Question

For the graph of cos(y)=x-4y, what is the range of the slopes of its tangent lines?

Answer 1

Any tangent line to the curve has a slope given by
`(\mathrm dy)/(\mathrm dx)`.

`\cos y=x-4y\implies-\sin y\,(\mathrm dy)/(\mathrm dx)=1-4\,(\mathrm dy)/(\mathrm dx)`

`\implies(\mathrm dy)/(\mathrm dx)=\frac1{4-\sin y}`

Since
`|\sin y|\le1`, the denominator ranges from 3 to 5, so the range of slopes is
`\frac15\le(\mathrm dy)/(\mathrm dx)\le\frac13`.