Pure paper definitions

For those annoying 1 mark questions

Under what conditions are small angle approximations valid?

θ is small and in radians

When is the infinite binomial expansion valid?

(1+bx/a)ⁿ is valid for |bx/a| < 1

How do you conclude a question where you use the sign change method to locate a root of a function?

There is a sign change between x=a and x=b and the function is continuous between x=a and x=b so the root lies between a and b. (Replace x, a, and b with values in question)

When does the Newton-Raphson iteration method fail?

When the gradient is small.

Why does the Newton-Raphson iteration method fail if x₀ is a stationary point?

The gradient will by 0, so f’(x)=0. You can’t divide by zero, so x₂ would be undefined.

What is the ‘limitation’ of a variable?

The range of values it can have

Is the application of the trapezium rule in the picture an underestimate or overestimate of the true value of the shaded area, and why?

It is an overestimate as the area of each trapezium is greater than the area of the corresponding strip.

When is the trapezium rule an overestimate?

When the curve is convex

When is the trapezium rule an underestimate?

When the curve is concave