Tangents, normals and gradient problems Problem set A Answers

1. (a) \(7e^x\)
2. (b) \(-4e^x\)
3. (c) \(-\frac13e^x\)
4. (d) \(\frac52e^x\)

1. (a) \(\frac{\sqrt3}{2}\)
2. (b) 10

\((\ln3,\,4\ln3)=(1.10,\,4.39)\)

\(y=4x-20\)

1. (a) \(y=-2x+\frac\pi3+2\sqrt3\), which is \(y=-2x+4.51\)
2. (b) \(y=2x+5-5\ln5\), which is \(y=2x-3.05\)

\(x>\frac14\)

1. (a) \(-1.25\) m/s
2. (b) The drone is descending: its height is decreasing at 1.25 metres per second.

1. (a) 4 million
2. (b) \(t=9\) days
3. (c) 1.5 million per day
4. (d) The model gives unlimited growth. In a real pond, food and space run out.

1. (a) \((25,5)\)
2. (b) \(\left(\frac\pi3,\sqrt3\right)=(1.05,1.73)\)

1. (a) \(y=3x-\frac{10}{3}\)
2. (b) \(e^x=\frac{10}{3}\) has only one solution

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

\(k=2\)

1. (a) \(\frac{3\pi^2}{4}=7.40\)
2. (b) \(\frac{\pi^2}{2}=4.93\)

1. (a) \((16,4)\)
2. (b) \(y=\frac x8+2\)