Maths for Engineers and Scientists 2

Maths for Engineers and Scientists 2

Differentiation and Integration

• Maclaurin and Taylor Series Approximations of Functions
• Optimisation: Stationary Points, Maxima/Minima and Modelling
• Integration by Substitution, Integration by Parts, Integration using Partial Fractions
• Area Under a Curve
• Average Value of a Function, Curve Length
• Solving Differential Equations: Separable Variables Method
• Numerical Integration: the Trapezium/Trapezoidal Rule

Complex Numbers

• Arithmetic of Complex Numbers
• Solving Quadratic Equations
• The Argand Diagram
• Polar Form of a Complex Number
• Exponential Form of a Complex Number
• De Moivre’s Theorem

Matrices

• Matrix Addition and Subtraction
• Multiplication by a Scalar
• Matrix Multiplication
• Solving Systems of Linear Equations
• The Determinant of a Matrix
• The Inverse of a Matrix

Differentiation Revision

• Given the function . Derivative of the function is defined as
• . The derivative of is
• a) the slope (or gradient) of the tangent line at the point on the graph of
• b) the rate of change of the function with respect to .

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Differentiation: Common Functions and Rules

• Differentiation: Common Functions and Rules

Product Rule:

• Product Rule:
• Quotient Rule : =
• Chain Rule :
• Example 1. Find the derivative of the function: a) ; b)
• c)

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Approximating Functions by Polynomials Maclaurin and Taylor Series

• Example 2. Write the equation of tangent line to the graph of the function at the point .

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The tangent line is a good approximation to for close to . However, at the tangent line is equal to while the curve

• The tangent line is a good approximation to for close to . However, at the tangent line is equal to while the curve
• We could improve the approximation by adding terms in , etc. This section is about doing this for a general function .

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Suppose that the function

• Suppose that the function
• (1)
• We want to find the numbers , etc.
• Putting in (1) gives
• Differentiating (1) gives
• Then

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• =
• Continuing this process we obtain
• This series is called Maclaurin series for .
• Example 3. Find the Maclaurin series for up to and including the term.

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Example 4. Find the Maclaurin series for up to and including the term.

• Example 4. Find the Maclaurin series for up to and including the term.
• Example 5. Find the Maclaurin series for up to and including the term.

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Example 6. Show that the Maclaurin series for up to and including the term is

• Example 6. Show that the Maclaurin series for up to and including the term is
• Example 7. Show that the Maclaurin series for up to and including the term is
• Example 8. Find the Maclaurin series for up to and including the term.

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Small angle approximations – always work in radians.

• Small angle approximations – always work in radians.
• The Maclaurin series can be used to find approximations for and when the angle in radians is small. If is small then will be very small and so terms in and of higher order can be ignored. Therefore if is small and measured in radians
• and

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Example 9. A wire of length is used to support a transmission tower. If the angle

• Example 9. A wire of length is used to support a transmission tower. If the angle
• in radians is small find a “rule of thumb” approximation for the height of the tower and the distance of the tower from where the support wire is anchored .

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Taylor series

• The Maclaurin series can be used to approximate a function at . If we want to approximate about a different value, say then we use the Taylor series.
• The Taylor series expansion of about the value is defined as

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Example 10. Find the Taylor series for at up to the term and use it to approximate .

• Example 10. Find the Taylor series for at up to the term and use it to approximate .

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