calculus - Help finding turning points to plot quartic and cubic functions

My teacher assigned us some graphing homework. I know how to graph all of them manually all except these two$f_1(x)=x^4-x^3-4x^2$$f_2(x)=\frac{1}{2}x^3+2x^2-8x-2$When i asked him how he said it was above our grade level and it was a calculus question so we should just use graphing calculators. So since I'm still curious, I'm asking here. Detailed answers with steps would be nice. The main thing I need to know is how to find the exact location of turning points....Read more

calculus - Does the fact that $\lim\limits_{n\to\infty} |f(n)/f(n-1)| = 1$ imply that $\lim\limits_{n\to\infty} f(n)$ exists?

This question was inspired by another question I asked on this site. For that question, I had thought that if the differences in subsequent evaluations of a function, $f(n)$, defined with a domain of only the natural numbers (including $0$) converged to $0$ as $n$ approached infinity, then this would show that $f(n)$ would converge to some value. I was wrong in this assumption. Another idea I had is that if the ratio of one evaluation to the previous converged to $1$, this would also imply existence of a limit. This seems more likely because of...Read more

What Field(s) Cover Calculus with Matrices and High Dimensional Geometry?

I am taking a course in machine learning and have found that the linear algebra and multivariable calculus from my engineering degree only take me part way in understanding some derivations. One specific example is differentiating things related to matrices (like differentiating wrt a matrix whose determinant appears in the function...) But this is by no means the only fuzzy bit. I have done just enough geometry, linear algebra and low dimensional calculus to have a notion that these things somehow extend into things involving matrices and thin...Read more

Can I Learning Algebra 1 and 2, Geometry, Trigonometry and Calculus in 1 years time?

I'm a Middle school student who is going to highschool soon and would like to learn all of the affromentioned things beforehand. I can work an average of 14 hours a day for a full year with my only other focus being working on language arts and science for 4 hours. This brings the amount of time that I can spend on math to 10 hours a day. I will be able to have approximately three to four hours of assistance everyday and also find it relatively easy to learn new concepts. My question than is if it is possible....Read more

reference request - suggest a textbook on calculus

Possible Duplicate: What are the recommended textbooks for introductory calculus? Hi, i am a software engg. and math was never my favourite. I somehow dragged thru the math i had to do in college to earn my degree. But my brother (7 years younger aged 15) does have an aptitude for math. i have seen him cruise thru his high school texts with little or no difficulty. He also says he wants to be a mechanical engineer (and backs up his claim by cleaning up and servicing my motorcycle incessantly) anyways he is 15 now and sadly his syllabus hasnt ...Read more

real analysis - Most Suitable Book after Kline's Calculus?

I've been working through Morris Kline's Calculus: An Intuitive and Physical Approach and it's an absolutely excellent book for self-studying applied single-variable (and some multi-variable) calculus but I'm starting to wonder what the best book to continue with would be? I wouldn't want to just review single-variable calculus in rigorous form as an introduction to analysis but I'm also not sure if going straight into Baby Rudin/Apostol Vol. II or anything of that sort is any wiser. Or perhaps it is, having the physical intuitions of single-va...Read more

calculus - weakly locally one-to-one?

Is there any standard name for this concept that is weaker than local one-to-one-ness?In some open neighborhood of $x_0$ there is no point $x\ne x_0$ such that $f(x)=f(x_0)$.Or, if you like: In some neighborhood of $x_0$, for every point $x$ in that neighborhood, $f(x)=f(x_0)$ only if $x=x_0$.Might one simply say that "$f$ is weakly locally one-to-one at $x_0$"?Trivial question, it might seem, if it's only about mathematics. Maybe it's about psychology of learning mathematics. I recently came across this error: If $f\;'(a)>0$, the $f$ ...Read more

reference request - What are the recommended textbooks for introductory calculus?

I've already taken my calculus sequence and I'm interested in brushing up and staying sharp on the basics. So far, my calculus background is limited to single-variable calculus, which I applied in my physics sequence and engineering statistics. I learned a little bit of vector math in physics as well, but I'm not strong in it. My mathematical strengths lie in the discrete mathematics.I'm particularly interested in studying for my IEEE Certified Software Development Associate exam - 10% of the exam is mathematics (calculus, differential equation...Read more

reference request - Rigorous Calculus Textbook

As a reader of I. M. Gelfand's algebra/trigonometry and A. P. Kiselev's geometry textbooks, I am struggling to find an equally rigorous calculus textbook. I've tried Stewart's "Calculus: Early Transcendentals", but it's way too (for lack of other words) fluffy. I enjoyed Gelfand/Kiselev books because of their succinctness and rigor, where the few exercises were never the same (i.e. solving 2 quadratic equations where $a$, $b$, and $c$ are just different numbers).I'm looking into Tom Apostol's book, but there the only editions I've seen have ter...Read more

calculus - Non-analytic smooth function

The Wikipedia page ( proves that$$f(x) =\begin{cases} \exp(-1/x), & \mbox{if }x>0 \\0, & \mbox{if }x\le0\end{cases}$$is a non-analytic smooth function. But I don't understand why the fact that "the Taylor series of $f$ at the origin converges everywhere [where does "everywhere" mean here?] to $0$" implies the Taylor series converges to $0$ when $x>0$? Can someone explain this in greater detail? I guess there should be a gap in my understanding of power series....Read more

calculus - Does the inverse function theorem provide a path to interpretation of more general infinitesimal quotients?

Let us first bring up inverse function theorem : If $y=f(x)$ and if $f'(x)$ exists at some point $x=a$, then exists in some neighborhood of $(a,f(a))$ an inverse function $f^{-1}(x)$ which around corresponding $b = f(a)$, such that it is differentiable and that this differential fulfills:$$(f^{-1})'(b) = \frac{1}{f'(a)}$$Could this bring a fruitful approach to define the reciprocal infinitesimal quotient below:$$\frac{\partial f^{-1}(y)}{\partial y}(b) = \frac{\partial x}{\partial f(x)}(a)$$Would it lead to anything meaningful, or will we run i...Read more