1. Miami Dade College -- Hialeah Campus. Calculus I Formulas. MAC 1. Limits and Derivatives. 2. Differentiation rules. 3. Applications of Differentiation. 4. Elementary Differential and Integral Calculus. FORMULA SHEET. Exponents xa · xb = xa+b, ax · bx = (ab)x, (xa)b = xab, x0 = 1. Logarithms lnxy = lnx + lny, lnxa. Differentiation Formulas d dx k = 0. (1) d dx. [f(x) ± g(x)] = f (x) ± g (x). (2) d dx. [k · f (x)] = k · f (x). (3) d dx. [f(x)g(x)] = f(x)g (x) + g(x)f (x) (4) d dx. (f(x) g(x).) = g(x)f (x).

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in Chapter X. of the Differential Calculus, on Maxima and Minima, and in Chapter .. whose Equation is given between Three Rectangular Co- ordinates, jc, y, z. Use differentiation and integration tables to supplement differentiation and integration techniques. Differentiation Formulas. 1. 2. 3. 4. 5. 6. 7. 8. 9. Vector Calculus. . Finding the zeros of equations; Numerical integration of differential Speigel, M.R., Mathematical Handbook of Formulas and Tables.

If f is differentiable at a, then f must also be continuous at a. As an example, choose a point a and let f be the step function that returns the value 1 for all x less than a, and returns a different value 10 for all x greater than or equal to a. Consequently, the secant lines do not approach any single slope, so the limit of the difference quotient does not exist. However, even if a function is continuous at a point, it may not be differentiable there. If h is positive, then the slope of the secant line from 0 to h is one, whereas if h is negative, then the slope of the secant line from 0 to h is negative one. In summary, a function that has a derivative is continuous, but there are continuous functions that do not have a derivative. Most functions that occur in practice have derivatives at all points or at almost every point. Early in the history of calculus , many mathematicians assumed that a continuous function was differentiable at most points. Under mild conditions, for example if the function is a monotone function or a Lipschitz function , this is true.

Start every new question number e. Q1, Q2 etc. Clearly label question attempted at the top of the page and on the front of the answer book Write clearly, starting a new equation on a new line, like we did on the board during lectures. Please don't try to cram everything within a few lines, ask for more paper if needed! Explain what you are doing where possible. Stay calm and good luck!

In higher dimensions , a critical point of a scalar valued function is a point at which the gradient is zero. The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point.

If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a " saddle point ", and if none of these cases hold i. Calculus of variations[ edit ] Main article: Calculus of variations One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface.

If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the shortest path is not immediately clear.

These paths are called geodesics , and one of the most fundamental problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This surface is called a minimal surface and it, too, can be found using the calculus of variations.

Physics[ edit ] Calculus is of vital importance in physics: many physical processes are described by equations involving derivatives, called differential equations. Physics is particularly concerned with the way quantities change and develop over time, and the concept of the " time derivative " — the rate of change over time — is essential for the precise definition of several important concepts.

You're bored. So I'm going to give you a book.

You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again. I was up in the back with this book: Bader knew I had studied "Calculus for the Practical Man" a little bit, so he gave me the real works—it was for a junior or senior course in college.

It had Fourier series , Bessel functions , determinants , elliptic functions —all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation.

It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals.

The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it.

Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me. From Wikipedia, the free encyclopedia. This article is about the integral rule.

For the convergence test of alternating series, see Alternating series test. This article needs additional citations for verification.

Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: See also: Intermediate Calculus Second ed.

New York: American Mathematical Monthly. Calculus 3 ed. Houston, Texas: Publish or Perish, Inc.

Calculus on Manifolds. Addison-Wesley Publishing Company. Retrieved from " https: