Tangents, normals and gradient problems Problem set B Answers

1. (a) \(9e^x\)
2. (b) \(-8e^x\)
3. (c) \(-\frac16e^x\)
4. (d) \(\frac73e^x\)

1. (a) \(\frac{\sqrt2}{2}\)
2. (b) 17

\((\ln2,\,6\ln2-3)=(0.693,\,1.16)\)

\(y=2x-22\)

1. (a) \(y=3x-\pi+3\sqrt3\), which is \(y=3x+2.05\)
2. (b) \(y=3x+8-24\ln2\), which is \(y=3x-8.64\)

\(x>\frac15\)

1. (a) \(-1.875\) m/min, which is \(-1.88\) m/min
2. (b) The balloon is descending: its height is decreasing at 1.88 metres per minute.

1. (a) 9 million
2. (b) \(t=9\) hours
3. (c) 1.75 million per hour
4. (d) The model gives unlimited growth. In a real culture, nutrients and space run out.

1. (a) \((36,6)\)
2. (b) \(\left(\frac\pi3,1\right)=(1.05,1)\)

1. (a) \(y=5x-\frac{26}{5}\)
2. (b) \(e^x=\frac{26}{5}\) has only one solution

1. (a) \(y-3\tan k=\frac{3}{\cos^2k}(x-k)\)
2. (b) \(y=4x+\sqrt3-\frac{2\pi}3\)

\(k=3\)

1. (a) \(\frac{5\pi^2}{4}=12.3\)
2. (b) \(\pi^2=9.87\)

1. (a) \((25,5)\)
2. (b) \(y=\frac x{10}+\frac52\)