Tangents and normals by hand and with technology Problem set A

Find the equation of the tangent to each curve at the given point. Give your answer in the form \(ax+by=c\), where a, b and c are integers.

1. (a) \(y=2x^2-5x+1\) at \(x=3\)
2. (b) \(y=x^3-2x^2+3x\) at \(x=-1\)
3. (c) \(y=1+\frac{8}{x}\) at \(x=4\)

Find the equation of the normal to each curve at the given point. Give your answer in the form \(y=mx+c\).

1. (a) \(y=5x-3\) at \(x=2\)
2. (b) \(y=x^3+x^2\) at \(x=1\)

Find the equation of the normal to each curve at the given point. Give your answer in the form \(y=mx+c\).

1. (a) \(y=x^4-x^2\) at \(x=1\)
2. (b) \(y=2-\frac{6}{x}\) at \(x=3\)

Find the equation of the normal to each curve at the given point. Give your answer in the form \(ax+by+c=0\), where a, b and c are integers.

1. (a) \(y=x^3-4x+1\) at \(x=2\)
2. (b) \(y=x+\frac{5}{x}\) at \(x=1\)

Use technology to find the equation of the tangent to each curve at the given point. Give your answer in the form \(y=mx+c\), with m and c to 3 significant figures.

1. (a) \(y=\sqrt{3x+1}\) at \(x=8\)
2. (b) \(y=4e^{-x}\) at \(x=1\)
3. (c) \(y=\frac{1}{x^2+1}\) at \(x=2\)

A hillside has a cross-section modelled by \(y=x^3-6x^2+9x+2\), where x and y are in metres. A straight drainage pipe is laid along the tangent to the hillside where \(x=2\), and a straight post is fixed along the normal at that point.

1. (a) Find the equation of the tangent, in the form \(ax+by=c\), where a, b and c are integers.
2. (b) Find the equation of the normal, in the form \(y=mx+c\).

The height of a seedling is modelled by \(h=9\ln(t+1)\), where h is in centimetres and t is the time in weeks after planting.

1. (a) Use technology to find the equation of the tangent at \(t=4\), in the form \(h=mt+c\), with m and c to 3 significant figures.
2. (b) Use the tangent to estimate h when \(t=5\).

A normal is drawn to the curve \(y=3x^2+\frac{2}{x}-1\) at the point where \(x=1\).

1. (a) Find the equation of the normal, in the form \(ax+by+c=0\), where a, b and c are integers.
2. (b) Find the x-coordinate of the point where this normal crosses the x-axis.

Use technology to answer these questions about the curve \(y=2x\ln x+3\).

1. (a) Find the equation of the tangent at \(x=3\), in the form \(y=mx+c\), with m and c to 3 significant figures.
2. (b) Write down the coordinates of the point where this tangent crosses the y-axis.