Expanding, splitting and increasing or decreasing Problem set A

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

1. (a) \(y=8ax-\frac{5b}{x^2}\)
2. (b) \(y=4b^2x^3+\frac{6a}{cx}\)

Expand, then differentiate with respect to x.

1. (a) \(y=(bx)^2\)
2. (b) \(y=(4ax)^2\)

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

1. (a) \(y=(x+p)(x+q)\)
2. (b) \(y=(px+q)(qx+p)\)

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

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

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

1. (a) \(f(x)=2x^{2} - 8x + 5\)
2. (b) \(f(x)=-x^{2} + 10x - 3\)

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

1. (a) \(f(x)=x^{2} - 6x\)
2. (b) \(f(x)=x^2+bx+c\), where \(b\) and \(c\) are constants

Differentiate \(s=8t-\frac{9}{t}\) with respect to \(t\).

If \(3x+y=5x^2-2\), find \(\frac{dy}{dx}\).

The heat energy H joules needed to warm a metal bar by T °C is modelled by \(H=cmT\), where c and m are constants.

1. (a) Find the rate at which H increases with T.
2. (b) Find this rate when \(c=0.9\) and \(m=2.5\). Give the units.

The temperature \(\theta\) °C of a wall at distance d metres from a fire is modelled by \(\theta=\frac{k}{d}\), where k is a constant. The cooling rate is the rate of change of \(\theta\) with respect to d.

1. (a) Find an expression for the cooling rate in terms of k and d.
2. (b) Find an expression for the cooling rate in terms of \(\theta\) and k.
3. (c) Point P is halfway between the fire and a point Q. Find the value of (cooling rate at P) ÷ (cooling rate at Q).

A model for the height H cm of a plant after d days is \(H=5+qd+2qd^2\), where q is a constant and \(d\geq0\).

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

Show that \(f(x)=3x^3+4x-7\) is increasing for all values of x.

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

1. (a) Find \(f'(x)\).
2. (b) Given that \(f'(2)=\frac14\), find the value of k.