Find derivative at origin using approximation

A question is this type if and only if it involves finding dy/dx at the origin or relating the small angle approximation to the gradient at a point.

2 questions · Standard +0.3

1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x
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Edexcel Paper 1 2019 June Q14
7 marks Standard +0.3
  1. The curve \(C\), in the standard Cartesian plane, is defined by the equation
$$x = 4 \sin 2 y \quad \frac { - \pi } { 4 } < y < \frac { \pi } { 4 }$$ The curve \(C\) passes through the origin \(O\)
  1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the origin.
    1. Use the small angle approximation for \(\sin 2 y\) to find an equation linking \(x\) and \(y\) for points close to the origin.
    2. Explain the relationship between the answers to (a) and (b)(i).
  3. Show that, for all points \(( x , y )\) lying on \(C\), $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { a \sqrt { b - x ^ { 2 } } }$$ where \(a\) and \(b\) are constants to be found.
OCR MEI Paper 2 2020 November Q10
9 marks Standard +0.3
10 In this question you must show detailed reasoning. The equation of a curve is $$y = \frac { \sin 2 x - x } { x \sin x }$$
  1. Use the small angle approximation given in the list of formulae on pages 2-3 of this question paper to show that $$\int _ { 0.01 } ^ { 0.05 } \mathrm { ydx } \approx \ln 5$$
  2. Use the same small angle approximation to show that $$\frac { d y } { d x } \approx - 10000 \text { at the point where } x = 0.01 \text {. }$$ The equation \(y = 0\) has a root near \(x = 1\). Joan uses the Newton-Raphson method to find this root. The output from the spreadsheet she uses is shown in Fig. 10.1. \begin{table}[h]
    \(n\)01234567
    \(\mathrm { x } _ { \mathrm { n } }\)10.9585090.9500840.9482610.947860.9477720.9477530.947748
    \captionsetup{labelformat=empty} \caption{Fig. 10.1}
    \end{table} Joan carries out some analysis of this output. The results are shown in Fig. 10.2. \begin{table}[h]
    \(x\)\(y\)
    0.9477475\(- 7.79967 \mathrm { E } - 07\)
    0.9477485\(- 2.90821 \mathrm { E } - 06\)
    \(x\)\(y\)
    0.947745\(4.54066 \mathrm { E } - 06\)
    0.947755\(- 1.67417 \mathrm { E } - 05\)
    \captionsetup{labelformat=empty} \caption{Fig. 10.2}
    \end{table}
  3. Consider the information in Fig. 10.1 and Fig. 10.2.