Tangents, normals and gradient problems Problem set B

Differentiate with respect to x.

1. (a) \(y=9e^x\)
2. (b) \(y=-8e^x\)
3. (c) \(y=-\dfrac{e^x}{6}\)
4. (d) \(y=\dfrac{7e^x}{3}\)

Find the gradient of each curve at the given value of x. Give exact answers.

1. (a) \(y=\sin x\) at \(x=\dfrac{\pi}{4}\).
2. (b) \(y=2e^x+5x\) at \(x=\ln 6\).

Find the point on the curve \(y=1+6x-2e^x\) where the gradient is 2. Give exact coordinates, then 3 significant figures.

Find the equation of the tangent to the curve \(y=x^2-8x+3\) that is parallel to the line \(y=2x+7\).

Find the equation of each tangent.

1. (a) \(y=6\sin x\), \(0<x<\dfrac\pi2\): the tangent parallel to \(y-3x=5\).
2. (b) \(y=e^x-5x\): the tangent perpendicular to \(x+3y=6\).

Find the interval in which \(f(x)=2\ln x-10x\) is a decreasing function.

A weather balloon descends after its gas is released. Its height above the ground is \(h=\frac{6t+54}{\sqrt t}\) metres, \(t\) minutes after the release.

1. (a) Find the rate of change of height when \(t=4\).
2. (b) What does the sign of your answer tell you?

The number of bacteria in a culture, in millions, is modelled by \(N=(3+\sqrt t)^2\), where \(t\) is the time in hours after the culture is set up.

1. (a) What is the number of bacteria at the start?
2. (b) How long does it take to reach 36 million?
3. (c) Find the rate of increase when \(t=16\).
4. (d) State one limitation of the model.

Find the coordinates of the point of tangency.

1. (a) The line \(y=\dfrac{x}{12}+3\) is a tangent to \(y=\sqrt x\).
2. (b) The line \(3y+3\sqrt3x=3+\sqrt3\pi\) is a tangent to \(y=2\cos x\), \(0<x<\dfrac\pi2\), at P.

The curve is \(y=5x-e^x\).

1. (a) Find the equation of the normal at the point where \(x=\ln\dfrac{26}{5}\).
2. (b) Show that this normal does not meet the curve at any other point.

The curve is \(y=3\tan x\). Find the equation of the tangent

1. (a) at the point where \(x=k\);
2. (b) when \(k=\dfrac\pi6\).

The tangent to \(y=\sqrt x\) at \(x=k\) is perpendicular to the tangent to \(y=\dfrac{36}{\sqrt x}\), also at \(x=k\). Find k.

The curve \(y=5\cos x\), for \(-\pi<x<\pi\), meets the x-axis at P and Q. The tangents at P and Q meet at R.

1. (a) Find the area of triangle PQR.
2. (b) The curve \(y=4\sin x\), for \(0\le x<2\pi\), meets the x-axis at A and B. The tangents at A and B meet at C. Find the area of triangle ABC.

A tangent to the curve \(y=\sqrt x\) passes through the point \((-5,2)\).

1. (a) Find the coordinates of the point where the tangent touches the curve.
2. (b) Find the equation of the tangent.