Tangents and normals by hand and with technology Problem set B

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) \(y=3x^2+2x-4\) at \(x=2\)
2. (b) \(y=x^3-x^2+5x\) at \(x=-2\)
3. (c) \(y=2-\frac{6}{x}\) at \(x=3\)

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

1. (a) \(y=7-4x\) at \(x=1\)
2. (b) \(y=x^3+2x\) 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) \(y=x^4-3x^2\) at \(x=-1\)
2. (b) \(y=3+\frac{4}{x}\) at \(x=2\)

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

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) \(y=\sqrt{2x+1}\) at \(x=12\)
2. (b) \(y=6e^{-x}\) at \(x=2\)
3. (c) \(y=\frac{1}{x^2+4}\) at \(x=1\)

The surface of a skate ramp has a cross-section modelled by \(y=x^3-9x^2+24x-10\), where x and y are in metres. A straight rail is fixed along the tangent to the ramp where \(x=3\), and a straight support 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 concentration of a medicine in a patient's blood is modelled by \(C=6\ln(t+2)\), where C is in mg/L and t is the time in hours after the dose.

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

A normal is drawn to the curve \(y=2x^2+\frac{6}{x}-3\) 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 \(y=3x\ln x+1\).

1. (a) Find the equation of the tangent at \(x=2\), 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.