still may. Let me know if you have any more feedback on the site! Lisa

23. Hello Lisa,

I looked through your website and I couldn’t really find anything that dealt with Binomial Series. I am having trouble with using the alternate definition of binomial coefficients to show that if n is a non-negative integer then (n k) = 0 whenever k > n. How does this allow me to express the binomial expansion as the infinite series (a+b)^n = (summation from k=0 to infinity) Σ (n k) (a^n-k)b^k

24. Hi Lisa,
Your website is awesome! I was wondering if you could tell me what I am doing wrong in this last step of this algebra word problem.
It says that each helper can make 2 large cakes or 35 small cakes per hour. The kitchen is available for 3 hours and 20 large cakes and 700 small cakes are needed. How many helpers are required? The answer is 10 helpers are needed.
Now using proportions I was able to determine that 10 helpers/hour are needed to make the 20 large cakes. Also 20 helpers are needed to make the 700 small cakes. So then adding the 10 helpers/hour+20 helpers/hour gives you 30 helpers/hour . But they give you 3 hours to complete the cakes so to get the answer of 10 they are dividing the 30 helpers by the 3 hours but the units do not cancel to give you helpers when you do that because its 30 helpers/hour/ 3 hours = 30 helpers/hr^2 which does not make any sense? Can you please explain how they are getting the unit just to be helper when they have the rate 30 helpers/hour and the 3 kitchen allotted hours? This question is really bugging me!

• Here’s how I’d do this: Since 20 large cakes are needed, we would need 10 hours (2 large cakes per hour). Since 700 small cakes are needed, we would need additional 20 hours (35 small cakes per hour). So we’ll need 30 hours total. Since 3 hours are available, we would need 10 helpers. Does that make sense? Lisa

25. Hi Lisa,
A few days ago I was searching internet to find useful math tutorial websites.
I found lots of them but this website is something else ! Excellent !
Are you going to support the text parts with tutorial videos ?
And I am looking for a website like your website that teaches physics to high school and college students from basics through advanced level. In any chance do you have any idea to introduce me a few?
I visited some websites like KHAN ACADEMY but I did not like them so much. To me, they were not attractive enough.
And the last thing do you have any idea how a student could be encouraged and motivated to learn pre-calculus or calculus while many of them know after graduating from college or even universities and when they get a job they will nearly never use what they had learned before!
To be honest with you I myself know very few doctors or nurses who use calculus techniques although they need to pass the related exams while studying at the college or universities.
Thank you.
Alireza.

• Alireza,
Thanks so much for your kind words! I’d love to do videos – I just want to finish the site (Calculus) first. And please let me know how I can make the site better.
Thanks again!
Lisa

26. Hi Lisa,
The Wronskian method is valid for all set of function or just for certain set of function? If just the certain set of function, can you list some of the example. Thank you.