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Thread: Overview of Calculus 2 anyone :)?

  1. #1
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    Overview of Calculus 2 anyone :)?

    Can anyone give me some examples of problems and properties, and theorems, I will be learning in Calculus 2?

    We finished Calculus 1 off with using the Disk and Shell and Washer's method to find volume of a solid.

    Course Description for Calculus 2:



    This course is the second in the Calculus sequence. Students will differentiate exponential, logarithmic, and inverse trigonometric functions, perform logarithmic differentiation, and learn techniques of integration. Evaluating improper integrals, indeterminate forms using L'Hopital's Rule, and infinite series are also included as well as polar coordinates, curves, conic sections, parametric equations, and separable first-order differential equations.

    This course is the second in the Calculus sequence. Students will differentiate exponential, logarithmic, and inverse trigonometric functions, perform logarithmic differentiation, and learn techniques of integration. Evaluating improper integrals, indeterminate forms using L'Hopital's Rule, and infinite series are also included as well as polar coordinates, curves, conic sections, parametric equations, and separable first-order differential equations.
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    Re: Overview of Calculus 2 anyone :)?

    Students will differentiate exponential, logarithmic, and inverse trigonometric functions, perform logarithmic differentiation, and learn techniques of integration.
    You will learn to differentiate such functions as y= e^{3x^2}, y= 2^{5x}, ln(7x^3), log_3(x), sin^{-1}(3x), arctan(x). You will learn to differentiate such things as y= x^x by writing it as ln(y)= x ln(x) and differentiating both sides with respect to x.

    Evaluating improper integrals,
    These are integrals such as \int_0^\infty \frac{dx}{1+ x^2} and \int_{-1}^1 \frac{dx}{\sqrt{x}}
    where the integrand is not defined at some point in the interval of integration.

    indeterminate forms using L'Hopital's Rule
    Such as lim_{x\to 2}\frac{x^2- 4}{e^{x- 2}- 1}
    L'Hopital's rule says that since numerator and denominator both go to 0 as x goes to 2, this limit is the same as the limit of the fraction you get by differentiating the numerator and denominator separately. Here the derivative of the numerator is 2x and the derivative of the denominator is e^{x- 2}. So \lim_{x\to 2}\frac{x^2- 4}{e^{x- 2}- 1}= \lim_{x\to 2}\frac{2x}{e^{x- 2}}= 4.

    and infinite series are also included
    Such as \sum_{n=0}^\infty \frac{1}{2^n} and \sum_{n=0}^\infty \frac{x^n}{n!}

    as well as polar coordinates, curves
    In 'Cartesian coordinates' a point is identified by (x, y) where x is measured along the x-axis from the origin to the foot of the perpendicular from the point to the x-axis and y is measured along the y-axis from the origin to the foot of the perpendicular from the point to the y-axis. In polar coordinates each point is identified by (r, \theta) where r is the distance measure along the straight line from the origin to the point and \theta is the angle measured counterclockwise from the positive x-axis to that line.

    conic sections
    conic sections are ellipses, circles, parabolas and hyperbolas, as well as the "degenerate conics", a straight line, two intersecting straight lines, two parallel straight lines, a single point, and the empty set (really "degenerate"!).
    All of those can be written as Ax^2+ Bxy+ Cy^2+ Dx+ Ey+ F= 0 for various values of A, B, C, D, E, and F.

    parametric equations
    Equations in which x, y, and z are written in terms of one or more "parameters". For example, if you think of t as "time" then the physics equation x= f(t), y= g(t), and z= h(t) are "parametric equations" with t the "parameter" describing the motion of some object. The circle with center at (0, 0) and radius r (in Cartesian coordinates x^2+ y^2= r^2 can be written with parametric equations (parameter \theta) x= r cos(\theta) and y= r sin(\theta). The sphere with center at (0, 0, 0) and radius \rho (In Cartesian coordinates x^2+ y^2+ z^2= \rho^2) can be written with parametric equations x= \rho cos(\theta)sin(\phi), = \rho sin(\theta)sin(\phi), and z= \rho cos(\phi) with parameters \theta and \phi

    and separable first-order differential equations
    A "differential equation" is an equation which has a function rather than a number as its "unknown" and contains various derivatives of that function. A "first-order differential equation" is one where you have only the first derivative and no higher order. A "separable first-order differential equation" is one where we can "separate" the variables so that one is on each side of the equation. For example, the "first-order differential equation" \frac{dy}{dx}= 2xy can be "separated" as \frac{dy}{y}= 2xdx which can then be integrated separately- ln(y)= x^2+ C where C is any constant. We can solve for y as y(x)= e^{x^2+ C}= C'e^{x^2} where C'= e^C
    Last edited by HallsofIvy; Dec 15th 2016 at 12:50 PM.
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