Calculus II - 2

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This document provides formulas for evaluating definite integrals of trigonometric functions. It presents integrals of the form ∫sin(x)dx, ∫cos(x)dx, ∫tan(x)dx, and ∫cot(x)dx, as well as integrals involving combinations of trigonometric functions. Formulas are given for both definite integrals where the limits are numeric values and indefinite integrals where the limits are not specified. Properties are also described for when the formulas work for both even and odd values of variables in the integrand.

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7.2 Trigonometric Integrals

Find

Find

Find

Find

Find

Find

Find

Find

= ( + )

= ( + + )

= ( + + ( + ))

= + + +

Also works for when both m
and n are EVEN.

Find

+ =

= ( )·( )·
= = ( )·( )·

= +

Also works for when one of
m and n is ODD.

Find

=
= =

=
= = ( )

= +

Also works for when m is ODD.

Find

=
= =

=
= = ( + )

= + +

Also works for when n is EVEN.

The document discusses properties of the roots of polynomial equations. It asks the reader to write down what they know about the sum and products of polynomial roots. It then asks the reader to find expressions for the sum of the roots, their product, and the sum of the products of two roots for a cubic polynomial in terms of its coefficients. Finally, it asks the reader to apply these results to a quadratic equation with specified roots.

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Factorise quadratic equations

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1) The document defines properties of exponential and logarithmic functions including: exponential functions follow exponent laws, logarithmic functions follow logarithmic laws, and the derivatives of exponentials and logarithms are the exponential/logarithmic functions themselves multiplied by the exponent/logarithm's argument. 2) Rules for limits of exponentials and logarithms as the argument approaches positive/negative infinity or zero are provided. 3) Graphs of the natural logarithm and logarithms with base a > 1 are similar shapes that increase without bound as the argument increases from 0 to infinity.

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This mathematical formula defines the change in temperature (ΔT) of an object over time as equal to the net rate of heat transfer into the object divided by its specific heat capacity. The net rate of heat transfer is calculated as the sum of the rates of heat transfer by conduction, convection and radiation, minus the sum of the rates of heat transfer out of the object by the same three mechanisms of heat transfer.

Poster

Jose Pasos

This document presents a model for capacity expansion with the risk of demand default. It formulates the problem using stochastic control theory and derives the Hamilton-Jacobi-Bellman equations. The optimal strategy is to immediately invest when demand exceeds the smooth free boundary curve G(y), and wait otherwise. The free boundary G(y) is the solution to an integral equation and separates the "wait" and "invest" regions. The model is illustrated with a basic example where demand follows geometric Brownian motion.

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).

Calculus II - 35

David Mao

The document discusses representations of lines, planes, and space curves in vector calculus. It defines the vector equation of a line, parametric equation of a line, and symmetric equation of a line. It also defines the vector equation and scalar equation of a plane. It then discusses representing space curves using parametric vector functions and calculating derivatives and integrals of vector functions, including rules for differentiation. It provides an example problem proving that the tangent line to a circle at a point is perpendicular to the line connecting that point to the circle's center.

Caculus II - 37

David Mao

Calculus II - 34

David Mao

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

David Mao

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

David Mao

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

David Mao

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. 2) Vectors as quantities that have both magnitude and direction. It shows examples of vector addition and subtraction using the triangle and parallelogram laws. 3) Key properties of vectors including that vector addition is commutative and distributive over vector addition, scalar multiplication distributes over vector addition, and vectors

Calculus II - 30

David Mao

Calculus II - 29

David Mao

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

David Mao

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

David Mao

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

David Mao

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

David Mao

This document discusses tests for determining if infinite series converge or diverge, including: 1) The Alternating Series Test, which can be used to determine if alternating series (with terms that are alternately positive and negative) converge. 2) Absolute and conditional convergence, where a series is absolutely convergent if the sum of the absolute values converges, and conditionally convergent if the series converges but the sum of absolute values does not. 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

David Mao

Calculus II - 23

David Mao

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 - 22

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This document discusses convergence of infinite series. It defines partial sums of a series and states that if the sequence of partial sums converges, then the series converges to that limit. If the partial sums do not converge or their limit does not exist, then the series diverges. It provides examples of geometric series, which converge if the common ratio is less than 1 in absolute value, and harmonic series, which diverges. It also discusses using known convergent series to determine if new series formed from their sums or products converge.

Calculus II - 21

David Mao

This document discusses infinite sequences and infinite series. Some key points: 1. It provides examples of determining if an infinite sequence converges or diverges based on the limit of its terms. 2. Properties of operations on convergent sequences are given, such as sums, products, and compositions. 3. Theorems regarding convergence of sequences are stated, such as if the limit of the sequence of partial sums exists. 4. Examples are given of showing monotonicity (increasing or decreasing behavior) of specific sequences. 5. Boundedness of sequences is defined, and the monotonic sequence theorem is stated. 6. Infinite series are introduced as the sum of the terms of a

Calculus II - 20

David Mao

1) A sequence is an ordered list of numbers written as {an}. A sequence converges to a limit L if the terms an get arbitrarily close to L as n increases. 2) Common properties of convergent sequences include that the sum, difference, and scalar multiple of two convergent sequences is also convergent. 3) Useful theorems include if {an} converges to L and {bn} converges to M, then {an + bn} converges to L + M. If |an| ≤ M for all n and M is a constant, then the sequence converges. 4) Examples of convergent sequences include the sequence of partial sums of a convergent series and the terms of a function evaluated at

Calculus II - 19

David Mao

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

David Mao

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.

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Calculus II - 2

1. 7.2 Trigonometric Integrals Find Find Find Find Find Find Find

2. Find

3. Find = ( + )

4. Find = ( + ) = ( + + )

5. Find = ( + ) = ( + + ) = ( + + ( + ))

6. Find = ( + ) = ( + + ) = ( + + ( + )) = + + +

7. Find = ( + ) = ( + + ) = ( + + ( + )) = + + + Also works for when both m and n are EVEN.

8. Find

9. Find + =

10. Find + = = ( )·( )·

11. Find + = = ( )·( )· = = ( )·( )·

12. Find + = = ( )·( )· = = ( )·( )· = +

13. Find + = = ( )·( )· = = ( )·( )· = + Also works for when one of m and n is ODD.

14. Find

15. Find = [ ( )+ ( )]

16. Find = [ ( )+ ( )] = ( ) + ( )

17. Find = [ ( )+ ( )] = ( ) + ( ) = +

18. Find

19. Find = = =

20. Find = = = =

21. Find = = = = = | |+

22. Find = = =

23. Find = = = =

24. Find = = = = = = +

25. Find = = = = = = + = + +

26. Find = = =

27. Find = = = =

28. Find = = = = = = ( )

29. Find = = = = = = ( ) = +

30. Find = = = = = = ( ) = + Also works for when m is ODD.

31. Find = = =

32. Find = = = =

33. Find = = = = = = ( + )

34. Find = = = = = = ( + ) = + +

35. Find = = = = = = ( + ) = + + Also works for when n is EVEN.

36. Find = = =

37. Find = = = =

38. Find = = = = =

39. Find = = = = = = +

40. Find = = = = = = + = ( + | + |) +

Calculus II - 2

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