Calculus II - 14
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Separable differential equations can be solved explicitly by factoring the expression for the derivative into a function of y and a function of x. Examples of separable equations provided include dy/dx = x, dy/dx = y + x, and dy/dx = y/x. These types of first order differential equations can be solved by separating the variables and integrating both sides.
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Chu v. Nguyen
Integral Exponents
Exponents
If n is a positive integer (a whole number, i.e., a number without decimal part) and x is a number, then
The number x is called the base and n is called the exponent.
The most common ways of referring to are “ x to the nth power,”
“ x to the nth,” or “the nth power of x.”
Integral Exponents (cont.)
For any non-zero number x and a positive integer n
and
Note: is not defined
and
Rules Concerning Integral Exponents
Following are five rules in which m and n are positive integers:
Rule 1: ; for example,
Rule 2: ; for example
or
Rules Concerning Integral Exponents (Cont.)
Rule 3: ; for example
or
Rule 4: ; for example
or
Rule 5: ; for example
or
Basic Rules for Operating with Fractions
Since dividing by zero is not defined, we assume that the denominator
is not zero.
Following are the eight basic rules for operating with fractions.
Rule 1: ; for example
Rule 2: ; for example
Rule 3: ; for example
Basic Rules for Operating with Fractions (cont.)
Rule 4: ; for example
Rule 5: ; for example
Rule 6: ; for example
Basic Rules for Operating with Fractions (cont.)
Rule 7: ; for example
Rule 8: ; for example
Notes: a*b +a*x may be expressed as a(b + x)
a*b + 1 may be written as a(b + ), and
m*x – y may be expressed as m(x - )
Square Root
Generally, for a>0 , there is exactly one positive number x such that
, we say that x is the root of a, written as
for
When n = 2, we say that x is the square root of “a” and is denoted by
or or
For example:
or
Practices
Carrying out the following operations:
24 ; 2-2 ; 2322, ; 252-5 ; and (2x3)5
; ; ; and
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2. Graphs are also included showing the relationships between trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions.
3. Trigonometric identities are defined relating trigonometric functions of sums and differences of angles.
Calculo
1. This document provides formulas and definitions for calculus, trigonometry, and hyperbolic functions.
2. It includes formulas for limits, derivatives, integrals, exponents, logarithms, trigonometric functions and their inverses, identities, and hyperbolic functions and their inverses.
3. Graphs are also included to illustrate various trigonometric and hyperbolic functions.
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Calculus II - 36
The document discusses arc length, curvature, and formulas for calculating them for curves in 2D and 3D. It provides the definitions of arc length, unit tangent vector, and curvature. Examples are given of finding the curvature of various curves, such as circles, helices, and parabolas, at given points using parametrizations of the curves. In general, the curvature of a plane curve f(x) = y is k(x) = f''(x) / [1+(f'(x))^2]^(3/2).
Caculus II - 37
The document discusses tangent, normal, and binormal vectors for curves. It defines these vectors for a space curve at a point and explains that they form an orthogonal TNB frame. It then gives examples of finding these vectors and the normal and osculating planes for a helix curve at a specific point.
Calculus II - 34
The cross product and dot product of vectors are introduced. Properties of the cross product include that it is only defined for 3D vectors and that the magnitude of the cross product equals the area of the parallelogram determined by the two vectors. Vector, parametric, and various forms of plane equations are presented along with examples of finding equations of lines and planes given conditions.
Calculus II - 33
The dot product of two vectors a and b is defined as a·b and is calculated by multiplying their corresponding components and adding them. It can be used to calculate the projection of one vector onto another and the cosine of the angle between the vectors. The cross product of two vectors a and b is defined as axb and produces a third vector that is perpendicular to both a and b. It has various properties including being anticommutative and having a magnitude equal to the area of the parallelogram determined by the two vectors.
Calculus II - 32
A vector is a quantity that has both magnitude and direction. It can be represented in a coordinate system with points (x,y,z) and the vector between two points is defined as (x2-x1, y2-y1, z2-z1). Vectors can be added, subtracted, and multiplied by scalars. The length of a vector is calculated using the Pythagorean theorem and vectors can be unitized. There are standard basis vectors i, j, k in 3D space and vectors can be represented as a combination of these basis vectors. The dot and cross products can be used to calculate properties of vectors.
Calculus II - 31
The document discusses 3D coordinate systems and vectors. It defines:
1) The distance formula in 3D space between two points (x1,y1,z1) and (x2,y2,z2) as the square root of (x2-x1)2 + (y2-y1)2 + (z2-z1)2.
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Calculus II - 30
The document discusses using Taylor series approximations to evaluate expressions. It finds the maximum error of approximating e-0.5 as 1 - 0.5 + 0.5^2, determines the value of x for which this approximation is accurate to within 0.01, and establishes that an 8th degree Taylor polynomial is needed to approximate e^x in [-1,1] with an error less than 0.001.
Calculus II - 29
The document discusses Taylor series and Maclaurin series. It provides the formulas for determining the coefficients of a Taylor series centered at a point a. It also discusses Taylor's inequality which relates the remainder of a Taylor series to the derivative of the function. Several examples are worked out of finding the Maclaurin series for common functions and proving they represent the functions for all values. Additional power series are derived from fundamental series like e^x, sin(x), and cos(x).
