Expanding, splitting and increasing or decreasing Problem set B

Differentiate with respect to x. The other letters are constants.

1. (a) \(y=3px+\frac{4q}{x^3}\)
2. (b) \(y=7q^2x^2-\frac{8p}{rx}\)

Expand, then differentiate with respect to x.

1. (a) \(y=(kx)^2\)
2. (b) \(y=(5px)^2\)

Expand, then find \(\frac{dy}{dx}\). The other letters are constants.

1. (a) \(y=(x+r)(x+k)\)
2. (b) \(y=(rx+k)(kx+r)\)

Split each fraction into powers of x, then differentiate with respect to x.

1. (a) \(y=\frac{x-9}{x}\)
2. (b) \(y=\frac{2+5x}{x^2}\)
3. (c) \(y=\frac{x^3+6x^4}{3x}\)

Find the values of x for which the function is decreasing.

1. (a) \(f(x)=3x^{2} - 18x + 4\)
2. (b) \(f(x)=-2x^{2} + 16x - 5\)

Find the values of x for which the function is increasing.

1. (a) \(f(x)=x^{2} + 4x\)
2. (b) \(f(x)=x^2+px+q\), where \(p\) and \(q\) are constants

Differentiate \(w=7p+\frac{10}{p}\) with respect to \(p\).

If \(x^2-y=4x+1\), find \(\frac{dy}{dx}\).

The thermal energy U joules of a sample of gas at temperature T kelvin is modelled by \(U=\frac52nRT\), where n and R are constants.

1. (a) Find the rate at which U increases with T.
2. (b) Find this rate when \(n=3\) and \(R=8.31\). Give the units and 3 significant figures.

The noise level N from a loudspeaker at distance d metres is modelled by \(N=\frac{k}{d}\), where k is a constant. The fade rate is the rate of change of N with respect to d.

1. (a) Find an expression for the fade rate in terms of k and d.
2. (b) Find an expression for the fade rate in terms of N and k.
3. (c) Point P is one third of the way from the loudspeaker to a point Q. Find the value of (fade rate at P) ÷ (fade rate at Q).

A model for the distance D km walked by a hiker after t hours is \(D=1+pt+3pt^2\), where p is a constant and \(t\geq0\).

1. (a) Find the rate of change of D with respect to t.
2. (b) Explain why you would expect D to be an increasing function. What does this tell you about p?

Show that \(f(x)=2x^3+9x+1\) is increasing for all values of x.

The constant k is positive. Let \(f(x)=\frac{x^3+kx}{3x^2}\).

1. (a) Find \(f'(x)\).
2. (b) Given that \(f'(3)=\frac19\), find the value of k.