smc-1090-60-Find point of tangency

Calculus, 2ADV C3 2024 HSC 29

Consider the curve , where .

At a particular point, the tangent and normal to the curve are given by and respectively.

The curve has a minimum turning point at .

Find the values of and . (4 marks)

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♦♦ Mean mark 35%.

Find the coordinates of the point on the curve at which the tangent to the curve is parallel to the line . (3 marks)

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The line is a tangent to the curve at a point .

Sketch the line and the curve on one diagram. (1 mark)

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Find the coordinates of . (3 marks)

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Find the value of . (1 mark)

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ii.

♦ Mean mark 40%
COMMENT: Given

, it follows

. Make sure you understand the arithmetic behind this (NB. Simply take the

of both sides).

iii.

♦♦ Mean mark 30%.