Calculus II - 28
The document discusses Taylor series and how they can be used to represent general functions as power series. It provides the formulas for determining the coefficients in the Taylor series expansion of a function f(x) about a point a. Examples are given of finding the Maclaurin series, which is the Taylor series expansion about x=0, for common functions and determining their radii of convergence. The key steps in proving that the Taylor series equals the original function within the radius of convergence are outlined using Taylor's inequality.
Calculus II - 27
This document discusses power series representations of functions and their radii of convergence. It provides examples of finding the power series for functions like 1/(1+x) and sin(x), as well as the general process for determining the power series representation of a function and the radius of convergence based on the interval of convergence. It also discusses rules for differentiating and integrating term-by-term for power series whose functions have a given radius of convergence.
Calculus II - 26
A power series is a series of the form Σan(x-c)n where an are the coefficients of the series. The sum of the series is a function of x. A power series centered at c or about c is of the form Σan(x-c)n. For a given power series, there are three possibilities for convergence: it converges only at c, converges for all x, or converges when |x-c|<R and diverges when |x-c|>R, where R is the radius of convergence.
Calculus II - 25
This document discusses tests for determining if infinite series converge or diverge, including:
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3) The Ratio and Root Tests, which can be used to determine if series converge absolutely (and thus converge) based on the ratios or roots of successive terms.
Calculus II - 24
The Comparison Test allows determining if a series converges or diverges by comparing it to standard p-series or geometric series. If a series is less than or equal to a convergent p-series, or greater than or equal to a divergent p-series, then the series has the same behavior. Examples demonstrate applying the test to various series.
Calculus II - 23
The document introduces the integral test, which can be used to determine if an infinite series is convergent or divergent. The integral test states that a series is convergent if the corresponding improper integral converges, and divergent if the improper integral diverges. Several examples demonstrate applying the integral test to various series by setting up and evaluating the corresponding improper integrals. The document also discusses using integrals to estimate the sum of a convergent series.
Calculus II - 19
There are three types of conic sections: parabolas, ellipses, and hyperbolas. A parabola is the set of points equidistant from a fixed point (focus) and line (directrix). An ellipse is the set of points where the sum of distances from two fixed points (foci) is a constant. A hyperbola is the set of points where the difference of distances from two fixed points (foci) is a constant. Equations of parabolas, ellipses, and hyperbolas can be written and their properties like foci, vertices and asymptotes can be determined from the equations.
Calculus II - 18
The document discusses formulas for calculating the area and length of regions defined by polar curves. It gives the formula for finding the area inside a polar curve between rays at specific theta values. Similarly, it provides the formula for calculating the arc length of a polar curve between two theta values. It then works through examples of using these formulas to find the area of a four-leaved rose curve and the area and length of a cardioid curve.
Calculus II - 17
Polar coordinates represent points in a plane using an angle and distance from an origin point, rather than Cartesian x and y distances. In polar coordinates, a point is defined by its radial distance r from the origin and its angle θ measured counterclockwise from the polar axis. Converting between polar and Cartesian coordinates involves simple formulas relating r, θ, x, and y. Polar equations can more simply describe curves like the cardioid than their Cartesian equivalents.
Calculus II - 16
The document discusses calculus concepts related to parametric curves including calculating tangents, areas, arc length, and surfaces of revolution. It provides the formulas for finding the tangent of a parametric curve, area under a parametric curve, arc length of a parametric curve, and surface area obtained by rotating a parametric curve around an axis. Examples are given for each concept using the parametric cycloid curve.
Calculus II - 15
The document discusses parametric equations and parametric curves. Parametric equations define curves using two equations, where x and y are defined in terms of a parameter t. Common examples of parametric curves are given. Parametric equations can be used to define circles by giving the x and y coordinates in terms of the center point and radius. The parameter t can be eliminated to obtain the Cartesian equation of the curve. Calculus can be applied to parametric curves to find tangents, areas, arc length, and surfaces of revolution. Formulas are given for finding the equations of tangent lines to parametric curves at a given value of t.
Calculus II - 13
This document discusses numerical methods for solving differential equations. It introduces direction fields, which provide a graphical approach to studying solutions, and Euler's method, which provides a numerical approach. Euler's method works by approximating the slope of the tangent line at each step using small step sizes to iteratively calculate successive approximations of the solution.
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Calculus II - 14
- 1. 9.3 Separable Equations For some certain types of differential equations, there exist explicit solution. A separable equation is a first order differential equation in which the expression for the derivative / can be factored as a function of and a function of : = ( )· ( )
- 2. = ( )· ( ) Examples: = = = +
- 3. = ( )· ( )
- 4. = ( )· ( ) = ( ) ( )
- 5. = ( )· ( ) = ( ) ( ) = ( ) ( )
- 6. = ( )· ( ) = ( ) ( ) = ( ) ( ) ! ction i t fu n An i mplic
- 7. Ex: =
- 8. Ex: = =
- 9. Ex: = = =
- 10. Ex: = = = = +
- 11. Ex: = = = = + = +
- 12. Ex: = = = = + = + = +
- 13. Ex: = +
- 14. Ex: = + = +
- 15. Ex: = + = + = +
- 16. Ex: = + = + = + = +
- 17. Ex: = + = + = + = + = +
- 18. Ex: = + = + = + = + = + = ( + )
Editor's Notes
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