Approximation calculus pdf notes

The following calculus notes are sorted by chapter and topic. They are in the form of pdf documents that can be printed or annotated by students for educational purposes. Linear approximations section of the applications of derivatives chapter of the notes for paul dawkins calculus i course at lamar university. Because of this we define the linear approximation to be. On occasion we will use the tangent line, lx, as an approximation to the function, f x, near x a. The trapezoid rule approximates the area under the graph of f. The classi cation of the equilibrium in the nonlinear system is the same as the classi cation of the origin in the linearization. Linear approximation is a powerful application of a simple idea.

This is a self contained set of lecture notes for math 221. Once i have a tangent plane, i can calculate the linear approximation. The number of cars per hour passing an observation point along a highway is called the traffic flow rate qt in cars per hour. These notes are intended to be a summary of the main ideas in course math 2142. Clearly, this is close to p 1 1, but we want better. Next, i will show where this sum actually occurs and why it is important. Lax will be good when x is close to a, and typically gets worse as x moves away from a. This example is intended to convince you that formulas a24 and some algebra are often easier than trying to apply a1 directly. We have already looked at tangent line approximations, which are not bad approximations of a solution. Linearization, or linear approximation, is just one way of approximating a tangent line at a certain point. Notes,whiteboard,whiteboard page,notebook software,notebook, pdf,smart,smart technologies ulc,smart board interactive whiteboard created date 112018 9. Differentials and approximations we have seen the notation dydx and weve never separated the symbols. Calculus definitions linearization and linear approximation in calculus.

Ap calculus ab and bc final notes lei maos log book. Each of the approximations is colored differently so. D once i have a tangent plane, i can calculate the linear approximation. In these cases we call the tangent line the linear approximation to the function at x a. Ap calculus class notes semester 1 sunapee middle high. This lesson contains the following essential knowledge ek concepts for the ap calculus course. Use the sum of rectangular areas to approximate the area under a curve. Example 9, 10, 11, and 12 real life applications of calculus newtons method a method for finding roots guess a first approximation to a solution of the equation fx 0. With the function in hand its now clear that we are being asked to use a linear approximation to estimate f 0. If you instead prefer an interactive slideshow, please click here. Ap calculus class notes semester 1 class notes will generally be posted on the same day of class.

I may keep working on this document as the course goes on, so these notes will not be completely. To get this formula you approximate the region by a large number of thin rectangles. Find materials for this course in the pages linked along the left. Math 221 1st semester calculus lecture notes for fall 2006. The approximation 4 can be thought of as coming from the formula for the. One needs to find a set of locations so that every wire has one of its ends covered.

Linearization and linear approximation calculus how to. We suggest that the presenter not spend time going over the reference sheet, but point it out to students so that they may refer to it if needed. Consider a wired network, with various locations and some wiring. Very small sections of a smooth curve are nearly straight.

Move all terms to the left, so that the equation becomes of the form fx 0 ln2. Calculus i linear approximations pauls online math notes. Click here for an overview of all the eks in this course. Single variable calculus, fall 2003 lecture notes the lecture notes represent a summary of the topics discussed and analyzed in class.

Pdf produced by some word processors for output purposes only. Ive tried to make these notes as self contained as possible and so all the information needed to read through them is either from an algebra or trig class or contained in other sections of the notes. Notions of this type appear everywhere, and will accompany you throughout your future career. Each of the approximations is colored differently so we can see how they actually work. When using linear approximation, we replace the formula describing a curve by the formula of a straight line.

For simpsons rule we are going to approximate the function with a quadratic and were going to require that the quadratic agree with three of the points from our subintervals. Use the first approximation to get a second, the second to get. Now, well give meaning to dy and dx as separate entities. Differentiation and approximation calculus of a single variable x. If neither eigenvalue has magnitude equal to 1, then the behavior of the system 3 near x.

Robbin december 21, 2006 all references to thomas or the textbook in these notes refer to. The flow rate is recorded at 15minute intervals between 7. From this graph we can see that near x a the tangent line and the function have nearly the same graph. Find the equation of the tangent line for f x x sin at x 0, then use it to approximate sin 0. Newtons method page 3 strategy for newtons method to obtain an approximate solution of an equation. To do this, look for a nearby input to this function that we can compute easily. Lecture notes single variable calculus mathematics.

Math 221 first semester calculus fall 2009 typeset. Archimedes was fascinated with calculating the areas of various shapesin other words, the amount of space enclosed by the shape. Seeing as you need to take the derivative in order to get the tangent line, technically its an application of the derivative like many tools or arguably, all of them, linearization isnt an exact science. Note, however, that rf cannot be our normal vector. These are notes for a one semester course in the di. While the taylor polynomial was introduced as far back as beginning calculus, the major theorem from taylor is that the remainder from the approximation, namely gx t rx, tends to 0 faster than the highestorder term in t rx. Included are detailed discussions of limits properties, computing, onesided, limits at infinity, continuity, derivatives basic formulas, productquotientchain rules lhospitals rule, increasingdecreasingconcave upconcave down, related rates, optimization and basic integrals. Here is a set of notes used by paul dawkins to teach his calculus i course at lamar university. Localism the linear approximation is only useful locally. The point for the linear approximation should also be somewhat clear. Lecture notes multivariable calculus mathematics mit. This is a quadratic approximation, so we dont care about anything higher than x2.

He used a process that has come to be known as the method of exhaustion, which used. A tutorial for solving nasty sums david gleich january 17, 2005 abstract in this tutorial, i will. As long as we are near to the point x0,y0 then the tangent plane should nearly approximate the function at that point. Calculus i linear approximations practice problems. One nice use of tangent planes is they give us a way to approximate a surface near a point. Math 221 1st semester calculus lecture notes version 2. Net or total change as the integral of a rate and substitution method. Area approximation and accumulation student sessionpresenter notes this session includes a reference sheet at the back of the packet. In this chapter, we will use local information near a point x b to. The notes were written by sigurd angenent, starting from an extensive collection of notes and problems compiled by joel robbin. Calculus i or needing a refresher in some of the early topics in calculus. Note that 4 becomes a special case of 5 if we take r.