Differentiating Inverse Trigonometric Functions

Ifthen we cannot finddirectly. Instead we take the sin of both sides to obtainand differentiate implicitly using the chain rule. We obtain Since originallywas given as a function ofwe would normally findas a function of We can do this forusing the identityWe rearrange this to makethe subject:Hence Ifthen we cannot finddirectly. Instead we take the […]

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Proving Divisibility With Induction

We can proved that a closed form for an expression id divisible for a certain number by proving that the difference between sucessive terms is divisible, as well as one of the terms. This is not always as simple as it sounds. Suppose we are to proveis divisible by 13 . The standard proof by […]

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Integrating Reciprocal Trigonometric and Hyperbolic Functions

All the the reciprocal trigonometric and hyperbolic functions can be integrated by manipulating them into the form differentiates to giveanddifferentiates to give giveso the numerator of the inegrand is the derivative of the denominator, hence differentiates to giveanddifferentiates to giveso the numerator of the integrand is the derivative of the denominator, hence SubstitutesoThe integral becomes […]

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The Relationship Between the Roots and Coefficients of Polynomials

There is an intimate relationship between the coefficients and the roots of polynomial equations, best illustrated by writing a polynomial in terms of its roots and expanding. Suppose thatWe can write this asIf the roots of this expression areandthenso that henceand Suppose thatWe can write this asIf the roots of this expression areand thenso that […]

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Odd and Even Functions

A functionis odd isand even if The function above is odd. Every odd function has rotational symmetry order two about the origin, so we can rotate it by is odd since The function above is even. Every even function is symmetric with respect to the y – axis, so it will be the same graph […]

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Finding Bounds for an Integral

The sum of a series may not be easily found, and it is desirable to be able to find upper and lower limits for its value. If series consists of terms that are decreasing or decreasing then it may be possible to easily find a bound by integration. To find the bounds fornote thatis an […]

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Sketching Graphs of y Squared

Given a graph ofwe can sketch the graphequivalent to by making the following observations. cannot be negative since we cannot take the square root of a negative number to obtain a real number. Ifis negative for any value ofthese values ofdo not appear in the domain ofand these points cannot be plotted. Any parts of […]

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Solving Constant Coefficient Linear Second Order Homogeneous Equations – Examples

Solving a constant coefficient equation – one of the form– may be accomplished by assuming a solution of the formThe solution will contain two arbitrary constants, which can be found given two boundary conditions onor some derivative of Example: Solve the equationsubject to the conditionswhenandwhen AssumingthenandSubstitute these into the original equation to getWe divide by […]

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