Trigonometric Identities

The best way to learn how to prove trigonometric identities is to do lots of example, there being ever so many identities to prove. Example: Prove Now divide every term in the numerator and denominator byto give Example: Prove the identity Divide every term in the numerator and denominator byto give Now perform long division […]

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Formulae Used in Trigonometric Proofs

There must be any number of trigonometric identities that can be proved. Really the only way to study them is to do lots of examples. It helps first to have a summary of the equations. There are five very important equations which can also be used to solve many trigonometric formulae: There are also some […]

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Applications of Simpson’s Rule

Simpson’s rule allows approximate calculations of definite integrals that might otherwise not be easily calculable. The approximation is given by wherethe number of strips, is even. Given an integral to estimate, we draw up a table of function values Example: Using Simpson’s Rule with six strips, estimate the value of the integral x-i 0 1 […]

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Integration by Substitution

Some integrals look intractible. Some of these intractible integrals can be made simpler and transformed into an integral that can be evaluated by making a suitable change of variables. We start with an integraland make the substitutionWe get Example: Find Substituteinto the integral Now we integrate the simplified integral: We are not finished yet. The […]

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Solids or Volumes of Revolution

We start with a graphIf the graph is rotated about theaxis it traces out a surfaces as shown. Between the surface and the– axis we may form a solid. We show here how to find the volume of this solid. We may picture the solid as being made up of slices of solid. For the […]

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Logarithms

It is helpful to state the rules of logarithms: (1) (2) Ifthen (3) (4) If b=x then bases seem to cancel giving (5) then (6) These rules may be used as shown in the following examples. Example: Solve Rule (2) gives The numerator and denominator both factorise to giveand we can cancel to give Rule […]

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Tangents and Normals

A tangent or normal to a curve is a line, taking the formwhereis the gradient and is the intercept. Given a functionwe can find the gradient atby finding the gradient functionand substituting the valueinto this expression. Sometimes however we don’t havesois not given explicitly as a function ofIn these cases typically we have to differentiate […]

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Exponential Quadratic Equations

Quadratic equations are easy to solve. You can factorise, or failing that, use the quadratic formula. If the quadratic formula returns no real solutions, the quadratic formula has no real solutions. Many equations can be transformed into quadratic equations by substitution and rearrangement. becomesby substituting becomesby substituting becomeson multiplying byand thenon substituting The quadratic equation […]

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Differential Equations

A differential equation is any equation with one or more differential terms –or similar. Givento findwe differentiateTo findwe differentiate We can differentiate any number of times: if we differentiate y n times, we have Examples of differential equations include: If a functionis a solution to the differential equation then we can substitutefor into the equation […]

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