9 Prove that the sum of a rational number and an irrational number is always irrational.
| 10 | | The volume of a spherical bubble is increasing at a constant rate. | | Show that the rate of increase of the radius, \(r\), of the bubble is inversely proportional to \(r ^ { 2 }\) \(\text { Volume of a sphere } = \frac { 4 } { 3 } \pi r ^ { 3 }\) |
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Jodie is attempting to use differentiation from first principles to prove that the gradient of \(y = \sin x\) is zero when \(x = \frac { \pi } { 2 }\)
Jodie's teacher tells her that she has made mistakes starting in Step 4 of her working. Her working is shown below.
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Step 1 Gradient of chord \(A B = \frac { \sin \left( \frac { \pi } { 2 } + h \right) - \sin \left( \frac { \pi } { 2 } \right) } { h }\)
Step 2 \(= \frac { \sin \left( \frac { \pi } { 2 } \right) \cos ( h ) + \cos \left( \frac { \pi } { 2 } \right) \sin ( h ) - \sin \left( \frac { \pi } { 2 } \right) } { h }\)
Step 3
$$= \sin \left( \frac { \pi } { 2 } \right) \left( \frac { \cos ( h ) - 1 } { h } \right) + \cos \left( \frac { \pi } { 2 } \right) \frac { \sin ( h ) } { h }$$
Step 4
For gradient of curve at \(A\),
let \(h = 0\) then
\(\frac { \cos ( h ) - 1 } { h } = 0\) and \(\frac { \sin ( h ) } { h } = 0\)
Step 5
Hence the gradient of the curve at \(A\) is given by \(\sin \left( \frac { \pi } { 2 } \right) \times 0 + \cos \left( \frac { \pi } { 2 } \right) \times 0 = 0\)
Complete Steps 4 and 5 of Jodie's working below, to correct her proof.
Step 4 For gradient of curve at \(A\),
Step 5 Hence the gradient of the curve at \(A\) is given by