Tangents, normals and gradient with technology Problem set B

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) \(y=x(\ln x)^2\) at \(x=3\)
2. (b) \(y=\frac{6}{1+x^2}\) at \(x=2\)
3. (c) \(y=3^x\) at \(x=2\)

Use technology to find the equation of the normal 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) \(y=\sqrt{x+7}\) at \(x=2\)
2. (b) \(y=\frac{x^2+3}{\sqrt{x}}\) at \(x=9\)

Use technology to find the equation of the normal 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) \(y=\ln 4x\) at \(x=4\)
2. (b) \(y=(x+2)e^{x}\) at \(x=0\)

Use technology to find the x-coordinate of the point on each curve that has the given gradient.

1. (a) \(y=\sqrt{x+2}\), gradient \(0.2\)
2. (b) \(y=e^{3x}\), gradient \(12\)

Use technology to find the x-coordinate of every point on each curve that has the given gradient.

1. (a) \(y=\frac{1}{x+2}\), gradient \(-0.5\)
2. (b) \(y=x^2-2\ln x\), gradient \(3\) (for \(x>0\))

A graph is drawn of \(y=x^4-5x\).

1. (a) Find an expression for \(\frac{dy}{dx}\).
2. (b) Write down the gradient of the tangent at \(x=0\).
3. (c) Hence find the equation of the normal to the curve at \(x=0\).

Let \(f(x)=x^5+\frac{2}{x}\).

1. (a) Find \(f'(x)\).
2. (b) Find the gradient of the tangent at \(x=1\).
3. (c) Hence show that the tangent at \(x=1\) passes through the origin.

The distance a cyclist has travelled, in kilometres, is modelled by \(d=t\ln(t+3)\), where t is the time in hours after the start.

1. (a) Use technology to find the equation of the tangent at \(t=1\), in the form \(d=mt+c\), with m and c to 3 significant figures.
2. (b) Find the value of t when the rate of change of d is 2 km per hour.

Use technology to answer these questions about the curve \(y=x+\frac{6}{\sqrt{x}}\).

1. (a) Find the x-coordinate of the point where the gradient of the curve is 0.5.
2. (b) Find the equation of the normal to the curve at this point, in the form \(y=mx+c\), with m and c to 3 significant figures.