Calculus 2 Pdf Notes
The volume of the solid is the sum of the volumes of its slices. Prove that the following sequence has a limit. Critical points will show up in most of the sections in this chapter, so it will be important to understand them and how to find them. To get a fuller understanding of some of the ideas in this section you will need to take some upper level mathematics courses. Infinite Limits of Integration.
We can use the linear approximation to a function to approximate values of the function at certain points. Table of Indefinite Integrals.
Single Variable Calculus
We will discuss many of the basic manipulations of logarithms that commonly occur in Calculus and higher classes. In other words, image segmentation techniques pdf we will be finding the largest and smallest values that a function will have. Work is the energy produced by a force F pushing a body along a given distance d.
Included is a brief discussion of inverse trig functions. Logarithmic differentiation gives an alternative method for differentiating products and quotients sometimes easier than using product and quotient rule.
In the following example the formula of integration by parts does not yield a final answer, but an equation verified by the integral from which its value can be derived. Now the idea is to divide the solid into shells and add up their volumes. The result is another function that can also be represented with another power series.
Analogously, the functions sin x, cos x, tan x, etc. Ak are coefficients to be determined. Sometimes we need to use the formula more than once. Finally multiply by the leading coefficient of Q x.
Each solution curve is a particular solution of the slope field. It can be interpreted as an infinite polynomial. Integrals of Products of Sines and Cosines. We will also take a conceptual look at limits and try to get a grasp on just what they are and what they can tell us. The initial condition can be used to determine the value of the constant in the solution of the equation.
CALCULUS NOTES (I)
Integrals of Hyperbolic Functions. Integrals of Secants and Tangents. Estimate to the third decimal place.
Examples in this section tend to center around geometric objects such as squares, boxes, cylinders, etc. Inverse Hyperbolic Functions. The examples in this section can all be done with a basic knowledge of indefinite integrals and will not require the use of the substitution rule. Suppose that an and bn are series with positive terms. The idea is to divide the solid into slices perpendicular to a given reference line.
Notes on Calculus
Newton's Method is an application of derivatives will allow us to approximate solutions to an equation. We will also look at computing limits of piecewise functions and use of the Squeeze Theorem to compute some limits.
Representation of Functions as Power Series We have already seen that a power series is a particular kind of function. An alternating series is a series whose terms are alternately positive and negative. Improper integrals are integrals in which one or both of these conditions are not met, i. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers. We show the derivation of the formulas for inverse sine, inverse cosine and inverse tangent.
Note that this section is only intended to introduce these concepts and not teach you everything about them. With few exceptions I will follow the notation in the book. As an example of application we find the net distance or displace- ment, and the total distance traveled by an object that moves along a straight line with position function s t. With the Mean Value Theorem we will prove a couple of very nice facts, one of which will be very useful in the next chapter.
The Fundamental Theorem of Calculus. We will also look at the first part of the Fundamental Theorem of Calculus which shows the very close relationship between derivatives and integrals.
We will determine the area of the region bounded by two curves. Rate of Change of a function and Tangent Lines to functions. We give two ways this can be useful in the examples. Summation Formulas Appendix C.
When did that individual die? This is often one of the more difficult sections for students. The Limit Comparison Test. In reality this model is unrealistic because envi- ronments impose limitations to population growth.
Just take the result in absolute value. Trigonometric Substitutions. So the right hand side of the equation can be interpreted as a slope field in the xy-plane. While it might not seem like a useful thing to do with when we have the function there really are reasons that one might want to do this.
The interval of convergence is the set of values of x for which the series converges. The first derivative will allow us to identify the relative or local minimum and maximum values of a function and where a function will be increasing and decreasing.
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