pure maths revision mix for y13

Front

What is the trapezium rule used for in calculus?

Back

The trapezium rule is a numerical method used to approximate the definite integral of a function. It works by dividing the area under the curve into trapezoids and summing their areas.

Front

How do you apply the trapezium rule with intervals of width 1.5?

Back

To apply the trapezium rule with intervals of width 1.5, divide the interval into segments of 1.5 units, calculate the function values at the endpoints of each segment, and use the formula: \( \text{Area} \approx \frac{h}{2} (f(a) + 2f(x_1) + 2f(x_2) + ... + f(b)) \) where \( h \) is the width of the interval.

Front

Expand the expression \( (1 + 3x)^{\frac{1}{3}} \) up to the term \( x^3 \).

Back

The expansion is \( 1 + x - \frac{1}{3}x^2 + \frac{5}{3}x^3 + ... \) and is valid for \( -\frac{1}{3} < x < \frac{1}{3} \).

Front

What is the formula for the trapezium rule?

Back

The trapezium rule formula is: \( \int_a^b f(x)dx \approx \frac{h}{2} (f(a) + 2f(x_1) + 2f(x_2) + ... + f(b)) \) where \( h \) is the width of the intervals.

Front

What is the significance of the value \( R \) in the equation \( \cos(x) - \sqrt{3}\sin(x) \equiv R\cos(x + a) \)?

Back

The value \( R \) represents the amplitude of the resultant vector formed by the coefficients of \( \cos(x) \) and \( \sin(x) \). It can be calculated using the formula \( R = \sqrt{a^2 + b^2} \) where \( a \) and \( b \) are the coefficients.

Front

How do you find the value of \( R \) in the equation \( \cos(x) - \sqrt{3}\sin(x) \equiv R\cos(x + a) \)?

Back

To find \( R \), use the formula: \( R = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2 \).

Front

What is the derivative of \( \frac{\sqrt{x} + 4x^3}{x^2} \)?

Back

The derivative can be simplified to \( x^{-\frac{3}{2}} + 4x \) using the quotient rule and power rule.

Front

What is the interval of validity for the expansion of \( (1 + 3x)^{\frac{1}{3}} \)?

Back

The interval of validity for the expansion is \( -\frac{1}{3} < x < \frac{1}{3} \).

Front

What is the approximate value of \( \int_0^8 \sqrt{1+x^2}dx \) using the trapezium rule with 5 ordinates?

Back

The approximate value is 33.946.

Front

What is the relationship between definite integrals and the area under a curve?

Back

Definite integrals represent the net area under a curve between two points on the x-axis, accounting for areas above and below the x-axis.