Tangents, normals and gradient with technology Problem set A

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) \(y=x^2\ln x\) at \(x=2\)
2. (b) \(y=\frac{10}{1+x^2}\) at \(x=3\)
3. (c) \(y=2^x\) at \(x=3\)

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) \(y=\sqrt{x+5}\) at \(x=4\)
2. (b) \(y=\frac{x^2+8}{\sqrt{x}}\) at \(x=4\)

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) \(y=\ln 5x\) at \(x=2\)
2. (b) \(y=(x+3)e^{x}\) at \(x=0\)

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

1. (a) \(y=\sqrt{x+3}\), gradient \(0.1\)
2. (b) \(y=e^{2x}\), gradient \(10\)

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

1. (a) \(y=\frac{1}{x+1}\), gradient \(-0.25\)
2. (b) \(y=x^2-\ln x\), gradient \(3\) (for \(x>0\))

A graph is drawn of \(y=x^4+2x\).

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 \(f(x)=2x^4+\frac{3}{x}\).

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 number of fish in a lake, in hundreds, is modelled by \(F=t\ln(t+1)\), where t is the time in years after the lake is stocked.

1. (a) Use technology to find the equation of the tangent at \(t=2\), in the form \(F=mt+c\), with m and c to 3 significant figures.
2. (b) Find the value of t when the rate of change of F is 1.5 hundred fish per year.

Use technology to answer these questions about the curve \(y=x+\frac{4}{\sqrt{x}}\).

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.